Primes of bad reduction of certain genus 2 curves
Speaker: Christophe Ritzenthaler (Executive Director of CIMPA)
Time: 10:30 - 11:30, November 05, 2025
Venue: Room 612, A6, Institute of Mathematics-VAST
Abstract: Given a curve over a number field, characterised by certain arithmetic conditions, it is a classical game to understand its reductions modulo a prime, even without explicitly knowing an equation for this curve. This has been done for complex multiplication (CM) curves of genus 1 and 2.
Here we study genus 2 curves whose Jacobian is the square of a CM elliptic curve. We will use the refined Humbert invariant introduced by Kani to bound and enumerate primes of bad reduction, then more sophisticated results from Kudla and Rapoport to control the exponents. Some related results on oriented supersingular curves or on the relations between invariants and modular forms may be discussed if time permits, but we will mainly try to highlight the many open questions that remain around this problem.
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