##### Department of Mathematics,

University of California San Diego

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### Math 295 - Mathematics Colloquium

## David Goss

#### Ohio State University

## Modularity in Characteristic p

##### Abstract:

The connection between elliptic curves over the rational numbers${f Q}$ and modular forms for $SL_2({f Z})$ is now very well known.This fundamental relationship both establishes that the L-series of onesuch elliptic curve has an analytic continuation and functional equationand gives representatives of the isogeny class of the elliptic curve(inside the Jacobians of modular curves).Now let $A:={f F}_q[T]$, where ${f F}_q$ is the finite field with$q$-elements, and let $k:={f F}_q(T)$. Let $K:={f F}_q((1/T))$with associated algebraic closure $ar{K}$. Mimicking the classicaldefinition of $f Z$-lattices inside the complex numbers $f C$,one has the notion of ${f F}_q[T]$-lattices inside $ar{K}$.Rank one lattices correspond to analogs of the exponential functionand rank two lattices uniformize analogs of elliptic curves. Theserank two \"Drinfeld modules\"" give rise to modular curves in exact analogywith elliptic curves. Remarkably

Host:

### February 13, 2003

### 3:00 PM

### AP&M 6438

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