days33/lfunction/tutorial

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)=∑∞n=1nsan,

the corresponding Euler product (if it exists) has the form

F(s)=∏p(1−psap)−1.

To define an L-series by an Euler product in Sage, one can use the LSeriesAbstract class. For example,

sage: L = LSeriesAbstract(conductor=1, hodge_numbers=[0], weight=1, epsilon=1, poles=[1], residues=[-1], base_field=QQ)

sage: L

returns an L-series Euler product with conductor 1, Hodge numbers [0], weight 1, epsilon 1, poles [1], residues [-1] over a Rational Field.

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:

ζ(s)=∑∞n=11ns=∏p(1−p−s)−1

2. Dirichlet L-function:

L(s,χ)=∑∞n=1nsχ(n)=∏p(1−psχ(p))−1

3. L-function of an Elliptic Curve (over Q):

L(E,s)=∑∞n=1nsan=∏p good reduction(1−app−s+p1−2s)−1∏p bad reduction(1−app−s)−1

Not all L-series have an associated Euler product, however. For example, the Epstein Zeta Functions, defined by

ζQ(s)=∑(u,v)/=(0,0)(au2+buv+cv2)−s,

where Q(u,v)=au2+buv+cv2 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