A venue for invited and local speakers to present their research on topics surrounding algebraic geometry and number theory, broadly conceived.

The seminar meets on Wednesdays, 16:00-17:15, in -101

## 2021–22–A meetings

### Upcoming Meetings

Date
Title
Speaker
Abstract
Dec 1 TBA Sa'ar Zehavi (TAU)
Dec 8 TBA David Ter-Borch Gram Lilienfeldt (HUJI)
Dec 15 TBA Dmitry Kerner (BGU)
Dec 22 TBA Ido Efrat (BGU)
Mar 2 TBA

### Past Meetings

Date
Title
Speaker
Abstract
Oct 27 TBA none
Nov 3 Prime torsion in the Tate-Shafarevich groups of abelian varieties over $\mathbb{Q}$ Ariel Weiss (BGU)

Very little is known about the Tate-Shafarevich groups of abelian varieties. On the one hand, the BSD conjecture predicts that they are finite. On the other hand, heuristics suggest that, for each prime $p$, a positive proportion of elliptic curves $E/\mathbb{Q}$ have $\Sha(E)[p] \ne 0$, and one expects something similar for higher dimensional abelian varieties as well. Yet, despite these expectations, it seems to be an open question whether, for each prime $p$, there exists even a single elliptic curve over $\mathbb{Q}$ with $\Sha(E)[p] \ne 0$. In this talk, I will show that, for each prime $p$, there exists a geometrically simple abelian variety $A/\mathbb{Q}$ with $\Sha(A)[p]\ne 0$. Our examples arise from modular forms with Eisenstein congruences. This is joint work with Ari Shnidman.

Nov 10 Rational points on ramified covers of abelian varieties, online lecture Ariyan Javanpeykar (Meinz)

Let X be a ramified cover of an abelian variety A over a number field k. According to Lang’s conjecture, the k-rational points of X should not be dense. In joint work with Corvaja, Demeio, Lombardo, and Zannier, we prove a slightly weaker statement. Namely, assuming A(k) is dense, we show that the complement of the image of X(k) in A(k) is (still) dense, i.e., there are less points on X than there are on A (or: there are more points on A than on X). In this talk I will explain how our proof relies on interpreting this as a special case of a version of Hilbert’s irreducibility theorem for abelian varieties.

Nov 17 TBA No talk

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Nov 24 Quadratic Chabauty and Beyond David Corwin (BGU)

I will describe my work (some joint with I. Dan-Cohen) to extend the computational boundary of Kim’s non-abelian Chabauty’s method. Faltings’ Theorem says that the number of rational points on curves of higher genus is finite, and non-abelian Chabauty provides a blueprint both for proving this finiteness and for computing the sets. We first review classical Chabauty-Coleman, which does the same but works only for certain curves. Then we describe Kim’s non-abelian generalization, which replaces abelian varieties in Chabauty-Coleman by Selmer groups (a kind of Galois cohomology) and eventually “non-abelian” Selmer varieties. Finally, we describe recent work in attempting to compute these sets using the theory of Tannakian categories.

Seminar run by Dr. Ishai Dan-Cohen