CNTA Schedule

Link to number theory and physics session


Note: The conference will start on Sunday afternoon June 17 at 1:50pm at the 1st Choice Centre for Sport and Wellness at the University of Lethbridge. It will end on Friday June 22 at noon.
Sunday, June 17 Monday, June 18 Tuesday, June 19 Wednesday, June 20 Thursday, June 21 Friday, June 22
8:30am
9:00am Ribenboim Prize:
(Presentation and Lecture)
9:00am - 10:00am
Break
9:20am - 9:40am
Contributed Talks
9:40am - 10:40am
Break
9:50am - 10:10am
Break
9:50am - 10:10am
Break
9:50am - 10:10am
10:00am Break
10:00am - 10:30am
Break
10:40am - 10:50am
Contributed Talks
10:50am - 12:10pm
Contributed Talks
10:50am - 11:10pm
11:00am
Free afternoon
11:30am -

Waterton Excursion (1:00pm)
and
CNTA Soccer (2:00pm)
12:00pm Registration
12:00pm - 2:00pm
Lunch
12:05pm - 2:00pm
Lunch
12:05pm - 2:00pm
End of conference
12:05pm -
Lunch
12:10pm - 2:00pm
1:00pm
2:00pm
Break
2:50pm - 3:10pm
Break
2:50pm - 3:10pm
Break
2:50pm - 3:10pm
Break
2:50pm - 3:10pm
3:00pm
4:00pm
Break
4:25pm - 4:40pm
Break
4:25pm - 4:35pm
Break
4:25pm - 4:35pm
Break
4:25pm - 4:35pm
Contributed Talks
4:35pm - 5:35pm
Contributed Talks
4:40pm - 5:40pm
5:00pm
Break
5:40pm - 5:50pm
Contributed Talks
5:50pm - 6:50pm
6:00pm
Banquet
6:30pm - 10:30pm
7:00pm Reception
7:00pm - 9:00pm
Public Lecture:
Andrew Granville
7:00pm - 8:00pm
8:00pm Reception
8:00pm - 10:00pm


Number Theory and Physics Session

Monday, June 18 Tuesday, June 19 Wednesday, June 20 Thursday, June 21 Friday, June 22

8:30am - 9:20am

10:15am - 11:05am

10:15am - 11:05am

10:15am - 11:05am

10:15am - 11:05am

10:45am - 11:35am

11:15am - 12:05pm

11:15am - 12:05pm

11:15am - 12:05pm

11:15am - 12:05pm


Contributed Talks

Sunday, June 17
4:40pm - 4:55pm
5:00pm - 5:15pm
5:20pm - 5:35pm
5:40pm - 5:50pm Break
5:50pm - 6:05pm
6:10pm - 6:25pm
6:30pm - 6:45pm

Monday, June 18th
10:50am - 11:05am
11:10am - 11:25am
11:30am - 11:45am
11:50am - 12:05pm

Tuesday, June 19
10:50am - 11:05am

Thursday, June 21
4:35pm - 4:50pm
4:55pm - 5:10pm
5:15pm - 5:30pm

Friday, June 22
9:40am - 9:55am
10:00am - 10:15am
10:20am - 10:35am


Speakers' Abstracts

The moduli space of cubic surfaces
Jeff Achter (Colorado State University)

For about a decade, it has been known that the moduli space of complex cubic surfaces is open in a certain arithmetic quotient of the complex $4$-ball. I explain this phenomenon by showing that, at least away from the prime $2$, the moduli space of stable cubic surfaces can be identified with a certain moduli space of principally polarized abelian fivefolds with endomorphism structure. This answers a recent question of Kudla and Rapoport, and naturally leads to some intriguing questions about the arithmetic and geometry of cubic surfaces.
Slides


Cubic Thue Inequalities
Shabnam Akhtari (CRM, Montréal)

Let $F(x , y)$ be an irreducible binary form and $h$ be a positive integer. We know that the inequality $$ |F(x , y)| \leq h $$ has only finitely many solutions in integers $x$ and $y$. I will talk about some old and new results on the number of solutions to the above inequality. In particular, we will see that there is an absolute upper bound for the number of such solutions that does not depend on $h$, provided that $h$ is small in terms of the discriminant of binary form $F$.


$p$-adic heights on elliptic curves
Jennifer Balakrishnan (Harvard University)

In 2006, the work of Mazur-Stein-Tate gave an effective algorithm to compute $p$-adic heights on elliptic curves over ${\mathbb Q}$. We begin with a brief overview of their algorithm and discuss a generalization to number fields. This is joint work with Mirela Çiperiani and William Stein.


Elliptic Curves failing a local-to-global principle for Isogenies
Barinder Banwait (University of Warwick)

A recent result of Andrew Sutherland implies that an elliptic curve defined over a number field $K$ must have a $K$-rational $5$-isogeny if its reduction at all good places admits a $5$-isogeny rational over its base field, under the assumption that $K$ not contain the square-root of $5$. I will explain how this assumption is necessary; namely, when $K$ contains the square-root of $5$, there are infinitely many elliptic curves failing this ``local-to-global principle'' for isogenies, and I will give a parameterisation for the $j$-invariants of such curves. This is part of joint work with John Cremona.


On the Nymann-Beurling criterion for the Riemann hypothesis
Sandro Bettin (University of Bristol)

The Nymann-Beurling criterion asserts that the Riemann hypothesis is true if and only if the inverse of the Riemann zeta function can be well approximated on the critical line on average by Dirichlet polynomials. We give a result on the optimal choice for this polynomial and highlight some properties of a cotangent sum related to this criterion.


The average size of the $2$-Selmer group of Jacobians of odd hyperelliptic curves
Manjul Bhargava (Princeton University)

By studying the rational and integral orbits on a certain natural representation of the odd split orthogonal group, we show that the average size of the $2$-Selmer group of the Jacobians of odd hyperelliptic curves over $\mathbb{Q}$ (of any given genus) is $3$. This implies that the average rank of the Jacobians of these hyperelliptic curves is bounded (by $3/2$). Via Chabauty methods, the result also then implies a uniform bound on the number of rational points on the majority of these curves. This is joint work with Dick Gross.


Applications of the Kuznetsov formula on $GL(3)$
Valentin Blomer (University of Gottingen)

We discuss the use of spectral summation formulae in number theory and develop a fairly explicit Bruggeman-Kuznetsov formula on $GL(3)$ with an emphasis on the analytic behaviour of the test functions on both sides. Various applications are given including a density estimate for the exceptional spectrum, a general large sieve for $GL(3)$ Fourier coefficients and bounds for $GL(3)$ $L$-functions.


The distribution of character sums
Jonathan Bober (University of Washington)

I will discuss experiments and results concerning the distribution of character sums over long intervals. In particular, I will focus on the frequency with which character sums are large and the location where the maximum occurs.


Topological recursion and modular forms
Vincent Bouchard (University of Alberta)

In recent years, a unifying theme has been found for a surprising number of counting problems. It appears that in many seemingly unrelated contexts, generating functions for enumerative invariants satisfy a particular topological recursion, based on the geometry of a complex curve. The motivation behind the appearance of the recursion comes from dualities in string theory. In this talk, I will first review the topological recursion and the string theory motivation for its appearance in enumerative geometry. Then I will report on work in progress on studying the topological recursion for families of elliptic curves. In this context, the recursion produces an infinite tower of quasi-modular forms. The question is: to what counting problem should these quasi-modular forms be related?


Detecting squarefree numbers
Andrew Booker (University of Bristol)

I will describe an algorithm for detecting squarefree integers without factoring.
Slides


Rational points on conic bundle surfaces via additive combinatorics
Tim Browning (University of Bristol)

Methods of Green and Tao can be used to prove the Hasse principle and weak approximation for some special intersections of quadrics defined over the rational numbers. This implies that the Brauer-Manin obstruction controls weak approximation on conic bundles with an arbitrary number of degenerate fibres, all defined over the rationals. This is joint work with Lilian Matthiesen and Alexei Skorobogatov.


Connecting homomorphisms associated to Tate sequences
Paul Buckingham (University of Alberta)

The Tate sequence is the result of a unification of local and global class field theory, and describes the cohomology of the $S$-units in a Galois extension of number fields. In the traditional construction, $S$ was assumed to be large enough that the $S$-class-group was trivial. A refinement of Ritter and Weiss removed that assumption, so that their Tate sequence involved both the $S$-units and the $S$-class-group, giving rise to connecting homomorphisms not previously studied. We will provide the first descriptions of some of these connecting homomorphisms, and discuss some consequences.


On the continued fraction expansions of algebraic numbers
Yann Bugeaud (Université de Strasbourg)

We survey recent results on the continued fraction expansions of algebraic numbers of degree at least three. We show that, if the sequence of partial quotients of a real number enjoys certain combinatorial properties, then this number must be transcendental. As a consequence, the infinite word composed of the partial quotients of an algebraic number of degree at least three cannot be too simple (it cannot have sublinear block complexity) and it cannot be generated by a finite automaton.


A New Algorithm for the Prouhet-Tarry-Escott Problem
Timothy Caley (University of Waterloo)

The Prouhet-Tarry-Escott (PTE) problem is a classical number theoretic problem which asks for integer solutions to sums of equal powers. Solutions to the PTE problem give improved bounds for the ``Easier'' Waring problem, but they are difficult to find using conventional methods. In this talk, we present a new algorithm to find solutions to this problem computationally.


Partial sum of Möbius function in function fields
Byungchul Cha (Muhlenberg College)

We study the growth rate of the summatory function of the Möbius function in the context of an algebraic curve over a finite field. Our work shows a strong resemblance to its number field counterpart, as described by Ng in 2004. We find an expression for a bound of the summatory function, which becomes sharp when the zeta zeros of the curve satisfy a certain linear independence property. Extending a result of Kowalski in 2008, we prove that most curves in the family of universal hyperelliptic curves satisfy this property. We also consider a certain geometric average of the bound using Katz and Sarnak's equidistribution theorem.


