# Number Theory

## Explicit results about primes in Chebotarev's density theorem

Let $L/K$ be a Galois extension of number fields with Galois group $G$, and let $C⊂G$ be a conjugacy class. Attached to each unramified prime ideal p in OK is the Artin symbol $\sigma p$, a conjugacy class in $G$. In 1922 Chebotarev established what is referred to his density theorem (CDT). It asserts that the number $\pi C(x)$ of such primes with $\sigma p=C$ and norm $Np≤x$ is asymptotically $\left|C\right|\left|G\right|\mathrm{Li} (x)$ as $x\rightarrow\infty$ where $\mathrm{Li} (x)$ is the usual logarithmic integral. As such, CDT is a generalisation of both the prime number theorem and Dirichlet's theorem on primes in arithmetic progressions. In light of Linnik's result on the least prime in an arithmetic progression, one may ask for a bound for the least prime ideal whose Artin symbol equals C. In 1977 Lagarias and Odlyzko proved explicit versions of CDT and in 1979 Lagarias, Montgomery and Odlyzko gave bounds for the least prime ideal in the CDT. Since 2012 several unconditional explicit results of these theorems have appeared with contributions by Zaman, Zaman and Thorner, Ahn and Kwon, and Winckler. I will present several recent results we have proven with Das, Ng, and Wong.

## Regular Representations of Groups

A natural way to understand groups visually is by examining objects on which the group has a natural permutation action. In fact, this is often the way we first show groups to undergraduate students: introducing the cyclic and dihedral groups as the groups of symmetries of polygons, logos, or designs. For example, the dihedral group $D_8$ of order 8 is the group of symmetries of a square. However, there are some challenges with this particular example of visualisation, as many people struggle to understand how reflections and rotations interact as symmetries of a square.

Every group G admits a natural permutation action on the set of elements of $G$ (in fact, two): acting by right- (or left-) multiplication. (The action by right-multiplication is given by $\left{t_g : g \in G\right}, where $t_g(h) = hg$ for every $h \in G$.) This action is called the "right- (or left-) regular representation" of $G$. It is straightforward to observe that this action is regular (that is, for any two elements of the underlying set, there is precisely one group element that maps one to the other). If it is possible to find an object that can be labelled with the elements of $G$ in such a way that the symmetries of the object are precisely the right-regular representation of $G$, then we call this object a "regular representation" of $G$.

A Cayley (di)graph $Cay(G,S)$ on the group $G$ (with connection set $S$, a subset of $G$) is defined to have the set $G$ as its vertices, with an arc from $g$ to $sg$ for every $s$ in $S$. It is straightforward to see that the right-regular representation of $G$ is a subset of the automorphism group of this (di)graph. However, it is often not at all obvious whether or not $Cay(G,S)$ admits additional automorphisms. For example, $Cay(Z_4, {1,3})$ is a square, and therefore has $D_8$ rather than $Z_4$ as its full automorphism group, so is not a regular representation of $Z_4$. Nonetheless, since a regular representation that is a (di)graph must always be a Cayley (di)graph, we study these to determine when regular representations of groups are possible.

I will present results about which groups admit graphs, digraphs, and oriented graphs as regular representations, and how common it is for an arbitrary Cayley digraph to be a regular representation.

## Class Numbers of Certain Quadratic Fields

Class number of a number field is one of the fundamental and mysterious objects in algebraic number theory and related topics. I will discuss the class numbers of some quadratic fields. More precisely, I will discuss some results concerning the divisibility of the class numbers of certain families of real (respectively, imaginary) quadratic fields in both qualitative and quantitative aspects. I will also look at the 3-rank of the ideal class groups of certain imaginary quadratic fields. The talk will be based on some recent works done along with my collaborators.

## Some specialization problems in Geometry and Number Theory

We shall survey over the general issue of `specializations which preserve a property', for a parametrized family of algebraic varieties. We shall limit ourselves to a few examples. We shall start by recalling typical contexts like Bertini and Hilbert Irreducibility theorems, illustrating some new result. Then we shall jump to much more recent instances, related to algebraic families of abelian varieties.

** Please note, this video was recorded using an older in room system and has substantially diminished video quality.**

## Polya’s Program for the Riemann Hypothesis and Related Problems

In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has only been proved for degrees d=1, 2, 3. We prove the hyperbolicity of 100% of the Jensen polynomials of every degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.

