Let K be a number field and let ρ_ℓ : G_K → GL(V_ℓ) be a strictly compatible family of ℓ-adic representations, according to Serre, associated with a pure, polarized, rational Hodge structure. In the lecture I will introduce Algebraic Sato-Tate and Sato-Tate conjectures in this general framework. I will explain how these conjectures are related to the motivic approach by Serre to generalize the classical Sato-Tate conjecture. Previously this work concerned abelian varieties and more generally, motives of odd weight in the Deligne's motivic category for absolute Hodge cycles. Now the results are extended to other motivic categories and to motives of arbitrary weight; the case of even weight introduces some parity considerations that do not appear for odd weight. This is joint work in progress with Kiran Kedlaya.
In my lecture, I will talk about results concerning linear relations in the Mordell-Weil group of a semi-abelian variety isogeneous to product of a torus and an abelian variety. I will show that to get these results one can use only finite number of reductions which amount to constructing Frobenius elements with special arithmetic properties in the ℓ-adic representation associated with the semi-abelian variety under investigation.
For an elliptic module of rank 2 and generic characteristic, with trivial endomorphism ring, we study the growth of the absolute discriminant of the endomorphism ring associated to its reduction modulo a prime. We prove that the absolute discriminant grows with the norm of the prime defining the reduction, and that, for a density one of primes, this growth is as close as possible to the natural upper bound. This is joint work with Mihran Papikian.
In this talk, we will focus on how one can deduce some geometric invariants of an abelian variety or a K3 surface by studying their Frobenius polynomials.
In the case of an abelian variety, we show how to obtain the decomposition of the endomorphism algebra, the corresponding dimensions, and centers.
Similarly, by studying the variation of the geometric Picard rank, we obtain a sufficient criterion for the existence of infinitely many rational curves on a K3 surface of even geometric Picard rank.
It is a conjecture often attributed to Serre that for any abelian variety defined over a number field there exists a nonzero density set of primes of ordinary reduction. For elliptic curves and abelian surfaces this has been known for a while and it is due to Katz, Ogus and Serre (recently Sawin has even determined the exact density of ordinary primes in the case of surfaces). I will discuss some current discoverings on the abundance of ordinary primes for certain types of abelian varieties of dimensions 3 and 4 which possess extra endomorphisms.
I will give a brief survey about torsion points on an abelian variety A over a number field K and their associated Galois representation before presenting some recent results and discuss future investigations. Topic involves naturally the uniform bound conjecture ("fixing K, varying A") and Mumford-Tate conjecture ("fixing A, varying K").
I report on ongoing joint work with Francesc Fite and Drew Sutherland on Sato-Tate groups of abelian threefolds. There are known to be 410 such groups, but it is not yet known how many groups occur for principally polarized abelian threefolds or for Jacobians; we report on progress towards answering these questions.
In the 1960's, Birch and Swinnerton-Dyer formulated several conjectures relating the rank r of the elliptic curve E to the order of the zero of the L-series attached to E at s=1. Their original conjecture connected the limiting behavior of the product over primes p<x of N_p/p, where N_p is the number of points of E (mod p) with the rank r of E. We will show that if the limit exists, then the value of the limit is as predicted by Birch and Swinnerton-Dyer. We will also make some remarks on how this is related to a conjecture of Nagao. This is a report on recent joint work with Seoyoung Kim.