Multiplicative Partitions
Marc Chamberland (Grinnell College)

While the standard additive partition of a number $n$ has been well-studied for over a century, very little has been done for multiplicative partitions. Let $a_n$ denote the number of multiplicative partitions of the natural number $n$. For example, $a_{12}$ = 4 since $$12 = 2\cdot 6 = 2\cdot 2\cdot 3 = 3\cdot 4.$$ The multiplicative partition is highly erratic because of its natural connection to prime numbers. This talk reports on the combinatorial structure of $a_n$, a fast algorithm for computing these terms, and certain evaluations of the corresponding generating function in terms of the Gamma function. Connections to the Riemann zeta function are also made.


Lower bounds on small fractional moments of Dirichlet $L$-functions
Vorrapan Chandee (CRM, Université de Montréal)

Moments of $L$-functions have been studied both for their application to the value distributions and zeros of $L$-functions, as well as for their own intrinsic interest. Lower bounds of the correct order of magnitude for large fractional moments of Dirichlet $L$-functions follow from a method of Rudnick and Soundararajan. I will explain why the method does not apply to small moments via the link to non-vanishing results, and outline a proof of the correct bound for small fractional moments using mollification. This talk is based on a joint work with Xiannan Li.


Cuspidal ${\mathbb Q}$-rational Torsion Subgroups of Jacobians of Modular Curves
Yaohan Chen (National Central University)

In this talk, we give an explicit basis for the group of modular units on $X(\Gamma)$ that have divisors defined over ${\mathbb Q}$, where $\Gamma$ is a congruence subgroup between $\Gamma_1(p^r)$ and $\Gamma_0(p^r)$ for a prime $p > 3$ and an integer $r > 0$. Then we utilize it to determine the order of the cuspidal ${\mathbb Q}$-rational torsion subgroup of $J(\Gamma)$ generated by the divisor classes of cuspidal divisors of degree $0$ defined over ${\mathbb Q}$ when $\Gamma = \Gamma_0(p^r),~ \Gamma_1(p^r)$.


Logarithmic derivatives of Artin $L$-functions
Peter Jaehyun Cho (University of Toronto)

Let $K$ be a number field of degree $n$, and $d_K$ be its discriminant. Then under the Artin conjecture, GRH and certain zero density hypothesis, we show that the upper and lower bound of the logarithmic derivative of Artin $L$-functions attached to $K$ at $s=1$ are $\log\log |d_K|$ and $-(n-1) \log\log |d_K|$, resp. Unconditionally we show that there are infinitely many number fields with the extreme logarithmic derivative values. They are families of number fields whose Galois closures have the Galois group as $C_n$, $2\leq n\leq 6$, $D_n$, $n=3,4,5$, $S_4$, and $A_5$.
Slides


Geometry of Numbers and quadratic forms
Pete Clark (University of Georgia)

It is classically known that the two and four squares theorems can be proved by Geometry of Numbers (GoN) methods. However, the literature contains remarkably little in the way of systematic results on the application of GoN methods to the representation of elements of a ring R by quadratic forms. I led a VIGRE research group on this topic in 2011-2012, and some insights emerged. I will explain a three part strategy for proving representation theorems for quadratic forms (over a rather general class of integral domains) by GoN methods and report on some instances of its successful implementation.


On K3 Surfaces of High Picard Rank
Adrian Clingher (University of Missouri)

I will discuss a special family of complex algebraic K3 surfaces polarized by the rank seventeen lattice H+E8+E7. In terms of Hodge theory, these surfaces are naturally related to principally polarized abelian surfaces. I will outline the geometry of the correspondence as well as present an explicit classification of these special K3 surfaces in terms of Siegel modular forms. Finally, if time permits, I will discuss some recent work extending these results to lattice polarizations of type H+E7+E7.


Many variable forms in the primes
Brian Cook (University of British Columbia)

We will discuss the solubility of the equation $\mathcal{F}(x)=v$ where $\mathcal{F}$ is an integral form of fixed degree $d$ in $n$ variables $x=(x_1,\ldots,x_n)$ under the condition that $x_i$ is prime for each $1\leq i \leq n$. The results presented shall focus mainly on the cases $d=2$ and $d=3$ under the assumption that $\mathcal{F}$ is sufficiently non-singular in relation to $d$.


A hierarchy of complexity in the context of Mahler's method
Michael Coons (Fields institute, the University of Waterloo, and the University of Newcastle)

In 1983, John Loxton (based off of work with Alf van der Poorten concerning the decimal expansion of algebraic irrationals) suggested an alternative theoretical approach to randomness in which he assigned a measure of computational complexity to a sequence by means of the following hierarchy:

  1. periodic sequences,
  2. sequences generated by finite automata,
  3. sequences generated by automata with one push-down store,
  4. sequences generated by non-deterministic automata with one push-down store, and
  5. sequences generated by Turing machines
In this talk, I will discuss a mathematical alternative to Loxton's hierarchy. In the context of this new hierarchy, I will discuss several recent results concerning the rational approximation of numbers connected to finite automata and generalizations within the context of Mahler's method.


Subconvexity bound for a twisted double Dirichlet series
Alexander Dahl (University of Toronto)

This talk investigates analytic properties of twisted multiple Dirichlet series of the type $$Z(s, w; \psi_1, \psi_2) = \sum_d L(s, \chi_d \psi_1) \psi_2(d) d^{-w},$$ for two real Dirichlet characters $\psi_1$, $\psi_2$, initially defined for $Re(s)$ and $Re(w)$ sufficiently large. Such double Dirichlet series come equipped with various functional equations. We discuss methods to obtain a uniform subconvexity estimate at the central point $s=w=1/2$ with respect to the conductors of $\psi_1$ and $\psi_2$. As Hoffstein and Kontorovich have shown in a recent preprint, such subconvex bounds imply certain non-vanishing results for ordinary quadratic Dirichlet $L$-functions.
Slides


A Cohen-Lenstra Phenomenon for Elliptic Curves
Chantal David (Concordia University)

Let $E$ be an elliptic curve over ${\bf Q}$. We consider the problem of counting the number of primes $p$ for which the reduction of $E$ over the finite field ${\bf F}_p$ is isomorphic to a fixed abelian group $G = {\bf Z} / N_1 {\bf Z} \times {\bf Z} / N_1 N_2 {\bf Z}$. Under some hypothesis on the distribution of primes in short intervals, we obtain an asymptotic formula for the average. The asymptotic involves the arithmetic factor $\# G / \# \mbox{Aut}(G)$, in accordance with the Cohen-Lenstra heuristics which predict that random groups $G$ occur with probability weighted by $1/\#\mbox{Aut}(G)$. We also show how can one obtain sharp unconditional upper bounds (including the arithmetic factor predicted by the Cohen-Lenstra Heuristics) for the average. This involves the use of a combinatorial sieve to replace short sums over primes by sums over integers with only large prime factors. This is joint work with Vorropan Chandee, Dimitris Koukoulopoulos and Ethan Smith (CRM, Montréal).
Slides


Computing discrete logarithms with pencils
Claus Diem (University of Leipzig)

We study the discrete logarithm problem in the degree $0$ class groups of curves $\mathcal{C}$ of a fixed genus over finite fields ${\mathbb F}_q$. We consider the usual index calculus method to compute discrete logarithms, and for this we want to compute suitable relations. We observe that any non-constant function on $\mathcal{C}$ leads to a relation. Moreover, we not only have one relation (given by the associated principal divisor) but in fact all preimages of closed points of $\mathbb{P}^1_{{\mathbb F}_q}$ in $\mathcal{C}$ are linearly equivalent. They form what is called a pencil. By using such pencils, we obtain the following heuristic result: Let a natural number $g \geq 5$ be fixed. Then the discrete logarithm problem for nearly all curves of genus $g$ can be solved in an expected time of \[ \tilde{O}(q^{2 - \frac{2}{g-2}}) \; . \] For nearly all curves of a fixed genus at least 5, this improves upon a previous algorithm which is based on the intersection of a plane model of the curve with lines. The improvement corresponds to the drop of the genus by 1.


A Recursive Formula for the Convolution Sum of Divisor Functions
Greg Doyle (Carleton University)

Let $\displaystyle q \in {\mathbb C}$ with $|q| < 1$, $\displaystyle u_n:=\frac{q^n}{1-q^n}$ for $n \in {\mathbb N}$, and $\theta \in [0,2\pi]$. Let $f,g:{\mathbb Z} \rightarrow {\mathbb C}$ be odd functions. We evaluate the product of two Lambert series \[\left(\sum_{n=1}^{\infty} u_n f(n) \cos n\theta\right)\left(\sum_{n=1}^{\infty}u_n g(n) \cos n\theta \right),\] which leads to a formal power series with coefficients $$\sum_{\substack{ax+by=n \\ a,x,b,y \in {\mathbb N}}} \left(f(a)\cos a\theta\right)\left(g(b) \cos b\theta\right).$$ Then we deduce a recursive formula for the convolution sum $\displaystyle\sum_{m=1}^{n-1}\sigma_r(m)\sigma_s(n-m)$, where $r,s$ are odd positive integers and $\sigma_r$, $\sigma_s$ are divisor functions.


The value distribution of incomplete Gauss sum
Emek Demirci Akarsu (University of Bristol)

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov's pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and Marklof. This talk is based on joint work with Jens Marklof.


Sums of reciprocals modulo composite integers
Karl Dilcher (Dalhousie University)

In 1938, as part of a wider study, Emma Lehmer derived a set of four related congruences for certain sums of reciprocals of positive integers over various ranges, modulo squares of odd primes. These were recently extended to congruences modulo squares of positive integers $n$, with certain restrictions on $n$. In this talk I will characterize those excluded $n$ for which the congruences still hold, and find the correct reduced moduli in the cases in which the congruences do not hold. (Joint work with John B. Cosgrave).
Slides


Arithmetic Progressions in the Polygonal Numbers
Scott M. Dunn (University of South Carolina)

In this talk, we investigate arithmetic progressions in the polygonal numbers with a fixed number of sides. We first show that four-term arithmetic progressions cannot exist. We then describe explicitly how to find all three-term arithmetic progressions. Finally, we show that not only are there infinitely many three-term arithmetic progressions, but that there are infinitely many three-term arithmetic progressions starting with an arbitrary polygonal number. Special attention is paid to the case of squares and triangular numbers.


