How Do Generic Properties Spread?MPIM

by Yu Fu (Caltech)

Monday Nov 10, 2025, 2:00 PM → 3:00 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Extra talk

Given a family of algebraic varieties, a natural question to ask is which properties of the generic fiber extend to the other fibers, and in what way they extend. Let us explore this topic, from an arithmetic point of view, in the following setting:

let A→XA→X be a one-parameter family of principally polarized abelian varieties over a number field KK, whose generic fiber has big monodromy.

How often does the specialization AxAx​ fail to be simple?

Equivalently, how often does the associated  mod ℓmodℓ Galois representation drop into a proper subgroup of GSp⁡2g(Fℓ)GSp2g​(Fℓ​)?

Can we give a quantitative estimate for the number of specializations of height at most BB where the monodromy drops?

I will give an answer to this question. Conceptually, this provides a quantitative illustration of the Mumford-Tate conjecture: failures of genericity are sparse and occur only on a height -density zero set of rational points.