Atkin polynomials for families of abelian varieties with real multiplication

Abstract

Generalizing the work of Atkin and Kaneko-Zagier in the elliptic case, we describe the non-ordinary locus of a genus-zero non-compact curve $Y$ in a Hilbert modular variety in terms of the zeros of generalized Atkin's orthogonal polynomials. The argument relies on the recent construction of lifts of partial Hasse invariants for $Y$. We further describe these orthogonal polynomials as denominators of Pad\'e approximants to the logarithmic derivatives of solutions of the Picard-Fuchs differential equations associated with $Y$. This provides a new link between Pad\'e approximation and the geometry of the non-ordinary locus, extending a classical observation of Igusa for the Legendre family and applying, in particular, to situations where the Picard-Fuchs equations do not admit modular solutions. As applications, we determine the three-term recurrence relations for Atkin polynomials attached to triangle curves via hypergeometric identities, and compute the supersingular locus of a double cover of the Teichm\"uller curve $W_{17}$. In the latter case, we conjecture that the associated supersingular polynomial is self-reciprocal, implying that supersingular points occur in pairs.