Sporadic balanced subgroups
Zebediah Engberg (Dartmouth College)

Let $d > 2$ be an integer. With respect to the usual representatives, any unit modulo $d$ lies in the interval $(0,d/2)$ or $(d/2,d)$. If $H$ is a subgroup of the unit group modulo $d$, we say $H$ is balanced if every coset of $H$ intersects these two intervals equally. Suppose $g > 1$ is a fixed integer. We are interested in the set of $d$ coprime to $g$ such that the cyclic group $\langle g \mod d\rangle$ is balanced. After discussing two infinite families of $d$ which arise, we will focus on the set of ``sporadic'' $d$ belonging to neither family.
Slides


Determining the Root Quadruple of an Apollonian Circle Packing
Stefan Erickson (Colorado College)

Apollonius's Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three. Apollonian circle packings are produced by repeating the construction of mutually tangent circles to fill all remaining spaces. A remarkable consequence of Descartes' Theorem is if the initial four tangent circles have integral curvatures, then all of the circles in an Apollonian circle packing will have integral curvatures. This process results a sequence of integers with fascinating arithmetic properties. Graham, et. al. demonstrated in 2003 a correspondence between the root quadruples and reduced binary quadratic forms. In this talk, we will present an improved reduction algorithm for finding the root quadruple of a given Descartes quadruple.


$L$-functions, periods and the relative trace formula for $GL(2n)$
Brooke Feigon (CCNY/CUNY)

An important result of Waldspurger relates central values of quadratic base change $L$-functions of cusp forms on $GL(2)$ to period integrals over the corresponding tori. This result was reproved by Jacquet using the relative trace formula. In this talk I will discuss joint work with Kimball Martin and David Whitehouse on extending Jacquet's approach to Waldspurger's result to higher rank groups.


Variations of Artin's conjecture on primitive roots
Adam Tyler Felix (Max Planck Gesellschaft)

Fix an integer $b$. In 1927, Artin conjecture about the density of primes $p$ for which $b$ generates the group of residues modulo $p$. We will discuss this conjecture and variations of it.


On a polynomial conjecture of Turán
Michael Filaseta (University of South Carolina)

This will be a survey talk on a conjecture of Turán which asserts that there is an absolute constant $C$ such that every polynomial $f(x) \in {\mathbb Z}[x]$ is within $C$ of being an irreducible polynomial (meaning that there is a $w(x) \in {\mathbb Z}[x]$ with $\deg w \le \deg f$ and the sum of the absolute value of the coefficients $w(x)$ at most $C$ satisfying $f(x) + w(x)$ is irreducible over the rationals). We will give a history of partial results on this still open conjecture, including the analogous question modulo primes as well as some items not yet in print.


How biased can a prime number race possibly be?
Daniel Fiorilli (Institute for Advanced Study)

In 1853, Chebyshev remarked that there are more primes of the form $4n+3$ than of the form $4n+1$ in the interval $[1,x]$, for many values of $x$. Rubinstein and Sarnak established under some technical hypotheses that the logarithmic density of $x$ for which Chebyshev's assertion is true is of $0.9959\ldots$. They also studied an even more biased race, and showed that the density of $x$ such that $Li(x) > \pi(x)$ is of $0.99999973\ldots$. Since their 1994 paper, many other densities have been computed and none of those numbers were found to exceed this last value. In this talk we will answer a question posed to us by Andrew Granville: Is $0.99999973\ldots$ the highest density one will ever obtain by studying fair prime number races?


On the average exponent of elliptic curves modulo $p$
Tristan Freiberg (KTH Royal Institute of Technology)

Given an elliptic curve $E$ defined over ${\mathbb Q}$ and a prime $p$ of good reduction, let $\tilde{E}({\mathbb F}_p)$ denote the group of ${\mathbb F}_p$-points of the reduction of $E$ modulo $p$, and let $e_p$ denote the exponent of this group. Assuming a certain form of the Generalized Riemann Hypothesis, we study the average of $e_p$ as $p \le X$ ranges over primes of good reduction, and find that the average exponent essentially equals $p\cdot c_{E}$, where the constant $c_{E} \in (0,1)$ depends on $E$. This is joint work with Pär Kurlberg.


Small representations by an indefinite ternary form
John Friedlander (University of Toronto)

We discuss recent work with H. Iwaniec wherein we study solutions of the equation $x^2 + y^2 - z^2 = D$. We are particularly concerned, with $D$ large and positive, in counting the solutions having $x, y, z$ all small relative to $D$.


The Selberg Class: Low degree functional equations
Ralph Furmaniak (Stanford University)

A new proof is presented of the known fact that the only Dirichlet series with functional equation of degree $\leq 1$ are Dirichlet polynomials and Dirichlet $L$-functions. This method also allows a classification of all Dirichlet series that extend to an entire function with growth rate at most that of a degree 1 $L$-function.


On the Correlation of completely multiplicative functions
Himadri Ganguli (Simon Fraser University)

Let $f(n)$ be an arithmetic function and $x > 0$, then we define the correlation function $C(f,x)=\sum_{n\le x}f(n)f(n+1)f(n+2)$. In this talk we present an asymptotic formula for $C(f,x)$ in the case when $f(n)$ is a completely multiplicative function and $|f(n)|\le 1$ for all $n \in {\mathbb N}$. Let $\lambda_y(n) $ denote the truncated Liouville function, which equals $+1$ or $-1$ according as $n$ has an odd or even number of prime divisors $p \le y$ counted with multiplicity. It follows from the main theorem that $C(\lambda_y,x)=o(x)$ whenever $ y=x^{o(1)}$. This speaks in favour of the Chowla conjecture that $C(\lambda,x)=o(x)$ where $\lambda$ is the classical Liouville function.


Arithmetic Dynamics
Dragos Ghioca (University of British Columbia)

We present several questions with both a dynamical and a geometric flavor, which are related to classical conjectures in number theory.
Slides


Communal Partitions of Integers
Darren Glass (Gettysburg College)

There is a well-known formula due to Andrews that counts the number of incongruent triangles with integer sides and a fixed perimeter, which is equivalent to counting the number of triples such that no single entry is more than half the sum of all three. In this talk, we consider the analogous question counting the number of $k$-tuples of nonnegative integers none of which is more than $1/(k-1)$ of the sum of all $k$ integers. We give an explicit generating function which counts these $k$-tuples in the case where they are ordered, unordered, or partially ordered. Finally, we discuss the application to Hurwitz spaces which motivated this question.


A quadratic large sieve over number fields
Leo Goldmakher (University of Toronto)

In 1995 Heath-Brown proved a large sieve inequality for quadratic characters, a tool which has proved to be extremely useful in applications. I will discuss recent work (joint with Benoit Louvel) in which we formulate and prove a generalization of his result to number fields. To do this, we introduce the notion of a `Hecke family', an infinite family of Hecke characters satisfying a rather special reciprocity law. I will describe such families and some related results on Hecke $L$-functions and multiple Dirichlet series.


Public Lecture:Pattern hunting in the prime numbers
Andrew Granville (Université de Montréal)

We will discuss patterns that have been found amongst the prime numbers, some of which we can prove are really there, and some of which are still a mystery....
Slides


A pretentious introduction to analytic number theory
Andrew Granville (Université de Montréal)

Since Riemann's 1859 monograph, the study of the distribution of prime numbers has been dominated by the study of the zeros of the Riemann zeta function and Dirichlet $L$-functions. Although there have been ad hoc elementary proofs of some of the key results, there has been no coherent alternative approach to that of Riemann. In this talk we will motivate and outline a new and different way to develop analytic number theory, without zeros, stemming from the concept of ``pretentiousness''. This is jointly developed with K. Soundararajan, and we will discuss further developments due to Dimitris Koukoulopoulos, Regis de la Breteche, Kevin Ford and others.
Slides


On the compositum of all extensions of a number field which have degree at most $d$
Bobby Grizzard (The University of Texas at Austin)

If $k$ is a number field, we denote by $k^{(d)}$ the field obtained by adjoining to $k$ all roots of polynomials in $k[x]$ of degree less than or equal to $d$. Such infinite algebraic extensions have been studied by Bombieri and Zannier, Checcoli, and Checcoli and Widmer, originating from questions about the Northcott property. We consider two questions about these fields. First, we ask whether $\mathbb{Q}^{[d]} = \mathbb{Q}^{(d)}$, where $\mathbb{Q}^{[d]}$ denotes the field obtained by adjoining algebraic numbers of degree exactly $d$. We answer this in the affirmative for $d \leq 4$. Secondly, we discuss whether finite degree subfields of $k^{(d)}$ can be generated by elements of bounded degree, as previously considered by Checcoli. We show that finite Galois subextensions of $k^{(d)}$ are generated by elements of bounded degree if and only if $d \leq 3$, while general finite degree subfields are generated by elements of bounded degree if and only if $d \leq 2$. This is joint work with Itamar Gal.


