Number Theory Tea 2021

Currently, we are meeting on Mondays 1PM Princeton time at https://princeton.zoom.us/j/92542250798

Upcoming talks

3/22: Gyujin Oh, *Modern approaches to Inverse Galois theory IV*

We will review the two notions of rigidity, one for local systems (``Galois representations'') and one for automorphic representations. We aim to explain why the two notions are analogous; along the way, we will talk about wild ramification/irregular singularities, Swan conductors, explicit local Langlands correspondence, etc.

3/29: Gyujin Oh, *Modern approaches to Inverse Galois theory V*

We will apply all the theories we've learned so far to actually construct special kinds of rigid local systems, those appearing in the works of Frenkel-Gross, Heinloth-Ngo-Yun, Yun, etc.

4/5: TBD, *TBD*

TBD

4/12: TBD, *TBD*

TBD

4/19: TBD, *TBD*

TBD

4/26: TBD, *TBD*

TBD

Past talks

1/4: Victor Wang, *Statistics of random hypersurfaces mod p*

The (resolution of the) Weil conjectures imply in particular that certain natural classes of varieties, such as smooth projective hypersurfaces over a finite field, satisfy a certain "square-root cancellation point count heuristic". The same heuristic fails for *some* singular hypersurfaces, but still seems to typically hold. The full truth seems fairly unexplored and far from known in general. I will discuss concrete examples suggesting that true "likelihood of failure" of the heuristic should be rather unlikely, i.e. have "codimension" significantly larger than 1 in general (i.e. "has a singularity" is far from the true "criterion for failure"). The main parts of the talk will be elementary.

1/11: Gyujin Oh, *Modern approaches to Inverse Galois theory I*

Inverse Galois problem asks whether every finite group can be realized as a Galois group over Q. The problem can be cast in more geometric terms, and we will start by first reviewing the basic theory. The new geometric problem lies in the realm of arithmetic geometry, where one can make use of tools used in the Langlands program. More surprisingly, the notion of rigidity connects the problem to geometric Langlands program. Over the next few talks we aim to present this connection.

1/18: Leo Lai, *Rohrlich's non-vanishing theorem*

It is often important to know that the central $L$-value of some (or many) twists of a modular form is non-zero. We will sketch the proof of one such result due to Rohrlich, using purely analytic methods.

2/8: Gyujin Oh, *Modern approaches to Inverse Galois theory II*

We saw at the end of the first talk that rigid local systems are related to the rigidity method in the inverse Galois problem. Katz on the other hand found a miraculous way of constructing basically all rigid local systems on P^1 minus several points via simple operations, including the crucial "middle convolution functor". We will review this theory, and discuss how it can be used for the inverse Galois problem as well as its l-adic version.

2/22: Gyujin Oh, *Modern approaches to Inverse Galois theory III*

The inverse Galois problem, as we've observed, is intimately related to the problem of constructing a local system over P^1 minus a few points with prescribed behavior around the punctures. This can also be thought as finding an appropriate "Galois representation" in geometric Langlands. Under this viewpoint, we can define the notion of "rigid automorphic data" analogous to the notion of rigid local systems. Even though the general "automorphic-to-Galois" construction of V. Lafforgue is not explicit, the attachment of Galois representations can be made explicit for rigid automorphic forms, in a way that the construction works for any function field (in particular over Q(t)!). We will review this work (due to Z. Yun) and make comparison with Katz's theory of rigid local systems.

3/1: Marco Sangiovanni, *Asymptotic distribution of Hecke eigensystems*

The aim of this talk is to explain a result due to Serre that proves that, as the level and weight varies, the Hecke eigensystems associated to classical cusp forms are equidistributed with respect to some explicit continuous measure. The main tool of the proof is the Eichler-Selberg trace formula which computes in an explicit way the traces of Hecke operators. I will also explain some interesting applications of this result; for example, the fact that the Ramanujan conjecture provides, at least asymptotically, an optimal bound for Hecke eigenvalues, and the fact that algebraic extensions associated to eigenforms have asymptotically "big degree".

3/8: Katy Woo & Noah Kravitz, *Hunting for Arithmetic Progressions in the Primes*

We will outline a proof of the Green-Tao Theorem, which says that the primes contain arbitrarily long arithmetic progressions. The argument involves three key components: an arithmetic regularity lemma, Szemerediâ€™s theorem for pseudorandom measures, and a construction of a measure based on the distribution of the primes.

3/15: Giorgos Kotsovolis, *Counting integer points using Equidistribution of Homogeneous spaces*

We will be discussing the counting method of Duke-Rudnick-Sarnak and Eskin-McMullen, an application of homogeneous dynamics in number theory. We will use the equidistribution of certain homogeneous spaces to count "integer" points inside "sufficiently round" shapes. The starting point will be to consider the Gauss Circle Problem, where we will prove a power saving error term O(R^{1-/delta}). We will then motivate the generalization by considering a theorem of Selberg, counting "integer points" inside geodesic circles in some fundamental domain, for some lattice in PSL2(R). We will prove this as a corollary of a more general theorem.