# Department of Pure Mathematics and Mathematical Statistics

I will present two seemingly unrelated results: (1) There exists a smooth projective curve C of genus 4, defined over some number field K, whose Jacobian J satisfies: End(J)=\Z and the image of the absolute Galois group of K in End(T_\ell (J)) is not open in GSp_8(Z_\ell). (2)The Hodge locus of positive period dimension of the universal family of degree d hypersurfaces in \mathbb{P}^{n+1} is algebraic as soon as n>2 and d>5.

The proof of both results builds on an ‘’Extended Zilber—Pink philosophy’’ for varieties supporting a variation of Hodge structure which predicts the behaviour of typical and atypical intersections (whose combination describes the whole Hodge locus). This is a joint work with B. Klingler and E. Ullmo.

19Oct
Oct 19th 2021
14:30 to 15:30

Online

## Speaker:

Gregorio Baldi (IHES)

## Series:

Number Theory Seminar