Automatic Resolution of Thue-Mahler Equations: Examples and Applications
Kyle Hambrook (University of British Columbia)

The development of Baker's theory on linear forms in logarithms of algebraic numbers in the 1960s led to the first effective method for solving Thue and Thue-Mahler equations. Unfortunately, due to the size of the bounds arising from Baker's method, a truly practical method for solving Thue-type equations remained out of reach. In his thesis, de Weger described how to use the lattice reduction algorithm of Lenstra, Lenstra, and Lovasz to reduce the bounds coming from Baker's method. Shortly thereafter, Tzanakis and de Weger published algorithms for solving arbitrary Thue and Thue-Mahler and illustrated them by solving a few example equations. Today, the main number theory software packages have good Thue solving routines, but there is no general routine available for solving Thue-Mahler equations. To date, only a handful of non-trivial Thue-Mahler equations have been solved. Nevertheless, there are many interesting problems that reduce to the resolution of Thue-Mahler equations. Hence there is a definite need for an automated Thue-Mahler solving routine. I will discuss my implementation of a fully automated Thue-Mahler equation solving routine. I will also discuss some interesting examples and some applications to problems like resolving certain exponential Diophantine equations and computing elliptic curves of prescribed conductor.
Slides


Special Values of Rankin-Selberg $L$-functions
Alia Hamieh (University of British Columbia)

Associated to a newform $F\in S_{2}(\Gamma_{0}(N))$ and a finite-order Hecke character $\chi$ of an imaginary quadratic field $K$, one defines the Rankin-Selberg $L$-function $L(F,\chi,s)$. Prof. Vatsal obtained various results pertaining to the non-vanishing modulo a prime $\lambda\in\overline{{\mathbb Q}}$ of the special values of these $L$-functions in the anticyclotomic tower of conductor $p^{\infty}$ over $K$ when the sign in the functional equation is $+1$. In this talk, we report on a work in progress to establish some generalizations of such results.
Slides


Sporadic Reinhardt Polygons
Kevin Hare (University of Waterloo)

Let $n$ be a positive integer, not a power of two. A Reinhardt polygon is a convex $n$-gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter. For each Reinhardt polygon, there exists an associated polynomial $P(z)$ such that the coefficients of $P(z)$ are $\{-1, 0, 1\}$, with the non-zero terms alternating, and further $\Phi_{2n}(z) \mid P(z)$. For almost all $n$, there are many Reinhardt polygons with $n$ sides, and many of them exhibit a particular periodic structure. While these periodic polygons are well understood, for certain values of $n$, additional Reinhardt polygons exist that do not possess this structured form. We call these polygons sporadic. In this talk we discuss these sporadic Reinhardt polygons.


The Iterated Carmichael Lambda Function
Nick Harland (University of British Columbia)

The Carmichael lambda function $\lambda(n)$ is defined to be largest order of an element in ${\mathbb Z}/n{\mathbb Z}.$ In this talk we will see some analytic properties of the iterated Carmichael lambda function, including some results on the function $L(n),$ defined to be the number of iterations of $\lambda$ required to reach one.
Slides


On the Factorization of the Trinomials $x^n+cx^{n-1}+d$
Joshua Harrington (University of South Carolina)

In this talk we discuss the factorization of trinomials of the form $x^n+cx^{n-1}+d\in{\mathbb Z}[x].$ We then use these results about trinomials to prove results about the factorization of polynomials of the form $x^n+ c(x^{n-1}+\cdots+x+1)\in{\mathbb Z}[x]$.
Slides


Zeros of pairs of quadratic forms
Roger Heath-Brown (University of Oxford)

When does a pair of quadratic forms have a common non-trivial zero? Work of Colliot-Thélène, Sansuc and Swinnerton-Dyer gives a good answer when the number of variables is 9 or more. In particular, over a totally imaginary number field, any such pair of forms has a non-trivial zero. For pairs of forms in 8 or fewer variables there can be $p$-adic constraints, but for 8 variables a variant of the previous method applies, providing that one can tackle a certain purely $p$-adic problem. The talk will give an account of this theory, and describe a resolution of the 8 variable case.


Minor arcs for Goldbach's problem
Harald Helfgott (CNRS/Ecole Normale Supérieure)

The ternary Goldbach conjecture states that every odd number $n\geq 7$ is the sum of three primes. The estimation of sums of the form $\sum_{p\leq x} e(\alpha p)$, $\alpha = a/q + O(1/q^2)$, has been a central part of the main approach to the conjecture since (Vinogradov, 1937). Previous work required $q$ or $x$ to be too large to make a proof of the conjecture for all $n$ feasible. The present paper gives new bounds on minor arcs and the tails of major arcs. For $q\geq 4\cdot 10^6$, these bounds are of the strength needed to solve the ternary Goldbach conjecture. Only the range $q\in \lbrack 10^5, 4\cdot 10^6\rbrack$ remains to be checked, possibly by brute force, before the conjecture is proven for all $n$. The new bounds are due to several qualitative improvements. In particular, this paper presents a general method for reducing the cost of Vaughan's identity, as well as a way to exploit the tails of minor arcs in the context of the large sieve.


Rational approximations for Euler's constant
Tatiana Hessami Pilehrood (Dalhousie University)

In this talk, we will discuss recent progress made towards constructing rational approximations to Euler's constant $\gamma.$ We present new rational approximations to $\gamma$ generated by a second-order inhomogeneous linear recurrence with polynomial coefficients and show their connection with a continued fraction for the Euler-Gompertz constant due to Stieltjes. (Joint work with Kh. Hessami Pilehrood)


The arithmetic of critical orbits
Patrick Ingram (Colorado State University)

In complex holomorphic dynamics, the orbits of critical points reveal much about the behaviour of a map under iteration. If the map and point are defined over a number field (or some other arithmetically interesting field), then we can ask about their arithmetic. We will survey some recent results on the arithmetic of critical orbits, focusing on the case of so-called post-critically finite rational maps, that is, those whose critical points all have finite forward orbit.


Tabulating Class Groups of Real Quadratic Fields
Michael Jacobson (University of Calgary)

Class groups of real quadratic fields have been studied since the time of Gauss, and in modern times have been used in applications such as integer factorization and public-key cryptography. Tables of class groups are used to provide valuable numerical evidence in support of a number of unproven heuristics and conjectures, including those due to Cohen and Lenstra. In this talk, we discuss recent progress in our efforts to extend existing, unconditionally correct tables of real quadratic fields. This includes incorporating ideas of Sutherland for computing orders of elements in a group, as well as constructing a unconditional verification algorithm using the trace formula of Maass forms based on ideas of Booker. This is joint work with C. Bian, A. Booker, A. Shallue, and A. StrÖmbergsson
Slides


Cohomology and zeta functions on $p$-adic Lie groups
Ming-Hsuan Kang (National Chiao Tung University)

In this talk, we introduce a new cohomology on lattice chains over a finite dimensional $p$-adic vector space. The Laplacian on this cohomology is related to the zeta function of complexes arising from a reductive $p$-adic Lie groups which acts on these lattices. From this, we are able to show an identity, which is an analogue of Ihara determinant formula for the zeta function of graphs.


Abelian surfaces over ${\mathbb Q}$ whose Tate-Shafarevich group has order five or seven times a square
Stefan Keil (Humboldt-Universität zu Berlin)

In 1965, Cassels has shown that the order of the Tate-Shafarevich group of an elliptic curve over ${\mathbb Q}$ is a square, if finite. For abelian varieties this is not true in general, which was wrongly believed for over 30 years. In 1996, Stoll constructed a principally polarized abelian surface with $\# Ш=2\square$. With the Cassels-Tate pairing one can prove that the order of $Ш$ of a PPAV is a square or twice a square. One might wonder what else can happen. In 2004, Stein gave an example of an abelian surface with $\# Ш=3\square$. We will show how to construct abelian surfaces with $\# Ш=5\square$ or $\# Ш=7\square$.


Cohomology and boundary components of Mumford-Tate domains
Matt Kerr (Washington University in St. Louis)

Mumford-Tate groups are the symmetry groups of Hodge theory, and their orbits (Mumford-Tate domains) are the moduli spaces for Hodge structures with given symmetries. This talk will describe recent work with G. Pearlstein on degenerations of Hodge structure within this context, and its implications for the study of period maps and cohomology of quotients of period domains. The examples we consider are related to Calabi-Yau varieties. More precisely, our main result concerns spaces of nilpotent orbits in "adjoint" Mumford-Tate domains: it tells how to compute the Mumford-Tate group of the limit mixed Hodge structure of a generic such orbit. This is used to present these spaces as iteratively fibered algebraic group orbits in a minimal way, and to decide whether certain quotients of them have canonical models. Before getting to this, we will also describe the rudiments of the cohomology of quotients of these domains -- the so-called automorphic cohomology -- and mention an application to automorphic representation theory.


On Zaremba's Conjecture
Alex Kontorovich (Yale University)

It is folklore that modular multiplication is ``random''. This concept is useful for many applications, such as generating pseudorandom sequences, or in quasi-Monte Carlo methods for multi-dimensional numerical integration. Zaremba's theorem quantifies the quality of this ``randomness'' in terms of certain Diophantine properties involving continued fractions. His 40-year old conjecture predicts the ubiquity of moduli for which this Diophantine property is uniform. It is connected to Markoff and Lagrange spectra, as well as to families of ``low-lying'' divergent geodesics on the modular surface. We prove that a density one set satisfies Zaremba's conjecture, using recent advances such as the circle method and estimates for bilinear forms in the Affine Sieve, as well as a ``congruence'' analog of the renewal method in the thermodynamic formalism. This is joint work with Jean Bourgain.


