Next Seminar:  January 15, 2026 at TU Graz

Time:  January 15, 13:00 - 18:00

Location: Kopernikusgasse 24, 3rd floor, HS F (Room NT03064)

Schedule: 

13:30-14:30: Dante Bonolis

Title: On the 2-torsion in class groups of number fields 

Abstract: In 2020, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao proved that for a finite extension K/Q of degree n ≥ 5, the size of the 2-torsion class group is bounded by # h2(K)=On,ɛ(DK(1/2-1/(2n)+ɛ)), where DK is the absolute discriminant of K. We improve their bound by proving that # h2(K)=On,ɛ(DK(1/2-1/(2n)-δK+ɛ)),  for a constant δK≥ 1/(28n)-3/(28n(n-1)).

14:30-15:00: Tea break

15:00-16:00: Lukas Spiegelhofer

Title: The valuation of binomial coefficients via cumulants

Abstract: Cumulants --- coefficients of the exponential generating function given by the logarithm of a characteristic function --- provide a powerful method for studying the distribution of 2-valuations of binomial coefficients. The number of elements in a given row n of Pascal's triangle having a given valuation j is uniformly close to a Gaussian [Spiegelhofer, Wallner, 2018]. Here, ``closeness'' depends on the number of blocks of 1s in the binary expansion of n. Using cumulants, we significantly refine this result, introducing a polynomial perturbation of the normal distribution, of arbitrary degree d. In the spirit of an asymptotic expansion, larger d eventually lead to finer approximations, as the number of blocks of 1s in the binary expansion of the row number grows.

16:00-16:15: Break

16:15-17:15:  Noy Soffer Aranov

Title: Escape of Mass of Sequences 

Abstract: One way to study the distribution of nested quadratic number fields satisfying fixed arithmetic relationships is through the evolution of continued fraction expansions. In the function field setting, it was shown by de Mathan and Teullie that given a quadratic irrational Θ, the degrees of the periodic part of the continued fraction of tn Θ are unbounded. Paulin and Shapira improved this by proving that quadratic irrationals exhibit partial escape of mass. Moreover, they conjectured that they must exhibit full escape of mass. We construct counterexamples to their conjecture in every characteristic. In this talk we shall discuss the technique of proof as well as the connection between escape of mass in continued fractions, Hecke trees, and number walls. This is part of joint works with Erez Nesharim and Uri Shapira and with Steven Robertson and is based on the recent preprints https://arxiv.org/abs/2503.18749, https://arxiv.org/abs/2510.19449, https://arxiv.org/abs/2510.19417.

18:00: Dinner