Abstract: Given an abelian variety A defined over a number field, a conjecture attributed to Serre states that the set of primes at which A admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fite 2021, etc).
In this talk, we will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including some cases of A with almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfan, Mantovan, Pries, and Tang.
Apart from ordinary reduction, we will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves. This is joint work with Mantovan, Pries, and Tang.
Number Theory Web Seminar: [ Ссылка ]
Original air date:
Thursday, February 16, 2023 (8am PST, 11am EST, 4pm GMT, 5pm CET, 6pm Israel Standard Time, 9:30pm Indian Standard Time)
Friday, February 17, 2023 (12am CST, 3am AEDT, 5am NZDT)