## Multivariate (phi, Gamma)-modules

The classical theory of (phi, Gamma)-modules relates continuous p-adic representations of the Galois group of a p-adic field with modules over a certain mildly noncommutative ring. That ring admits a description in terms of a group algebra over Z_p which is crucial for Colmez's p-adic local Langlands correspondence for GL_2(Q_p). We describe a method for applying a key property of perfectoid spaces, the analytic analogue of Drinfeld's lemma, to the construction of "multivariate (phi, Gamma)-modules" corresponding to p-adic Galois representations in more exotic ways. Based on joint work with Annie Carter and Gergely Zabradi.

## On the local Langlands conjectures

### Abstract

The Langlands program, initiated in the 1960s, is a set of conjectures predicting a unification of number theory and the representation theory of groups. More precisely, the Langlands correspondence provides a way to interpret results in number theory in terms of group theory, and vice versa.

In this talk we sketch a few aspects of the local Langlands correspondence using elementary examples. We then comment on some questions raised by the emerging "mod p" Langlands program.

### Biography

Professor Ollivier works in the Langlands Programme, a central theme in pure mathematics which predicts deep connections between number theory and representation theory. She has made profound contributions in the new branches of the "p-adic" and "mod-p" Langlands correspondence that emerged from Fontaine's work on studying the p-adic Galois representation, and is one of the pioneers shaping this new field. The first results on the mod-p Langlands correspondence were limited to the group GL2(Qp); but Dr. Ollivier has proved that this is the only group for which this holds, a surprising result which has motivated much subsequent research.

She has also made important and technically challenging contributions in the area of representation theory of p-adic groups, in particular, in the study of the Iwahori-Hecke algebra. In joint work with P. Schneider, Professor Ollivier used methods of Bruhat-Tits theory to make substantial progress in understanding these algebras. She has obtained deep results of algebraic nature, recently defining a new invariant that may shed light on the special properties of the group GL2(Qp).

Rachel Ollivier received her PhD from University Paris Diderot (Paris 7), and then held a research position at ENS Paris. She subsequently was an assistant professor at the University of Versailles and then Columbia University, before joining the UBC Department of Mathematics in 2013.

Rachel is the recepient of the 2015 UBC Mathematics and Pacific Institute for the Mathematical Sciences Faculty Award.

More information on this event is available on the event webpage

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## Lifts of Hilbert modular forms and application to modularity of Abelian varieties

The Langlands program predicts that for every n-dimensional Abelian variety over Q there is an automorphic representation of GSpin(2n+1) over Q whose L-function coincides with the L-function coming from the Galois representation on the Tate module of the Abelian variety. Recently, Gross has refined this prediction by identifying specific properties that one should find in a vector in the automorphic representation. In joint work with Lassina Dembele, we have found some examples of automorphic representations of GSpin(2n+1) over Q whose L-functions match those coming from certain n-dimensional Abelian varieties over Q, all built from certain Hilbert modular forms. We are in the process of checking if these examples contain vectors with the properties predicted by Gross. In this talk I will explain the lifting procedure we are using to manufacture GSpin automorphic representations and describe the examples we are focusing on as we hunt for the predicted vectors in the representation space.

## Abelian Varieties Multi-Site Seminar Series: Drew Sutherland

Let A be an abelian variety of dimension g over a number field K. The Sato-Tate group ST(A) is a compact subgroup of the unitary symplectic group USp(2g) that can be defined in terms of the l-adic Galois representation associated to A. Under the generalized Sato-Tate conjecture, the Haar measure of ST(A) governs the distribution of various arithmetic statistics associated to A, including the distribution of normalized Frobenius traces at primes of good reduction. The Sato-Tate groups that can and do arise for g=1 and g=2 have been completely determined, but the case g=3 remains open. I will give a brief overview of the classification for g=2 and then discuss the current state of progress for g=3.

## OM representation of prime ideals and applications in function fields

Let $A$ be a Dedekind domain, $K$ the fraction field of $A$, and $f\in A[x]$ a monic irreducible separable polynomial. Denote by $\theta\in K^{\mathrm{sep}}$ a root of $f$ and let $F=K(\theta)$ be the finite separable extension of $K$ generated by $\theta$. We consider $\mathcal{O}$ the integral closure of $A$ in $L$. For a given non-zero prime ideal $\mathfrak{p}$ of $A$ the Montes algorithm determines a parametrization (OM representation) for every prime ideal $\mathfrak{P}$ of $\mathcal{O}$ lying over $\mathfrak{p}$. For a field $k$ and $f\in k[t,x]$ this yields a new representation of places of the function field $F/k$ determined by $f$. In this talk we summarize some applications which improve the arithmetic in the divisor class group of $F$ using this new representation.