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| 3. L-function of an Elliptic Curve: $$L(E, s) = \sum_{n = 1}^\infty \frac{a_n}{n^s} = \prod_{\substack{p \\ E \ \mathrm{has good reduction at} \ p} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{\substack{p \\ p \ \mathrm{\substack{p \\ E \ \mathrm{does not have good reduction at} \ p} \left(1 - a_p p^{-s}\right)^{-1}$$ | 3. L-function of an Elliptic Curve: $$L(E, s) = \sum_{n = 1}^\infty \frac{a_n}{n^s} = \prod_{\substack{p \\ E \ \mathrm{has good reduction at} \ p}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{\substack{p \\ p \ \mathrm{\substack{p \\ E \ \mathrm{does not have good reduction at} \ p}} \left(1 - a_p p^{-s}\right)^{-1}$$ |
Tutorial Outline!
Introduction
Definition (Amy and Cassie)
- - Dirichlet L-series and zeta functions (Amy) - for elliptic curves (Cassie) - for modular forms (Cassie)
Basic Functions (Amy)
- - not everything, but hit the highlights
Euler Product (Lola)
- - translating between Euler product and Dirichlet series
An Euler product is an infinite product expansion of a Dirichlet series, indexed by the primes. For a Dirichlet series of the form
F(s) = \sum_{n = 1}^\infty \frac{a_n}{n^s},
the corresponding Euler product (if it exists) has the form F(s) = \prod_p \left(1 - \frac{a_p}{p^s}\right)^{-1}
. In many cases, an L-series can be expressed as an Euler product. By definition, if an L-series has a Galois representation then it has an Euler product. Some examples of common L-series with Euler products include: 1. Riemann zeta function:
\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_p \left(1 - p^{-s}\right)^{-1}
2. Dirichlet L-function:
L(s, \chi) = \sum_{n = 1}^\infty \frac{\chi(n)}{n^s} = \prod_p \left(1 - \frac{\chi(p)}{p^s}\right)^{-1}
3. L-function of an Elliptic Curve:
L(E, s) = \sum_{n = 1}^\infty \frac{a_n}{n^s} = \prod_{\substack{p \\ E \ \mathrm{has good reduction at} \ p}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{\substack{p \\ p \ \mathrm{\substack{p \\ E \ \mathrm{does not have good reduction at} \ p}} \left(1 - a_p p^{-s}\right)^{-1}
Not all L-series have an associated Euler product, however. For example, the Epstein Zeta Functions, defined by
\zeta_Q(s) = sum_{(u,v) \neq (0,0)} (au^2 + buv + cv^2)^{-s},
where Q(u,v) = au^2 + buv + cv^2 is a positive definite quadratic form, has a functional equation but, in general, does not have an Euler product.
Functional Equation
Taylor Series
Zeros and Poles
Analytic Rank
Precision Issues
Advanced Topics:
- - creating a new L-series class
Finding L-series from incomplete information