NTS ABSTRACTFall2025

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Sep 11

The least prime in the Chebotarev density theorem

Robert Lemke Oliver (UW-Madison)

The Chebotarev density theorem is a powerful tool in number theory, in part because it guarantees the existence of primes whose Frobenius lies in a given conjugacy class in a fixed Galois extension of number fields.  However, for some applications, it is necessary to know not just that such primes exist, but to additionally know something about their size, say in terms of the degree and discriminant of the extension.  In this talk, I'll discuss forthcoming work with Cho and Zaman on the least prime with a specified Frobenius in a fixed Galois extension, with a particular focus on S_n extensions.  Our approach is comparatively elementary, but when combined with existing results based on the zeros of L-functions, it leads to the strongest known bounds in this setting.

Oct 16

Qiao He (Columbia)

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.