Sato-Tate groups of abelian varieties

14-Oct-2022, 12:40-13:40 (18 months ago)

Abstract: Let A be an abelian variety of dimension g defined over a number field K. As defined by Serre, the Sato-Tate group ST(A) is a compact subgroup of the unitary symplectic group USp(2g) equipped with a map that sends each Frobenius element of the absolute Galois group of K at primes p of good reduction for A to a conjugacy class of ST(A) whose characteristic polynomial is determined by the zeta function of the reduction of A at p. Under a set of axioms proposed by Serre that are known to hold for g <= 3, up to conjugacy in Usp(2g) there is a finite list of possible Sato-Tate groups that can arise for abelian varieties of dimension g over number fields. Under the Sato-Tate conjecture (which is known for g=1 when K has degree 1 or 2), the asymptotic distribution of normalized Frobenius elements is controlled by the Haar measure of the Sato-Tate group.

In this talk I will present a complete classification of the Sato-Tate groups that can and do arise for g <= 3.

This is joint work with Francesc Fite and Kiran Kedlaya.

algebraic geometry

Audience: researchers in the discipline

Comments: This is a hybrid talk. To request Zoom link please write to sertoz@bilkent.edu.tr


ODTU-Bilkent Algebraic Geometry Seminars

Organizer: Ali Sinan Sertöz*
*contact for this listing

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