Feb 28 (Fri), 2025, 16:00--17:00, Room 1423

Title: Schubert cells and Whittaker functionals for $\mathrm{GL}(n,\mathbb{R})$: Existence via integration by parts

Speaker: Doyon Kim (Mathematical Institute of the University of Bonn)

Abstract: The “multiplicity-one theorem” proved by Piatetski-Shapiro and Shalika states that the space of Whittaker functionals on an irreducible representation of GL(n, R) is at most one-dimensional. In this talk, I will discuss a new proof of the existence of Whittaker functionals on the principal series representations of GL(n, R) which uses the analytic theory of distributions. This reduces the analysis of Whittaker functionals to integration by parts and gives an explicit proof of the analytic continuation of Jacquet integrals.

Feb 04 (Tue), 2025, 11:00--12:00, Room 1424

Title: Algebraic points on modular curves I: Linear bounds on prime order torsion

Speaker: Maarten Derickx (University of Zagreb)

Abstract: Let S(d) denote the set of primes p such that there exists an elliptic curve over a number field of degree d with a point of order p. Oesterlé showed that the largest prime in S(d) is smaller than (3^{d/2}+1)^2, giving an exponential upper bound. However, explicit calculations for small values of d have shown that this bound is far from sharp. In this talk we will present some conjectures on the Mordell-Weil ranks of modular jacobians that imply a linear bound on the largest prime in S(d). We provide numerical evidence for these conjectures by verifying them for p < 100,000. This talk is based on work in progress together with Michael Stoll.

Feb 04 (Tue), 2025, 14:00--15:00, Room 1424

Title: Possible degrees of morphisms to elliptic curves

Speaker: Petar Orlic (University of Zagreb)

Abstract: The problem of finding infinitely many degree d points on curves is related to determining all possible degrees of rational morphisms to elliptic curves. In this talk, we will present a method to determine all such possible degrees, mainly concentrating on the modular curve X_0(N).

Feb 04 (Tue), 2025, 15:30--16:30, Room 1424

Title: Algebraic points on modular curves II: Classification of torsion over quartic fields

Speaker: Maarten Derickx (University of Zagreb)

Abstract: In a recent preprent [1] Filip Najman and I classified the groups that can occur as the torsion subgroup of an elliptic curve over a quartic number field. This work builds on a long series of papers starting by Mazurs famous result on the classification of torsion subgroups of elliptic curves over Q. It is also the next step after the classification over cubic number fields in [2] by Derickx et. al., which was obtained by developing powerful computational techniques. There are several important improvements with regard to the classification over cubic fields that made this result possible. One of these is based on the conjectures with Michael Stoll in the first talk, by verifying these conjectures explicitly one can drastically decrease the number of torsion groups that need to be considered. This approach requires much less computation than what was used in [2] and hopefully will make the classification of torsion over quitic fields within reach. [1] https://arxiv.org/abs/2412.16016 [2] https://arxiv.org/abs/2007.13929