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.

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.