## Representations in Arithmetic

### Scientific Committee:

Rachel Ollivier (University of British Columbia)

Sujatha Ramdorai (University of British Columbia)

Peter Schneider (Universität Münster)

2018 Schedule:

Short term Visitors:

Jan 24-26: Benedict Gross, Harvard University with a special lecture on Jan 25, 2018:

Values of the zeta function at negative integers, from Euler to the trace formula

Although the zeta function $\zeta(s)$ is often named after Riemann, it was Euler who discovered many of its remarkable properties. After making his name on the evaluation of $\zeta(2)$, Euler was able to obtain similar formulas at all positive even integers, and defined putative values at negative integers, where the series does not converge. Euler showed these values at negative integers were all rational numbers. A comparison with the values at positive integers led him to guess the functional equation relating $\zeta(s)$ to $\zeta(1-s)$ (which was proved about one hundred years later by Riemann). I will begin by exposing some of this work, then show how the values at negative integers can be used to compute the dimension of certain spaces of automorphic forms. In a special case the dimension turns out to be 1, and this leads to a construction of local systems with exceptional Galois groups on the projective line (minus two points) over a finite field.

March 4-10: Antonio Lei, Université Laval: Lectures on March 5, 7, 9, 2018:

Iwasawa Theory for elliptic curves with super singular reduction.

Let E/Q be an elliptic curve. In Iwasawa Theory, we study the behaviours of E over a tower of number fields. For example, it is known that the Mordell Weil ranks of E over all p-power cyclotomic extensions of Q are bounded when p does not divide the conductor of E. Surprisingly, the techniques required to show this are very different depending on the number of points on the finite curve when we consider E reduced modulo p. The easier case is when E has "ordinary" reduction at p and the more difficult case is when E has "supersingular" reduction at p. I will review the Iwasawa-theoretic tools used to study the behaviours of E over cyclotomic fields in these two cases. I will also discuss some recent developments on the Iwasawa theory of elliptic curves over quadratic extensions of Q.

March 13-17:Adrian Iovita, Concordia University; Lecture on March 12 and 15, 2018.

p-Adic modular forms have first been defined by J.-P. Serre as q-expansions and have later been interpreted geometrically by N. Katz as sections of certain modular line bundles over the ordinary locus of the relevant modular curves. Katz also defined overconvergent modular forms of integer weights as overconvergent sections

of the modular line bundles of that weight. Many years later H. Hida and respectively R. Coleman defined ordinary, respectively finite slope overconvergent modular forms of arbitrary, p-adic weight as q-expansions and using these Coleman and Mazur constructed at the end of the 90's the famous eigencurve. Recently, together with Andreatta, Pilloni and Stevens we have been able to geometrically redefine the overconvergent modular forms of Hida and Coleman and so we were able to generalize these constructions to Hilbert and Siegel modular forms.

March 26- 29: Vaidehee Thatte, Queen's University; Lectures on March 27 and 28, 2018

Ramification Theory for Arbitrary Valuation Rings in Positive Characteristic.

In classical ramification theory, we consider extensions of complete discrete valuation rings with perfect residue fields. We would like to study arbitrary valuation rings with possibly imperfect residue fields and possibly non-discrete valuations of rank ≥ 1, since many interesting complications arise for such rings. In particular, defect may occur (i.e. we can have a non-trivial extension, such that there is no extension of the residue field or the value group). We present some new results for Artin-Schreier extensions of arbitrary valuation fields in positive characteristic p. These results relate the \higher ramification ideal" of the extension with the ideal generated by the inverses of Artin-Schreier generators via the norm map. We also introduce a generalization and further refinement of Kato's refined Swan conductor in this case. Similar results are true in mixed characteristic (0; p).

March 28: Ila Varma, Columbia University; Lecture on March 28, 2018

Counting D_4 quatic fields orered by conductor

We consider the family of D_4-quartic fields ordered by the Artin conductors of the corresponding 2-dimensional irreducible Galois representations. In this talk, I will describe ways to compute the number of such D_4 fields with bounded conductor. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination of geometry-of-number techniques, and applying Kummer theory together with L-function methods. Both of these strategies fall
short in the case of D_4 fields since counting quartic fields containing a quadratic subfield of large discriminant is difficult. However, when ordering by conductor, these techniques can be utilized due to additional algebraic structure that the Galois closures of such quartic fields have, arising from the outer automorphism of D_4. This result is joint work withAli Altug, Arul Shankar, and Kevin Wilson.

Other Visitors:

Feb 24 - March 7: Shen-Ning Tung, Universität Duisburg-Essen

March 3 - 12: Praham Hamidi, University of Waterloo

March 25- 29: Debanjana Kudu, University of Toronto

2017 Schedule:

The focus period has two components.

1. A focus semester on the mod p Langlands program for p-adic groups and (derived) Hecke algebras. It is centered around the following events and activities:

Visit of Peter Schneider (Universität Münster) as PIMS Distinguished Visitor, from February to July 2017.

Short term visitors:

- Marie-France Vignéras (Paris 7). February-March

- Otmar Venjakob (Universität Heidelberg). March

- Elmar Grosse-Klönne (Humboldt-Universität zu Berlin). March

- Laurent Berger (Ecole Normale Supérieure de Lyon). April-May

Seminars held on Thursdays at 3:30pm in ESB 4127. The schedule will be continuously updated.

- Feb 202128: Kiran Kedlaya (UC San Diego)

- March 2nd: Marie-France Vignéras (Paris 7)

- March 9th: Niccolo' Ronchetti (Stanford)

- March 16th: Otmar Venjakob (Universität Heidelberg)

- March 30th: Elmar Grosse-Klöne (Humboldt-Universität zu Berlin)

- April 27th: Laurent Berger (Ecole Normale Supérieure de Lyon)

- May 4th: Sandra Rozensztajn (Ecole Normale Supérieure de Lyon)

- May 25th: Florian Herzig (University of Toronto)

Introductory lectures  by Kiran Kedlaya in February (see abstracts above).

Lectures by Pierre Colmez ( Institut de Mathématiques de Jussieu), Wieslawa Niziol (Ecole Normale Supérieure de Lyon) and Jared Weinstein (Boston University) from May 15 to May 19

-May 15th: Jared Weinstein (Boston University)

-May 16th: Jared Weinstein (Boston University)

-May 17th: Jared Weinstein (Boston University)

-May 18th: Jared Weinstein (Boston University)

-May 15th: Pierre Colmez (Institut de Mathématiques de Jussieu)

-May 16th: Pierre Colmez (Institut de Mathématiques de Jussieu)

-May 17th: Wieslawa Niziol (Ecole Normale Supérieure de Lyon)

-May 18th: Wieslawa Niziol (Ecole Normale Supérieure de Lyon)

Final scientific report available here.