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(A)typical intersections and where to find them

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  • UserGregorio Baldi (IHES)
  • ClockTuesday 19 October 2021, 14:30-15:30
  • HouseOnline.

If you have a question about this talk, please contact Rong Zhou.

https://maths-cam-ac-uk.zoom.us/j/92192959944?pwd=OVJ3d0pJS3RRaWhKOVBuMFJIM3FiQT09

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.

This talk is part of the Number Theory Seminar series.

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