##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Alyson Deines

#### Center for Communications Research

## Elliptic Curve Parameterizations by Modular curves and Shimura curves

##### Abstract:

A crowning achievement of Number theory in the 20th century is a theorem of Wiles which states that for an elliptic curve $E$ over $\mathbb{Q}$ of conductor $N$, there is a non-constant map from the modular curve $X_0(N)$ to $E$. For some curve isogenous to $E$, the degree of this map will be minimal; this is the modular degree. The Jacquet-Langlands correspondence allows us to similarly parameterize elliptic curves by Shimura curves. In this case we have several different Shimura curve parameterizations for a given isogeny class. Further, this generalizes to elliptic curves over totally real number fields. In this talk I will discuss these degrees and I compare them with $D$-new modular degrees and $D$-new congruence primes. This data indicates that there is a strong relationship between Shimura degrees and new modular degrees and congruence primes.

Host: Kiran Kedlaya

### February 25, 2016

### 1:00 PM

### AP&M 7321

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