Oct 27

TBA

none


Nov 3

Prime torsion in the TateShafarevich groups of abelian varieties over $\mathbb{Q}$

Ariel Weiss (BGU)

Very little is known about the TateShafarevich 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 krational 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


Nov 24

Quadratic Chabauty and Beyond

David Corwin (BGU)

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

Dec 1

TBA

Sa'ar Zehavi (TAU)


Dec 8

Experiments with Ceresa classes of cyclic Fermat quotients

David TerBorch Gram Lilienfeldt (HUJI)

We give two new examples of nonhyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, we find that the central value of the Lfunction of the relevant motive is nonvanishing, consistent with the conjectures of Beilinson and Bloch. We speculate on a possible explanation for the existence of these torsion Ceresa classes, based on some computations with cyclic Fermat quotients. This is joint work with Ari Shnidman.

Dec 15

Finite determinacy of maps. Group orbits vs the tangent spaces

Dmitry Kerner (BGU)

Consider a morphism of germs of Noetherian schemes, f: (X,x)> (Y,y). When is it ‘stable’ under perturbations by higher order terms? I.e. when can such a perturbation be undone by a group action, e.g. by the local coordinate changes.
This question has been extensively studied for real/complex analytic (or C^k) maps
(k^n,o)> (k^m,o). The idea is to reduce the orbit study, Gf, to the study of the tangent space, T_G f.
The classical methods used vector field integration and infinite dimensional Lie groups, thus obstructing extensions to the zero/positive characteristic. During the last years we have developed a purely algebraic approach to this problem, extending the results to arbitrary characteristic.
The key tool is the ‘Lietype pair’. This is a group G, its wouldbe tangent space T_G, and certain maps between G, T_G, approximating the classical exponential/logarithm.
(joint work with G. Belitskii, A.F. Boix, G.M. Greuel.)

Dec 22

Filtrations of profinite groups as intersections and absolute Galois groups

Ido Efrat (BGU)

The general structure of absolute Galois groups of fields as profinite groups is still a mystery. Among the very few
known properties of such groups are several “Intersection Theorems”, describing subgroups in standard filtrations
of absolute Galois groups as the intersection of all normal open subgroups with quotient in a prescribed list of
finite groups. These theorems are based on deep cohomological properties of absolute Galois groups. We will
present a general “Transfer Theorem” for profinite groups, which explains what lies behind these intersection
theorems.

Dec 29

TBA

Amit Ophir (HUJI)


Jan 5

Theta cycles

Daniel Disegni (BGU)

I will discuss results and open problems in an emerging theory of ‘canonical’ algebraic cycles for all motives enjoying a certain symmetry. The construction is inspired by theta series, and based on special subvarieties in arithmetic quotients of the complex unit ball.
The ‘theta cycles’ seem as pleasing as Heegner points on elliptic curves: (1) their nontriviality is detected by derivatives of complex or padic Lfunctions; (2) if nontrivial, they generate the Selmer group of the motive. This supports analogues of the Birch and SwinnertonDyer conjecture. I will focus on (2), whose proof combines the method of Euler systems and the local theta correspondence in representation theory.

Mar 2

TBA