When the sieve works
Dimitris Koukoulopoulos (Université de Montréal)

The simplest sieve problem asks for estimates on $S(x;P)$, the number of integers up to $x$ that have no prime factor from a set of primes $P\subset[1,x]$. A standard probabilistic heuristic predicts that this number is about $x\cdot\prod_{p\in P}(1-1/p)$. The two extreme examples, when $P=\{p\le y\}$ or when $P=\{y < p\le x\}$, have been studied extensively in the literature, corresponding to counting integers with no small prime factors or smooth numbers, respectively. While in the first case the prediction of the heuristic is accurate, in the second case it fails dramatically as $\log x/\log y\to\infty$. In this talk we investigate what happens for sets $P$ that have both some small and some big prime factors. Using tools from additive combinatorics, we show that if $P^c\cap[x^{1/u},x]$ is not too sparse for some $u$, then $S(x;P)$ is about $x\cdot u^{-O(u)}\prod_{p\in P}(1-1/p)$. This is joint work with Andrew Granville and Kaisa Matomäki.
Slides


Homotopy theory and Diophantine geometry
Minhyong Kim (University of Oxford)

In the 1980's Grothendieck wrote a letter to Faltings proposing that the arithmetic theory of fundamental groups should have substantial applications to the study of Diophantine problems. We will discuss the influence of that letter, and the progress that has been made on Grothendieck's programme over the last few decades.
Slides


On $\lambda$-invariants of number fields
Manfred Kolster (McMaster University)

For an odd prime $p$ we prove a Riemann-Hurwitz type formula for odd eigenspaces of the standard Iwasawa modules over $F(\mu_{p^{\infty}})$, the field obtained from a totally real number field $F$ by adjoining all $p$-power roots of unity. We use a new approach based on the relationship between eigenspaces and étale cohomology groups over the cyclotomic ${\mathbb Z}_p$-extension $F_{\infty}$ of $F$. The systematic use of étale cohomology greatly simplifies the proof and allows to generalize the classical result about the minus-eigenspace to all odd eigenspaces. This is joint work with A. Movahhedi.


On the diophantine equation $-1^k +2 ^k + \ldots + (-1)^{x} x^k=g(y)$
Dijana Kreso (Graz University of Technology)

In 2000, Bilu and Tichy succeeded in fully joining polynomial decomposition theory with the classical theorem of Siegel on finiteness of integral points on curves of genus greater than 0, to give a complete ineffective finiteness criterion for diophantine equations of the form $f(x)=g(y)$, where $f(x)$ and $g(x)$ are polynomials with rational coefficients. Using the Theorem of Bilu and Tichy we characterized those polynomials $g(x)\in {\mathbb Q}[x]$ for which the diophantine equation $$-1^k +2 ^k + \ldots + (-1)^{x} x^k=g(y)$$ has infinitely many rational solutions with a bounded denominator, provided $k\geq 7$ and $\deg g\geq 2$. To the proofs we classified polynomial decompositions of Euler polynomials over the set of complex numbers, which we find to be a result of independent interest. In this talk we present the aforementioned results, which come from a joint work with Csaba Rakaczki from the University of Miskolc in Hungary.


Preperiodic points of quadratic polynomial maps defined over number fields
David Krumm (University of Georgia)

We describe a method for computing the sets of preperiodic points of quadratic maps over number fields, and give several examples of new structures obtained by this method. This is joint work with John Doyle and Xander Faber.


Grimm's Conjecture and Smooth Numbers
Shanta Laishram (Indian Statistical Institute)

Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i \mathop{|} n+i$. This function is related to a celebrated conjecture of C. A. Grimm which is considered very difficult. In this talk, we establish upper and lower bounds for $g(n)$ by relating its study to the distribution of smooth numbers. Standard conjectures concerning smooth numbers in short intervals imply $g(n) =O(n^\epsilon)$ for any $\epsilon > 0$. We also prove unconditionally that $g(n) =O(n^\alpha)$ with $0.45 < \alpha < 0.46$. The study of $g(n)$ has some interesting implications for gaps between consecutive primes. This is a joint work with R. Murty.
Slides


Mahler measures of some K3 surfaces
Matilde Lalin (Université de Montréal)

We study the Mahler measure of the three-variable Laurent polynomial $x + \frac{1}{x} + y + \frac{1}{y} + z + \frac{1}{z} -k$ where $k$ is a parameter. The zeros of this polynomial define (after desingularization) a family of $K3$-surfaces. In favorable cases, a singular $K3$-surface is obtained and the Mahler measure is related to its $L$-function. This was first studied by Marie-Jose Bertin. In this talk we present some new formulas. This is joint work with Marie-Jose Berin, Amy Feaver, Jenny Fuselier, and Michelle Manes.


Counting rational points on singular cubic surfaces
Pierre Le Boudec (Institut de Mathématiques de Jussieu)

Let $V$ be a projective cubic surface. Manin's conjecture predicts the asymptotic behaviour of the number of rational points of bounded height on $V$. In this talk, we will focus on a specific example of singular cubic surface for which Manin's conjecture can be proved.


Modularity of Fano Varieties
Jacob Lewis (University of Vienna)

We prove that Landau-Ginzburg models mirror to rank-1 Fano threefolds are families of Shioda-Inose surfaces, thereby explaining the observed modular properties of their associated regularized quantum differential equations. We conjecturally extend modularity to Fano varieties of any rank, and discuss this conjecture on toric examples.


Additive twists of Fourier coefficients of $GL(3)$ Maass forms
Xiannan Li (University of Illinois at Urbana Champaign)

We find cancellation in sums of Fourier coefficients of $GL(3)$ Maass forms twisted by additive characters uniformly in the $GL(3)$ form. This had previously been proven in the special case of symmetric square lifts of classical Maass forms of full level by Xiaoqing Li and Matt Young.


Derivatives of $p$-adic $L$-functions
Benjamin Lundell (University of Washington)

We will discuss a new approach to proving the Ferrero-Greenberg formula for the derivative of a Kubota-Leoplodt $p$-adic $L$-function at $s=0$. The aim is to provide a proof which uses two-variable $p$-adic $L$-functions in a manner analogous to the Greenberg-Stevens proof of the Mazur-Tate-Teitelbaum conjecture for elliptic curves. In the Kubota-Leopldt setting, we use the Katz two-variable $p$-adic $L$-function attached to an imaginary quadratic field $K$. This is joint work with Ralph Greenberg and Shaowei Zhang.


Automorphic Forms and Combinatorial Asymptotics
Karl Mahlburg (Louisiana State University)

I will describe the use of various types of automorphic forms (including modular, mock modular, and Jacobi theta functions) in the study of the asymptotic behavior of combinatorial objects such as unimodal sequences and stacks.


K3 surfaces associated with Seiberg-Witten curves
Andreas Malmendier (Colby College)

The heterotic string compactified on $T^2$ has a large discrete symmetry group $SO(2, 18; Z)$, which acts on the scalars in the theory in a natural way. In the first part of the talk we will give a more complete construction of these models in the special cases in which either there are no Wilson lines - and $SO(2, 2;Z)$ symmetry - or there is a single Wilson line - and $SO(2, 3; Z)$ symmetry. In those cases, the modular forms have be analyzed in detail and there turns out to be a precise theory of K3 surfaces with prescribed singularities which corresponds to the structure of the modular forms. In the second part of the talk, we will show how families of Jacobian elliptic K3 surfaces of Picard rank 17 and 18 are in turn related to Seiberg-Witten curves of gauge theory by quadratic twist. We will work out precise relations between the function theory and geometry on these K3 surfaces. Using special functions we will compute explicitly the periods and period relations for their two-forms and many-valued modular forms, and describe the transcendental lattices and the Riemann matrices for associated Kuga-Satake varieties explicitly in terms of the periods. (This is joint work with David Morrison and Charles Doran.)


Rational conformal field theory, congruence modular forms, and the bounded denominator conjecture.
Christopher Marks (University of Alberta)

It has been understood for quite some time that certain modular functions encode (in their Fourier coefficients) the data of quantum mechanical systems arising in two-dimensional conformal field theory, and mathematical physicists "know" that the modular functions attached to physical theories belong to congruence subgroups of the modular group $SL_2(Z)$. Nonetheless, no mathematically rigorous proof of this "fact" has appeared thus far. After discussing (very briefly) the connection between these modular functions and the physical systems they model, I will explain how this congruence property follows from a much more fundamental mathematical phenomenon, which was first observed by Atkin and Swinnerton-Dyer in the early 1970s, and is known today as the bounded denominator conjecture for modular forms. I will then explain how the theory of vector-valued modular forms and modular differential equations provides a novel and somewhat practical way to probe this conjecture, and present some recent evidence I have accumulated in support of the conjecture, in the three-dimensional setting.


Average values of some non-multiplicative functions
Greg Martin (University of British Columbia)

Of the many functions of arithmetic interest, those that are multiplicative have average values that can be calculated with a variety of standard techniques. Recently, we have had the occasion to study several functions that, despite arising from questions that are multiplicative in nature, are not themselves multiplicative functions. These functions include: the least character nonresidue of Dirichlet characters; the least prime with a particular factorization type in cubic number fields; and the number of reductions (modulo primes) of an elliptic curve that have a specified number of points. We describe these results and the methods required to evaluate their average values. The results are joint work with Paul Pollack and also with Ethan Smith.
Slides


Almost-prime $k$-tuples
James Maynard (University of Oxford)

For $i=1,\dots,k$, let $L_i(n)=a_in+b_i$ be integer linear functions such that $\prod_{i=1}^k L_i(n)$ has no fixed prime divisor. Then it is conjectured that there are infinitely many integers $n$ for which all of the $L_i(n)$ ($1\le i\le k$) are simultaneously prime, and weighted sieves allow us to show that there are infinitely many integers $n$ for which $\prod_{i=1}^k L_i(n)$ has at most $r_k$ prime factors (for some constant $r_k$ depending only on $k$). We describe a new weighted sieve which improves these bounds when $k\ge 3$.
Slides


A New Computation of Viswanath's Constant
Karyn McLellan (Dalhousie University)

The random Fibonacci sequence is defined by $t_1 = t_2 = 1$ and \[ t_n = \pm t_{n-1} + t_{n-2}, \] for $n \geq 3$, where the $\pm$ sign is chosen at random with $P(+) = P(-) = \frac{1}{2}$. We can think of all possible such sequences as forming a binary tree $T$. Viswanath (1999) has shown that almost all random Fibonacci sequences grow exponentially at the rate $1.13198824\ldots$. He was only able to find 8 decimal places of this constant through the use of random matrix theory and a fractal measure, although Bai (2007) has extended the constant by 5 decimal places. Numerical experimentation is inefficient because the convergence is so slow. In this talk we will discuss a new computation of Viswanath's constant which at present gives only 8 decimal places of accuracy, but we feel this can be improved. It is based on a formula due to Kalmár-Nagy, and uses an interesting reduction of the tree $T$ developed by Rittaud.
Slides


Elliptic Curves, KR-theory, and T-duality
Stefan Mendez-Diez (University of Alberta)

We will use the classification of real elliptic curves to describe string theory compactifications on torus orientifolds. We will show that the various string theories can be broken into 3 groups, based on the species of the possible antiholomorphic involutions, via T-duality. We will describe how KR-theory can be used to determine the D-brane content and add insight into the various T-dualities involved.


