Unlikely intersections on Shimura varieties

Author / Creator

G, Asvin, author

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Online

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Summary

Recently, there has been much work on the question of just-likely intersections on Shimura varieties. These works are concerned with showing that there are an infinite number of "conincidences" up-...

Recently, there has been much work on the question of just-likely intersections on Shimura varieties. These works are concerned with showing that there are an infinite number of "conincidences" up-to Hecke translations in the case where the dimensions of the subvarieties are complementary. However, all these results only consider intersections between an (arithmetic) curve and a special divisor. A recent AIM project formulated more general conjectures concerning intersections of arbitrary subvarieties in Shimura varieties (up-to Hecke translations) and this thesis takes the first steps towards proving such a general conjecture. In the first part of this thesis, we prove the conjecture (appropriately formulated) for products of the modular curve. Moreover, we also consider unlikely intersections and show that, contrary to expectations, such inter- sections can occur infinitely often in positive characteristic. In the second part of this thesis, in joint work with Qiao He and Ananth Shankar, we prove the conjecture for Hilbert modular surfaces in positive characteristic (with mild restrictions).

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