Determining monodromy groups of abelian varieties
Given an abelian variety over a number field one can associate to it its motivic Galois group, its Mumford-Tate group (defined by means of Hodge decomposition), its \(\ell\)-adic monodromy groups (where \(\ell\) is a rational prime), and its Sato-Tate group. The author assumes the Mumford-Tate conjecture and shows that from two well chosen Frobenius polynomials of the abelian variety (corresponding to two ``random'' primes), one can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. He also proposes a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular one obtains a probabilistic algorithm to compute the dimension of the Hodge classes of the abelian variety for any fixed degree.
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