The pair correlation of the zeros of the Riemann zeta-function
Micah B. Milinovich (University of Mississippi)

Assuming the Riemann Hypothesis, using Montgomery's Theorem and properties of extremal functions of exponential type, P. X. Gallagher proved bounds for the number of pairs $1/2+i \gamma$ and $1/2+i\gamma'$ of zeros of the zeta-function with $0<\gamma,\gamma' \le T$ and satisfying $0 < \gamma'-\gamma \le 2 \pi \beta/ \log T$ when $\beta$ is a positive integer or half integer and $T$ is large. Using different (but related) techniques, I will indicate how to extend Gallagher's work to all $\beta>0$. This is joint work with Emanuel Carneiro and Vorrapan Chandee.
Slides


Siegel's mass formula and averages of Dirichlet $L$-functions over function fields
Jorge Morales (Louisiana State University)

We give a self-contained proof of Siegel's mass formula for definite ternary quadratic forms over the polynomial ring ${\mathbb F}_q[t]$, where ${\mathbb F}_q$ is the finite field with $q$ elements and $q$ is odd. The mass formula is obtained by estimating averages of representation numbers over a fixed degree and does not involve computations with local densities. Our method is specific to the function field setting and reveals an interesting and seemingly new relation with work by Hoffstein and Rosen on averages of Dirichlet $L$-functions over function fields. This is joint work with P. Maciak (EPFL).
Slides


Negative Pisot and Salem numbers as roots of Newman polynomials
Michael Mossinghoff (Davidson College)

A Newman polynomial has all its coefficients in $\{0, 1\}$ and constant term 1; every root of a Newman polynomial has modulus between $1/\tau$ and $\tau$, where $\tau$ denotes the golden ratio. We show that every negative Pisot number in $(-\tau, -1)$ with no positive conjugates, and every negative Salem number in the same range obtained by using Salem's construction on small negative Pisot numbers, is satisfied by a Newman polynomial. We also verify this for all negative Salem numbers $\alpha > -\tau$ having small degree. We then raise the question of whether there exists a real constant $C > 1$ so that if $\alpha$ is an algebraic unit having no positive real conjugates and Mahler's measure of $\alpha$ is less than $C$, then a Newman polynomial must exist having $\alpha$ as a root. We show, however, that one cannot possibly take $C = \tau$ in this statement by constructing a number of integer polynomials having no positive real roots and Mahler's measure less than $\tau$, which do not divide any Newman polynomial. This is joint work with Kevin Hare.
Slides


Modular forms for triangle groups
Hossein Movasati (IMPA)

In this talk we first describe a solution of the Halphen equation which has modular properties with respect to a group which is not discrete in general. We show that in the particular case of triangular groups, the Halphen equation gives us a basis of the algebra of modular and quasi-modular forms. Some statements and conjectures about the q-expansion of such modular forms will be also presented. This is a work under preparation jointly with Ch. Doran, T. Gannon, Kh. M. Shokri.


The error term in the Sato Tate conjecture
Kumar Murty (University of Toronto)

With the recent proof of the Sato-Tate conjecture, there is some interest in understanding what type of error term one might expect. Assuming some reasonable hypotheses on the symmetric power L-functions, we exhibit an explicit error term.


Special values of $L$-series
M. Ram Murty (Queen's University)

We give a general overview of what is known and unknown about special values of $L$-series attached to automorphic representations and Galois representations.


False complex multiplication on elliptic curves
Filip Najman (University of Zagreb)

We introduce the notion of false complex multiplication on elliptic curves, and describe certain classes of so called ${\mathbb Q}$-curves with this property. As one of the consequences, we prove that any elliptic curves over any quadratic field with a point of order 13 or 18 and any elliptic curve over any quartic field with a point of order 22 has even rank.
Slides


The Artin-Wedderburn Theorem and local Tate duality
Rachel Newton (University of Cambridge)

The cup product map which occurs in the statement of local Tate duality has a natural interpretation in terms of constructing a central simple algebra from two 1-cocycles. I will show how to use the proof of the Artin-Wedderburn Theorem to strip away the matrix part of the central simple algebra, thereby exposing the division ring which gives its class in the Brauer group.


The mock modularity of $q$-hypergeometric series
Robert Osburn (University College Dublin)

An intriguing and almost completely unsolved problem is to understand the overlap between classes of $q$-hypergeometric series and modular forms. This challenge was the subject of George Andrews' plenary address at the Millennial Conference on Number Theory and has its origin in Ramanujan's last letter to G. H. Hardy on January 12, 1920 whereby 17 mock theta functions were introduced. In this talk, we discuss recent work concerning the explicit construction of new individual examples and infinite families of mock theta functions. This is joint work with Jeremy Lovejoy (Paris 7).
Slides


Hilbert's tenth problem over number fields
Jennifer Park (Massachusetts Institute of Technology)

Hilbert's tenth problem over ${\mathbb Q}$ (or, any number field $K$) asks the following. Given a polynomial in several variables with coefficients in ${\mathbb Q}$ (resp. $K$), is there a general algorithm that decides whether this polynomial has a solution in ${\mathbb Q}$ (resp. $K$)? Unlike the classical Hilbert's tenth problem over ${\mathbb Z}$, this problem is still open. To reduce this problem to the classical problem, we need a definition of ${\mathbb Z}$ in ${\mathbb Q}$ (resp. ring of integers in $K$) using only an existential quantifier. This problem is still open. I will present a definition of the ring of integers in a number field, which uses only one universal quantifier, which is, in a sense, the simplest logical description that we can hope for. This is a generalization of Koenigsmann's work, which defines ${\mathbb Z}$ in ${\mathbb Q}$ using one universal quantifier.


An upper bound for the average number of amicable pairs
James Parks (Concordia University)

Amicable numbers have been known since Pythagoras and are defined to be two different numbers so related that the sum of the proper divisors of each is equal to the other number. In 2009, Silverman and Stange provided an elliptic curve analogue to amicable numbers. Let $E$ be an elliptic curve over ${\mathbb Q}$. They defined a pair $(p, q)$ of rational primes to be an amicable pair for $E$ if $E$ has good reduction at these primes and the number of points on the reductions $\widetilde{E}_p$ and $\widetilde{E}_q$ satisfy $\#\widetilde{E}_p({\mathbb F}_p) = q$ and $\#\widetilde{E}_q({\mathbb F}_q) = p$. Let $Q_E(X)$ denote the number of amicable pairs $(p, q)$ for $E/{\mathbb Q}$ with $p\leq X$. Then they conjectured that $Q_E(X) \asymp X/(\log X)^{-2}$ if $E$ does not have complex multiplication. In this talk I will provide an upper bound for the average of $Q_E(X)$ over the family of all elliptic curves which is very close to the conjectural asymptotic of Silverman and Stange.
Slides


Near-optimal estimates for multidimensional Weyl sums
Scott Parsell (West Chester University)

We report on recent joint work with Sean Prendiville and Trevor Wooley, in which we investigate generalizations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems. By extending Wooley's efficient congruencing method to this multidimensional setting, we obtain sharp estimates for mean values in which the number of variables required is within a constant factor of best possible. We mention several concrete examples of systems to which our new bounds apply and discuss some consequences for associated Diophantine problems.
Slides


Transition Mean Values, Eisenstein Series and Nondifferentiable Functions
Ian Petrow (Stanford University)

The average value of the Jacobi symbol in both entries is computed easily when the sum in one entry is much longer than the other. When both sums are of proportional size, it was discovered in 2000 that the leading constant in the asymptotic average varies as ``Riemann's nondifferentiable function''. In recent work, we have found that the same phenomenon occurs for averages of shifted convolution sums of Hecke eigenvalues of classical modular forms. In both cases we explain the strange behavior in the transition region through multiple Dirichlet series, Eisenstein series and automorphic distributions.
Slides


On the number of solutions of some binomial Thue inequalities
István Pink (University of Debrecen)

Let $a$, $b$, $n$ be positive integers with $1 \leq a < b$ and $n \ge 3$. Consider the binomial Thue equation \begin{equation} \label{BdW} \vert ax^n-by^n\vert=1. \end{equation} Bennett and de Weger (1998) and Bennett (2001) proved that equation (\ref{BdW}) has at most one solution in positive integers $(x,y)$. In our talk we give the following extension of the above result. Consider the binomial Thue inequality \begin{equation} \label{BPR} \vert ax^n-by^n\vert \leq 3. \end{equation} We prove that apart from certain exceptions inequality (\ref{BPR}) possesses at most one solution in positive integers $(x,y)$. The proof combines a result of Evertse (1983) with the hypergeometric and Baker's method. This result is a joint work with M. A. Bennett and Zs. Ràbai.
Slides


The sum of divisors of $n$, modulo $n$
Paul Pollack (University of British Columbia)

The natural number $n$ is called perfect if $n$ is the sum of all of its proper divisors. More generally, $n$ is called multiply perfect if $\sigma(n) \equiv 0 \pmod{n}$. We discuss recent upper bounds on the number of solutions $n \leq x$ to the more general congruence $\sigma(n) \equiv a\pmod{n}$. This is joint work with Aria Anavi, Carl Pomerance, and Vladimir Shevelev.
Slides


Balanced subgroups of the multiplicative group
Carl Pomerance (Dartmouth College)

