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Mathematics Colloquium


Date:  Tuesday, November 29, 2016

Title:  Quaternion algebras and supersingular locus of Hilbert modular varieties

Abstract:  There is a deep and famous result of Deuring and Serre (refined by Ribet) in arithmetic geometry saying that one has a canonical bijection between isomorphism classes of supersingular elliptic curves over the algebraic closure of F_p and those of oriented maximal Eichler orders of discriminant p, where p is a prime number. The theorem immediately gives a formula for the number of such elliptic curves. However, it has much more significant applications in Galois representations and modular forms, if one interprets it in terms of the geometry of modular curves. This result was later generalized to some other Hilbert modular varieties, for example, by Nicole in genus 2 and by Yu in genus 4. In this talk, we will generalize this result to arbitrary Hilbert modular varieties, by providing a global description of the supersingular locus. We will also discuss its applications in number theory. This is a joint work with Yichao Tian at Bonn.

Speaker:  Yifeng Liu
Institution:  Northwestern University


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