Legendre polynomials and the elliptic genus

Abstract

The famous Legendre polynomials are intricately connected with the theory of elliptic integrals and with the formal groups associated to Jacobi quartic curves Y2 = 1 26X2 + &X4. For instance, Honda [4] has obtained new congruences for these polynomials using his theory of formal groups. Congruences modulo p2 involving Legendre polynomials are at the root of the construction of elliptic cohomology [S]. On the other hand, Legendre polynomials may be interpreted as matrix coefficients for finite-dimensional representations of the algebraic group X(2) [3]. In this article, we use well-known facts about the reduction modulo p of such representations to find a representation-theoretic statement which implies the Schur congruences for the Legendre polynomials (see Section 1). We use similar methods to obtain a new derivation for some of Honda’s congruences. In Section 3, we prove a statement about the reduction modulo a large prime number p of a certain representation of an orthogonal group to obtain a congruence (Corollary 3.2) for Gegenbauer polynomials. We also give a congruence for Tchebycheff polynomials (Proposition 3.4). The expression of the Legendre polynomials as matrix coefficients allows us to write down an intriguing formula (Proposition 2.1) for the logarithm of the formal group associated to a Jacobi quartic. This formula shows that this formal power series is an “expectation value” (in the sense of quantum theory) for the representation of GL(2) in the space of formal power series in two variables X and Y. The role of the vacuum vector is played by the formal power series exy. The parameter t for the formal power series appears as the entry of a scalar (2, 2)-matrix. This suggests that some