NTS ABSTRACTFall2025

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Sep 11

Enumerating Galois extensions of number fields

Robert Lemke Oliver (UW-Madison)

We provide an asymptotic formula for the number of Galois extensions of a number field with absolute discriminant bounded by some X.  The key behind this result is a new upper bound on the number of Galois extensions with a given Galois group G of order at least 5.  In particular, we give the first general bound with an exponent that decays with the order of G.  This improves over the previous best bound due to Ellenberg and Venkatesh.

Sep 18

Isogenies between reductions of elliptic curves with complex multiplication

Jiacheng Xia (UW Madison)

Given two elliptic curves over a number field, Charles proved that there are infinitely many primes where the reductions of these two curves are geometrically isogenous. We study a refined problem for a given pair of CM elliptic curves $E_1$ and $E_2$: for a positive integer $m$, how many primes are there where the reductions of $E_1$ and $E_2$ are related by a cyclic isogeny of degree $m$? We establish a polynomial lower bound in $m$ for such a counting problem. Gross-Zagier type theorems and higher Green functions connect our counting problem to the Fourier coefficients of certain incoherent Eisenstein series, which can in turn be approximated by those of certain elliptic cusp forms which are not necessarily eigenforms. One further step is to establish an explicit Deligne bound for a general cusp form of arbitrary weight and level, which might be novel and of independent interest. This is a joint work with Edgar Assing, Yingkun Li, and Tian Wang.

Sep 25

Effective Brauer-Siegel theorems for Artin L-functions

Asif Zaman (UToronto)

Given a number field K other than Q, in a now classic work, Stark pinpointed the possible source of a so-called Landau--Siegel zero of the Dedekind zeta function of K and used this to give effective upper and lower bounds on the residue of the Dedekind zeta function at s=1.               I will discuss an extension of Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin L-functions at s=1 that are, up to the value of implied constants, as strong as could reasonably be expected given current progress toward the generalized Riemann hypothesis.  The bounds are completely unconditional, and rely on no unproven hypotheses about Artin L-functions. This is joint work with Peter Cho and Robert Lemke Oliver.

Oct 2

Some cases of the Serre weight conjecture over CM fields

Daniel Le (Purdue)

A generalization of Serre's conjecture predicts that every odd irreducible mod p Galois representation comes from the cohomology of some mod p local system on an appropriate locally symmetric space. The weight part of the conjecture (not yet formulated in general) predicts the set of local systems for which the Galois representation arises in terms of the restriction of the Galois representation to the decomposition group at p. We establish the weight part of Serre's conjecture for automorphic mod p Galois representations over a CM field when the local Galois representation is semisimple and generic under a number of technical hypotheses. Our arguments incorporate some recent developments in modularity lifting and the Emerton--Gee stack. This is joint work with Bao Le Hung.

Oct 23

Intersection of Hecke correspondences and a general conjecture

Qiao He (Columbia)

A classical and beautiful result of Gross-Keating relates the intersection of three Hecke correspondences on the integral model of $X_0(1)\times X_0(1)$ with derivative of certain Eisenstein series. Such results can be regarded as an example of arithmetic Siegel-Weil formula and can serve as an ingredient for an arithmetic Gan-Gross-Prasad formula. In this talk, we consider a variant of this formula for the self-product of $X_0(N)$ where $N$ is square-free and  the self-product of Shimura curves respectively. Moreover, we explain how these results confirm a general conjecture for GSpin Shimura varieties with vertex levels. This talk is based on joint works with Baiqing Zhu.

Oct 30

Beilinson-Bloch-Kato conjecture for polarized motives

Hao Peng (MIT)

The Beilinson—Bloch—Kato conjecture is a far-fetching generalization of the (rank part of the) BSD conjecture for modular elliptic curves. The conjecture is partially proved for U(N)*U(N+1)-motives in the work of Y. Liu, Y. Tian, L. Xiao, W. Zhang, and X. Zhu. Using theta correspondence, we prove that their result implies the BBK conjecture for U(2n)-motives, e.g. odd symmetric powers of non-CM modular elliptic curves, in the rank zero case. Similar trick works in the orthogonal case. If time permits, we talk about the work in progress partiallu proving the BBK conjecture for O(N)*O(N+1)-motives when analytic rank is at most one.