Non-Archimedean Integration and Elliptic Curves over Function Fields

✍ Scribed by Ignazio Longhi

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 247 KB
• Volume: 94
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2001.2735

✦ Synopsis

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 m E on P 1 (F . ). By choosing an appropriate embedding of a quadratic unramified extension K/F into the matrix algebra M 2 (F), m E 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.