Abstract

We show how the Gross–Kudla formula about triple product L-functions allows us to construct degree-zero elements of the supersingular module annihilated by the winding ideal. Using the method of Parent, we apply those results to the study of rational points on modular curves, determining a set of primes of analytic density 1 − 9 ∕ 2¹⁰ for which the quotient of X₀(p^r) (r > 1) by the Atkin–Lehner operator w_p^r has no rational points other than the cusps and the CM points.

Keywords

rational points on modular curves, supersingular module, special values of L-functions

Mathematical Subject Classification

Primary: 14G05, 11G05, 11G18

Secondary: 14G10, 11R52

Authors