For an integer $n > 2$, the unit group modulo $n$ has an even number of elements, with half of them having representatives in $(0,n/2)$ and the other half having representatives in $(n/2,n)$. It is ``balanced''. Say a subgroup $H$ of this unit group is ``balanced'' if each coset of $H$ is evenly split between the bottom half and the top half. Suppose $g > 1$ is a fixed integer. We are concerned with the distribution of numbers $n$ coprime to $g$ for which the cyclic subgroup $\langle g\rangle$ in the unit group mod $n$ is balanced. This has an application to the statistical study of the rank of the Legendre curve over function fields. (Joint work with Douglas Ulmer.)
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Uniform boundedness of rational points
Bjorn Poonen (MIT)

We propose new conjectures that combine and generalize existing uniform boundedness statements concerning rational points on varieties, torsion points on abelian varieties, and preperiodic points.
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The $p$-torsion invariants of Jacobians over finite fields
Rachel Pries (Colorado State University)

An elliptic curve defined over a finite field of characteristic $p$ can be ordinary or supersingular. One can distinguish between these by counting the number of its $p$-torsion points or by calculating its $L$-function. This distinction can be generalized for abelian varieties of higher dimension, giving rise to several important invariants such as the $p$-rank, Newton polygon, and Ekedahl-Oort type. Very little is known about which invariants occur for Jacobians of curves. In the first part of the talk, I will survey existence and non-existence results, with the aim of explaining several open conjectures. In the second part of the talk, I will describe new results in the case when the Jacobian has an automorphism of order $p$.
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On Shimura Decomposition and Tunnell-like Formulae
Soma Purkait (University of Warwick)

Let $k$ be an odd integer and $N$ be a positive integer divisible by $4$. Let $g$ be a newform of weight $k-1$, level dividing $N/2$ and trivial character. We give an explicit algorithm for computing the space of cusp forms of weight $k/2$ that are `Shimura-equivalent' to $g$. Applying Waldspurger's theorem to this space allows us to express the critical values of the $L$-functions of twists of $g$ in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the $n$-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms.


Limitations to mollifying $\zeta(s)$
Maksym Radziwill (Stanford University)

We establish limitations to how well one can mollify the Riemann zeta function on the critical line. For a very long Dirichlet polynomial $A(s)$ we obtain a non-trivial lower bound for the mean-square of $1 - \zeta(s)A(s)$ on the critical line; the lower bound depends on the length of $A$ but only very weakly on the coefficients of $A$. Conjecturally, our lower bound is a constant factor away from the truth. When $A$ is a long mollifier, our lower bound captures the contribution of the off-diagonal terms that would arise if one were to expand $1 - \zeta(s)A(s)$ into a Dirichlet series to compute the mean-square of $1 - \zeta(s)A(s)$.
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Quantitative forms of Landau's equivalence theorem
Olivier Ramaré (CNRS/Université Lille 1)

There are many explicit asymptotic evaluations concerning of the number of primes up to $x$ but much less concerning the summatory function of the MÖbius function. Both problems are known to be equivalent, but a strong quantitative form of this equivalence is missing. We will present some recent work in this direction, as well as some consequences regarding $\sum_{n\le x}\mu(n)$, $\sum_{n\le x}\mu(n)/n$, $\sum_{n\le x,(n,r)=1}\mu(n)/n$, $\sum_{n\le x,(n,r)=1}\mu(n)\log(x/n)/n$, and finally $\sum_{n\le x}\Lambda(n)/n$. All these results will improve on earlier ones. In the last mentioned case, we will even get an extremely sharp estimate.
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A new effective version of Lindemann-Weierstrass theorem by methods of algebraic independence
Damien Roy (University of Ottawa)

The Lindemann-Weierstrass theorem asserts that the exponentials of ${\mathbb Q}$-linearly independent complex algebraic numbers are algebraically independent over ${\mathbb Q}$. It is one of the most satisfactory results in transcendental number theory. The first complete proof was published by Weierstrass in 1885, using Hermite's method, based on the construction of Pade's approximants. A first measure of algebraic independence was given by Mahler in 1932 and later improved by several authors. At present the best known completely explicit measure is that of Sert in 1999, using Laurent's interpolation determinants. An alternative approach based on methods of algebraic independence was first proposed by Chudnovsky. In this spirit in 1998, Ably proved a measure of algebraic independence using an effective version of Philippon's algebraic independence criterion. The goal of this talk is to present another proof of the Lindemann-Weierstrass theorem by an algebraic independence method, using only resultants and a particularly simple construction of an auxiliary function. It leads to an effective estimate which is stronger than that of Ably but weaker than that of Sert.


Identities for $L$-functions
Michael Rubinstein (University of Waterloo)

I will present some interesting identities for $L$-functions that have application to high precision computation, special values, and which are beautiful to look at.


Distribution of lattices and the Frobenius problem
Wolfgang Schmidt (University of Colorado)

It has been conjectured that the ``average value'' of the Frobenius number $F(A)$ of a coprime $m$-tuple of positive integers $a_1,\ldots,a_m$ is about $$P(A) = (a_1\cdots a_m)^{1/(m-1)}.$$ We will estimate how often $ F(A) / P(A)$ is large. In our arguments we will use results on the distribution of sublattices of ${\mathbb Z}^m$.


The value distribution of the Epstein zeta function
Anders SÖdergren (Institute for Advanced Study)

We will discuss the value distribution of the Epstein zeta function $E_n(L,cn)$ for real $c$ and a random lattice $L$ of covolume 1 and large dimension $n$. We will present applications of our results to, for example, the distribution of heights of flat tori, and if time permits, we will also highlight the main ingredients in the proofs, including a new bound on the remainder term in the generalized circle problem.
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Limiting Distributions in Number Theory
Majid Shahabi (University of Lethbridge)

One way to study number theoretical functions is through limiting distributions. In 1935, Wintner proved that, under the assumption of the Riemann hypothesis, $$e^{-y/2}(\pi(e^y)-\text{Li}(e^y))$$ has a limiting distribution. In 1994, Rubinstein and Sarnak showed that, under the assumption of the Riemann hypothesis for Dirichlet $L$-functions $L(s,\chi)$ mod $q$, \begin{equation*} ye^{-y/2}\times(\varphi(q)\pi(e^y,q,a_1)-\pi(e^y),\dots,\varphi(q)\pi(e^y,q,a_r)-\pi(e^y)) \end{equation*} has a limiting distribution. In 2004, Ng found necessary conditions that imply $e^{-y/2}M(e^y)$ has a limiting distribution. In this talk, we state a theorem which gives general conditions under which a function with an ``explicit formula'' possesses a limiting distribution. Our theorem generalizes the previously mentioned results. Moreover, we provide several new examples of functions with limiting distributions. Furthermore, we study large deviations of such limiting distributions.


Mordell-Weil Generators for Cubic Surfaces
Samir Siksek (University of Warwick)

Let C be a smooth plane cubic curve over the rationals. The Mordell-Weil Theorem can be restated as follows: there is a finite subset B of rational points such that all rational points can be obtained from this subset by successive tangent and secant constructions. It is conjectured that a minimal such B can be arbitrarily large; this is indeed the well-known conjecture that there are elliptic curves with arbitrarily large ranks. This talk is concerned with the corresponding problem for cubic surfaces.


The sensual Apollonian circle packing
Katherine Stange (Stanford University)

We explain the similarity between the parallelogram law for a quadratic form and the Descartes rule for the curvatures in an Apollonian circle packing. Conway described the values of a binary quadratic form on a topograph representing $\mathbb{P}^1(\mathbb{Q})$. The curvatures of the circles in an Apollonian circle packing are related to values of a Hermitian form on a similar topograph for $\mathbb{P}^1(\mathbb{Q}(i))$.


Exceptional units and cyclic resultants
Cameron Stewart (University of Waterloo)

Let $\alpha$ be a non-zero algebraic integer of degree $d$ over the rationals. Put $K=Q(\alpha)$ and let $O(K)$ denote the ring of algebraic integers of $K$. We shall discuss estimates for the number of positive integers $n$ for which $\alpha^n-1$ is a unit in $O(K)$ and for the largest positive integer $n$ for which $\alpha^j-1$ is a unit for $j$ from 1 to n.


Computing the image of Galois
Andrew Sutherland (MIT)

Let $E/{\mathbb Q}$ be an elliptic curve and let $\ell$ be a prime. The action of the absolute Galois group of ${\mathbb Q}$ on the $\ell$-torsion points of $E$ yields a representation $$ \rho_{E,\ell}\colon{\rm Gal}(\overline{{\mathbb Q}}/{\mathbb Q})\longrightarrow {\rm Aut}(E[\ell])\simeq{\rm GL}_2({\mathbb Z}/\ell{\mathbb Z}). $$ Under a well-known conjecture of Serre, there is a constant $\ell_{\rm max}$ such that $\rho_{E,\ell}$ is surjective for all $E$ without complex multiplication and all $\ell > \ell_{\rm max}$; it is generally believed that $\ell_{\rm max}=37$ suffices. Under this conjecture, there is a finite list of groups that arise as the image of a non-surjective representation $\rho_{E,\ell}$ for which $E$ does not have complex multiplication. As a first step toward determining this list, I will describe a highly efficient algorithm that, given a family $\mathcal{E}$ of elliptic curves $E/{\mathbb Q}$ and a bound $\ell_{\max}$, computes the image of $\rho_{E,\ell}$ for all $E\in\mathcal{E}$ and $\ell\le\ell_{\max}$. Applying this algorithm to the elliptic curves in Cremona's tables and the Stein-Watkins database (over 100 million curves) yields 45 non-conjugate groups that arise as the image of $\rho_{E,\ell}$ for some $E$ without complex multiplication and $\ell\in\{2,3,5,7,11,13,17,37\}$. I will also discuss generalizations to treat composite values of $\ell$, and consider the general setting of abelian varieties over number fields.
Slides


Squarefree values of trinomial discriminants
Mark Thom (University of British Columbia)

The discriminant of a trinomial of the form $x^n \pm x^m \pm 1$ has the form $\pm n^n \pm (n-m)^{(n-m)}m^m$ when $n$ and $m$ are co-prime. We determine necessary and sufficient conditions for identifying primes whose squares never divide the discriminants arising from coprime pairs $(n,m)$. These conditions are adapted into an exhaustive search method, which we use to corroborate a heuristic estimate of the density of all such primes among the odd primes. The same results are used to produce a heuristic estimate of the density of squarefree values of these discriminants. This is joint work with David Boyd and Greg Martin.


