Schedule for: 19w2262 - Alberta Number Theory Days XI (ANTD XI)

Note: the Lecture rooms are available after 16:00.

Beverages and a small assortment of snacks are available in the lounge on a cash honour system.

A buffet breakfast is served daily between 7:00am and 9:00am in the Vistas Dining Room, the top floor of the Sally Borden Building. Note that BIRS does not pay for meals for 2-day workshops.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

There is a classical but not very well-known connection between counting objects that are in some sense `in the same class' but not isomorphic, and volume computations. I will start by recalling the analytic class number formula, and the Minkowski-Siegel mass formula for the "number" of quadratic forms in a genus, as well as Tamagawa's reformulation of these results as a volume computation. Then I will discuss a similar formula for the number of elliptic curves in an isogeny class, and we will see that it can again appear in two versions: one is due to Gekeler (2003) and comes from probabilistic and equidistribution considerations, and the other is due to Langlands and Kottwitz and is based on a volume computation. Finally, I will talk about our recent generalization of Gekeler's result to counting principally polarized Abelian varieties, by `reverse engineering' the Langlands-Kottwitz formula.

Solving norm equations is a classical problem that has a long history in number theory. While there is a considerable body of research dedicated to them over number fields, norm equations over function fields are significantly less studied. The main algorithm for solving norm equations over function fields, due to Gaal and Pohst, requires representing units and performing exponentiation on them. The standard representation of units of function fields is typically exponential in the size of the field, so powering them requires a very large amount of space and time. In order to practically solve norm equations over function fields, it is favourable to have a shorter representation for units. In this talk, we describe how using a notion referred to as a compact representation can speed up the Gaal-Pohst algorithm and significantly reduce the amount of required memory.

Erdős once asked about the function M(n) which counts the number of distinct products in an nxn multiplication table. We review the current known asymptotic behavior of this function and present computational results evaluating M(n) for n < 2^30. We describe the algorithms used and give a proof of their run-times and space constraints.

Recently Lipnowski and Tsimerman defined a 'Cohen-Lenstra' type measure on $R$-modules for certain rings $R$, and by extending methods of Ellenberg-Venkatesh-Westerland proved that moments of the Picard group of hyperelliptic curves (viewed as a module over the Frobenius) 'almost' converge to the moments of this measure. Under certain conditions on the ring $R$ knowledge of these moments can determine a distribution. We study when such conditions occur, as well as other questions related to the above measure.

This talk will begin with a survey of several results in (birational) anabelian geometry where the cohomology ring of the given field (broadly interpreted) is the main player. If time permits, we will discuss the state of the art in the subject, as well as potential future applications in arithmetic/geometric geometry.

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.

Group photo: 15:10 - 15:20 (Location TBD)

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.

For a number field $K$, Ihara has introduced an invariant $\gamma_K$, called the Euler-Kronecker constant, which is closely related to the values of the logarithmic derivative of $L$-functions at $1$. More precisely, $$\gamma_K=\lim_{s\rightarrow 1^+} \left( \frac{\zeta_K^\prime(s)}{\zeta_K(s)}+ \frac{1}{s-1} \right),$$ where $\zeta_K(s)$ is the Dedekind zeta function of $K$. In past few years, the size and the sign of these constants have extensively been investigated for certain families of number fields. In this talk we outline Ihara's approach in a systematic study of these constants and as a sample result we describe our joint work with Alia Hamieh (UNBC) on the existence of a distribution function for the Euler-Kronecker constants of certain cubic extensions of $\mathbb{Q}(\sqrt{-3})$. Also as another example we describe the role these constants played in our recent joint work with Forrest Francis (UNSW Canberra) regarding the inequalities involving Euler's function that are equivalent to GRH.

In this talk, we survey some recent results on the distribution of values of various families of $L$-functions in the critical strip. We also describe a value-distribution theorem for the logarithms and logarithmic derivatives of a family of $L$-functions attached to cubic Hecke characters. As a corollary we establish the existence of an asymptotic distribution function for the error term of the Brauer-Siegel asymptotic formula for a certain family of cubic extensions of $\mathbb{Q}(\sqrt{-3})$. We also deduce a similar result for the Euler-Kronecker constants of this family. This is joint work with Amir Akbary.

2-day workshop participants are welcome to use BIRS facilities (Corbett Hall Lounge, TCPL, Reading Room) until 15:00 on Sunday, although participants are still required to checkout of the guest rooms by 12 noon. There is no coffee break service on Sunday afternoon, but self-serve coffee and tea are always available in the 2nd floor lounge, Corbett Hall.

Generating functions of various enumerative invariants in algebraic geometry often satisfy Virasoro constraints. It was recently discovered by Kontsevich and Soibelman that the recursive relations of the Eynard-Orantin topological recursion can be recast as a set of Virasoro constraints, which they called Airy structures. I will talk about our generalization of their notion to "higher Airy structures", and realize the Bouchard-Eynard topological recursion as a set of W-algebra constraints. I will also mention some other higher Airy structures that we can produce in other geometric contexts.

An important result of Bach and Sorenson gives upper bounds on the first prime in the Chebotarev Theorem. This result has important applications in primality testing and class group computations. The result itself, though dependent on the GRH, is not expected to be tight and any improvement would lead to corresponding improvements in the performance of algorithms based on it. In this talk I will discuss several aspects of my recent work on computationally investigating/verifying conjectural improvements to these results including the resulting discovery of a family of fields for which we can prove a strong lower bound on the first prime in Chebotarev.