AGNT Ical Atom

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.

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.

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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.

We give two new examples of non-hyperelliptic curves whose Ceresa cycles have torsion images in the intermediate Jacobian. For one of them, we find that the central value of the L-function of the relevant motive is non-vanishing, 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.

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 ‘Lie-type pair’. This is a group G, its would-be 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.)

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.

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 p-adic L-functions; (2) if nontrivial, they generate the Selmer group of the motive. This supports analogues of the Birch and Swinnerton-Dyer conjecture. I will focus on (2), whose proof combines the method of Euler systems and the local theta correspondence in representation theory.