Sums of distinct divisors
Lola Thompson (Dartmouth College)

Following Srinivasan, an integer $n\geq 1$ is called practical if every natural number in $[1,n]$ can be written as a sum of distinct divisors of $n$. This motivates us to define $f(n)$ as the largest integer with the property that all of $1, 2, 3, \dots, f(n)$ can be written as a sum of distinct divisors of $n$. (Thus, $n$ is practical precisely when $f(n)\geq n$.) We think of $f(n)$ as measuring the ``practicality'' of $n$; large values of $f$ correspond to numbers $n$ which we term practical pretenders. In this talk, we will describe the distribution of the practical pretenders, as well as the maximal order of $f$ when restricted to non-practical inputs; these results improve upon theorems of Saias and Hausman and Shapiro in the literature on practical numbers. This talk is based on joint work with Paul Pollack.


Modular forms over imaginary quadratic fields: torsion and regulators.
Akshay Venkatesh (Stanford University)

The story of modular forms over imaginary quadratic fields is different in flavor to the story over the rational numbers, because of the (experimentally observed) abundance of torsion. I will explain what this means, and show some data from H. Sengun's computations. I will then discuss analytic questions that arise when one tries to rigorously establish this abundance -- in particular, a conjecture of mine and Bergeron about the growth of "regulators".


On the Arithmetic of BPS states
Johannes Walcher (McGill University)

The most recent calculations of Hodge theoretic normal functions on families of Calabi-Yau 3-folds predict, via mirror symmetry, enumerative invariants with values in the number field over which the algebraic cycle is defined. The integral structure unveils a new, arithmetic, twist of the dilogarithm, the D-logarithm.


Transcendence of the Gamma function on imaginary quadratic fields
Chester Weatherby (Queen's University)

Little is known about the transcendence of special values of the Gamma function. In this talk we examine the Gamma function at points from an imaginary quadratic field. As a corollary of our analysis, we gain knowledge about values of infinite products of rational functions.
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Testing Asymptotic Estimates for Cubic Function Fields
Colin Weir (University of Calgary)

A now classical result, states that the number of cubic fields up to a discriminant bound $X$, grows linearly in $X$. However, in the general programme described by Ellenberg and Venkatesh (2010) for formulating these types of estimates, they point out the ``alarming gap between theory and experiment'' in asymptotic predictions for number fields. In the case of cubic number fields, this inconsistency lead Roberts (2001) to conjecture the secondary term in the asymptotic formula (which is now theorem). In the function field setting, however, there is practically no experimental data to potentially identify a similar such gap. We present a Kummer theoretic method for constructing and tabulating cubic function fields. We also compare the data of our algorithm with the known (and conjectural) asymptotic estimates for the number of cubic function fields.


Shafarevich Theorems for dynamical systems
Lloyd West (CUNY Graduate Center)

The Shafarevich conjecture for curves states that for a number field or function field $K$ and a given finite set of places $S$ of $K$ there are only finitely many isomorphism classes over $K$ of curves of genus $g$. The conjecture was proved for characteristic zero function fields by Parshin and Arakelov in the early seventies and extended to positive characteristic by Szpiro. Via the ``Parshin-trick,'' the number fields version of the Shafarevich conjecture plays a central role in Faltings's proof of the Mordel Conjecture. Within the topic ``arithmetic dynamics'', Szpiro and Tucker have recently proved a version of Shafarevich's conjecture for rational maps of the projective line defined over number fields. Another version was subsequently proved by Petsche. I have begun work with Szpiro to prove a function field version. In this talk I will introduce the above results of Szpiro and Tucker and discuss possible ways to prove the function field version.


Lattice-polarized K3 surfaces and Humbert surfaces
Ursula Whitcher (University of Wisconsin)

We study the Picard-Fuchs equations of a family of lattice-polarized K3 surfaces. Each surface in our family is uniquely associated with a principally polarized abelian surface. Loci where the Picard-Fuchs equations drop in rank correspond to Humbert surfaces; thus, we may characterize Humbert surfaces using differential equations. This talk describes joint work with Charles Doran and Hossein Movasati.


Conjugacy classes in GSp$_4$ and an application to the enumeration of abelian surfaces
Cassie Williams (Colorado State University)

The Frobenius endomorphism of an abelian variety $A/{\mathbb F}_q$ acts as a symplectic similitude on the torsion subgroups $A[\ell^n](\overline{{\mathbb F}}_q)$. In 2003, Gekeler considered the distribution of these endomorphisms for elliptic curves in the groups GL$_2({\mathbb Z}/\ell^r)$ and found a relationship, via isogeny and the class number, to the Euler factors of the $L$-function of a quadratic imaginary field. We have extended Gekeler's heuristic for the distribution of Frobenius elements from elliptic curves to abelian surfaces by identifying conjugacy classes in GSp$_4({\mathbb Z}/\ell^r)$ and relating their sizes to the class number via isogeny.
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Compact Representation of the Generator of a Certain Ideal
Hugh Williams (University of Calgary)

Suppose we have a real quadratic number field of discriminant $D$. If we have a reduced principal ideal $I$, it usually requires an exponential (in $\log D$) amount of time to write out a generator of $I$ in the conventional way. However, there exists a representation of this generator, called a compact representation, which can be written out in polynomial time. In this talk I discuss several new techniques for finding a compact representation when we are given an approximate value of the logarithm of the absolute value of a generator and an integral basis of $I$.
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Near optimal estimates for exponential sums
Trevor D. Wooley (University of Bristol)

Exponential sums of large degree play a prominent role in the analysis of problems spanning the analytic theory of numbers. In 1935, I. M. Vinogradov devised a method for estimating their mean values very much more efficient than the methods available hitherto due to Weyl and van der Corput, and subsequently applied his new estimates to investigate the zero-free region of the Riemann zeta function, in Diophantine approximation, and in Waring's problem. Recent applications from the 21st century include sum-product estimates in additive combinatorics, and the investigation of the geometry of moduli spaces. Over the past 75 years, estimates for the moments underlying Vinogradov's mean value theorem have failed to achieve those conjectured by a factor of roughly $\log k$ in the number of implicit variables required to successfully analyse exponential sums of degree $k$. In this talk we will sketch out the ideas underlying our recent work which comes within a stone's throw of the best possible conclusions.
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Unitary untouchable numbers
Hee-Sung Yang (Dartmouth College)

In 1973, Erdős proved that a positive proportion of numbers are untouchable; that is, not of the form $s(n)$, where $s(n)=\sigma(n)-n$ is the sum of the proper divisors of $n$. We investigate the analogous question where $\sigma$ is replaced with similar divisor functions, such as the unitary sum-of-divisors function $\sigma^*$ (which sums those divisors $d$ of $n$ coprime to $n/d$). This problem is mentioned in Guy's book on unsolved problems.


Solvability in polynomials of Pell equations in a pencil
Umberto Zannier (Scuola Normale Superiore)

The classical Pell equation $X^2-DY^2=1$, to be solved in integers $X,Y\neq 0$, has a known analogue for function fields (studied already by Abel), where now $D=D(t)$ is a complex polynomial of even degree and we seek solutions in nonzero complex polynomials $X(t),Y(t)$. In this context solvability is no longer ensured by simple conditions on $D$ and may be considered `exceptional'.
In the talk we shall let $D(t)=D_\lambda(t)$ vary in a pencil. When $D_\lambda(t)$ has degree $\le 4$, it may be seen that for infinitely many $\lambda\in{\mathbb C}$ there are nontrivial solutions. On the other hand, it is not so when $D_\lambda$ has degree $ 6$ (provided natural assumptions are verified).
Such a finiteness result, obtained jointly with D. Masser, represents a conjecture of Pink for `Unlikely Intersections' in a simple abelian scheme over a curve. We shall survey on the whole context and also comment on related `Likely Intersections' associated to a `quasi-Pell' equation.


Averages of shifted convolutions of $d_3(n)$
Liangyi Zhao (Nanyang Technological University)

We investigate the first and second moments of shifted convolutions of the generalized divisor function $d_3(n)$.


On quadratic Waring-Goldbach problems
Lilu Zhao (University of Hong Kong)

It is conjectured that all sufficiently large integers satisfying a congruence condition can be represented as a sum of four squares of primes. This problem is still unsolved so far, while there are some approximations to the conjectured result. Brüdern and Fouvry established that all large integers $n\equiv 4\pmod{24}$ can be written as sums of four squares of $P_{34}$ numbers. Heath-Brown and Tolev showed that each large integer $n\equiv 4\pmod{24}$ can be written as the sum of one square of a prime number and three squares of $P_{101}$ numbers. Liu, Liu and Zhan proved that sufficiently large even numbers can be represented as the sum of four squares of primes and $k$ powers of two. Those results have been improved by a number of researchers. We shall present some new improvement upon above results. We prove all large integers $n\equiv 4\pmod{24}$ can be written as sums of four squares of $P_{4}$ numbers by developing the method of Heath-Brown and Tolev. We also show that $k=39$ is acceptable for the powers of two by developing the method of Wooley on quadratic Waring-Goldbach problems.
Slides


A generalization of Vinogradov's mean value theorem in function fields
Xiaomei Zhao (University of Waterloo)

Let ${\mathbb F}_q[t]$ denote the polynomial ring over the finite field ${\mathbb F}_q$ having $q$ elements. We obtain new upper bounds for the number of solutions of a system of symmetric equations in ${\mathbb F}_q[t]$, which can be viewed as a multi-dimensional version of the system considered in Vinogradov's mean value theorem in ${\mathbb F}_q[t]$. We then apply these bounds within a variant of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number of solutions of a system connected to the problem concerning rational spaces on hypersurfaces in function fields.
Slides