Number Theory Seminar

Let $A$ be a $g$-dimensional abelian variety defined over a number field $F$. It is conjectured that the set of ordinary primes of $A$ over $F$ has positive density, and this is known to be true when $g=1, 2$, or for certain abelian varieties with extra endomorphisms. In this talk, I will prove the positive density statement for new cases of abelian varieties, including certain modular abelian varieties. The proof is carried out in the general setting of compatible systems of Galois representations, and consequently also implies a positive density result for the set of ordinary primes of certain modular forms of weight $2$. This is joint work with Pengcheng Zhang.