University of Alberta Number Theory/Representation Theory Seminar
Winter/Spring 2019

In the 1970s, Serre formulated his remarkable conjecture that every two-dimensional mod-$p$ Galois representation of the absolute Galois group of $\mathbb{Q}$, which is odd and irreducible, should come from a modular form. He later refined his conjecture, giving a precise recipe for the weight and level of the modular form. Both the "weak form" and "strong form" of Serre's conjecture are now theorems, due to the work of many mathematicians (Khare-Wintenberger, Kisin, Edixhoven, Ribet, and others). In this talk, we will discuss how to generalize Serre's weight recipe when the Galois representation is replaced by a homomorphism from an absolute Galois group to the Langlands dual of a rank 2 unitary group. This is joint work with Stefano Morra.

The main questions in arithmetic dynamics are motivated by analogous classical problems in arithmetic geometry, especially the theory of elliptic curves. We study one such question, which is an analogue of Serre's open image theorem regarding $\ell$-adic Galois representations arising from elliptic curves. We consider the action of the absolute Galois group of a field on pre-images of a point $\alpha$ under iterates of a rational map $f$ (points that eventually map to $\alpha$ as we apply $f$ repeatedly). These points can be given the structure of a rooted tree in a natural way. This determines a homomorphism from the absolute Galois group of the field to the automorphism group of this tree, called an arboreal Galois representation. As in Serre's open image theorem, we expect the image of this representation to have finite index in the automorphism group except in certain cases.

Let $K$ be a field of characteristic $0$ and let $m\geq 1$. A power series $f(x_1,\ldots,x_m)\in K[[x_1,\ldots,x_m]]$ is said to be D-finite if all the partial derivatives of $f$ span a finite dimensional vector space over the field $K(x_1,\ldots,x_m)$. Examples in the univariate case include rational functions, algebraic functions such as $\sqrt{1+x}$, the exponential function, and $\log(1+x)$. The logarithmic Weil height is a function $h:\ \bar{\mathbb{Q}}\rightarrow \mathbb{R}^{+}$ that plays an important role in diophantine geometry. Motivated by earlier work of van der Poorten-Shparlinski and Bell-Chen, we study problems related to the growth of the height of coefficients of D-finite power series in $\bar{\mathbb{Q}}[[x_1,\ldots,x_m]]$. This is joint work in progress with Jason Bell and Umberto Zannier.

This talk uses two examples to show how Arthur parameters determine a category of equivariant perverse sheaves and how the microlocal perspective on this category reveals the Arthur packet attached to the Arthur parameter. The full picture is explained in Arthur packets for p-adic groups by way of microlocal vanishing cycles, with examples, joint with Andrew Fiori, Ahmed Moussaoui, James Mracek and Bin Xu, to appear in Memoirs of the American Mathematical Society. The talk includes some remarks on an emerging categorical local Langlands correspondence.

The essential dimension of a finite group over a field $K$, defined by Buhler and Reichstein, is the smallest dimension of a linearizable $G$-variety over $K$ with a faithful $G$-action. By a result of Karpenko and Merkurjev, if $G$ is a finite $p$-group and $K$ contains a primitive $p$-th root of unity, then this number is equal to the smallest dimension of a faithful $K$-representation of $G$. Using the orbit method we prove qualitative and quantitative results on the essential dimensions of finite $p$-groups. This talk is based on a joint project with M. Bardestani and K. M. Karai.

In the 1980's the idea emerged that it could be useful to tie together a family of related $L$-functions to create a double, or multiple, Dirichlet series, which could be used to study the average behavior of the original family of $L$-functions. The local structure of these multiple Dirichlet series shows a rich connection to the theory of automorphic forms (i.e., Whittaker functions on $p$-adic groups and their covers are the fundamental objects), representation theory, and as noticed recently, to algebraic geometry.

In this talk, I will focus on the most important case, namely the multiple Dirichlet series associated to moments of $L$-functions. I will discuss the connection between the local parts of these series and the compactifications of certain moduli spaces of curves, and how this information can be combined with the (conjectural in general) analytic continuation of the multiple Dirichlet series to obtain precise asymptotics for moments, for example, of the classical family of quadratic Dirichlet $L$-functions.

Recent work has shown that to each irreducible smooth complex representation $\pi$ of a $p$-adic group $G$ we may associate a type, which is a pair $(K,\rho)$ (for $K$ a compact open subgroup and $\rho$ an irreducible representation $K$) from which one may recover the (Bernstein component of) $\pi$. These pairs are not unique, even up to conjugacy. For the case of supercuspidal representations, the conjecture of "unicity of types" originally proposed that the restriction of $\pi$ to a compact open subgroup could contain at most one type.

Using tools from Bruhat-Tits theory, we show this version of unicity fails, yet deduce from our methods a strategy for proving a refined version of the conjecture. This is ongoing joint work with Peter Latham of King's College, London.

A result of Bernstein from the 80s gives an equivalence between modules over the Iwahori-Hecke algebra and the category of smooth $G$-representations generated by their Iwahori-fixed vectors. Here $G$ is a $p$-adic Lie group and the coefficient field is $\Bbb{C}$. This equivalence usually fails if we take a coefficient field $k$ of characteristic $p$, as in the mod $p$ local Langlands correspondence. Schneider found a way to remedy the situation by instead relating the derived category $D(G)$ to modules over a certain differential graded variant of the (pro-$p$) Iwahori- Hecke algebra. The goal of the talk is to interpret Schneider's equivalence in the case $G=I$ as a version of Koszul duality for the Iwasawa $k$-algebra $\Omega(G)$. This makes use of Lazard's theory of $p$-valuations, which played a key role in his proof that $\Omega(G)$ is Noetherian. The rough idea is that the filtration on $\Omega(G)$ somehow corresponds to an $A_{\infty}$-structure on the Yoneda extension algebra. When $G$ is abelian the $A_{\infty}$-structure is trivial (the converse was recently shown by Carl Wang-Erickson); this generalizes a result of Schneider for $\Bbb{Z}_p$.