• Corpus ID: 247793180

Unlikely and just likely intersections for high dimensional families of elliptic curves

@inproceedings{Asvin2022UnlikelyAJ,
title={Unlikely and just likely intersections for high dimensional families of elliptic curves},
author={G Asvin},
year={2022}
}
• G. Asvin
• Published 30 March 2022
• Mathematics
Given two varieties V , W in the n-fold product of modular curves, we answer afﬁrmatively a question (formulated by Shou-Wu Zhang’s AIM group) on whether the set of points in V that are Hecke translations of some point on W is dense in V . We need to make some (necessary) assumptions on the dimensions of V , W but for instance, when V is a divisor and W is a curve, no further assumptions are needed. We also examine the necessity of our assumptions in the case of unlikely intersections and show…

References

SHOWING 1-9 OF 9 REFERENCES
O-minimality and certain atypical intersections
• Mathematics
• 2014
We show that the strategy of point counting in o-minimal structures can be applied to various problems on unlikely intersections that go beyond the conjectures of Manin-Mumford and Andr\'e-Oort. We
Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields
• Mathematics
• 2019
Given a K3 surface $X$ over a number field $K$, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite, assuming that $X$ has everywhere potentially good reduction.
Frobenius distribution for pairs of elliptic curves and exceptional isogenies
Let E and E be two elliptic curves over a number field. We prove that the reductions of E and E at a finite place p are geometrically isogenous for infinitely many p, and draw consequences for the
Exceptional splitting of reductions of abelian surfaces
• Mathematics
• 2017
Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces
Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
• Mathematics
Inventiones mathematicae
• 2022
Let $\mathscr{X} \rightarrow C$ be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve $C$ in characteristic $p \geq 5$. We prove that the geometric Picard rank jumps
Equidistribution of hodge loci
• ii. arXiv preprint arXiv:2103.15717,
• 2021
Hypersymmetric abelian varieties
• Pure and Applied Mathematics Quarterly,
• 2006