Bad Reduction of Genus Two Curves with complex multiplication

Abstract

We study genus two curves whose Jacobians have complex multiplication by a biquadratic CM field. In this setting the Jacobian decomposes as a product of two elliptic curves with complex multiplication by orders in the same imaginary quadratic field. We derive upper bounds for the primes of stable bad reduction of such curves in terms of the discriminants of the CM orders. To do so, we give an explicit description of principally polarized abelian surfaces with CM by a biquadratic field via small CM cycles on an orthogonal Shimura variety of signature (3,2). We use it to prove an arithmetic intersection formula between such CM cycles with Humbert surfaces in terms of coefficients of incoherent Eisenstein series and representation numbers of ternary positive definite quadratic forms. Finally, employing the arithmetic properties of higher Green functions, we show that primes of stable bad reduction always exist.