Non-Archimedean Integration and Elliptic Curves over Function Fields

Abstract

Let F be a global function field of characteristic p and E/F an elliptic curve with split multiplicative reduction at the place ∞: then E can be obtained as a factor of the Jacobian of some Drinfeld modular curve. This fact is used to associate to E a measure μE on P1(F∞). By choosing an appropriate embedding of a quadratic unramified extension K/F into the matrix algebra M2(F), μE is pushed forward to a measure on a p-adic group G, isomorphic to an anticyclotomic Galois group over the Hilbert class field of K. Integration on G then yields a Heegner point on E when ∞ is inert in K and an analogue of the L-invariant if ∞ is split. In the last section, the same methods are extended to integration on a geometric cyclotomic Galois group.