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Tutorial Outline!

Introduction

Definition (Amy and Cassie)

- Dirichlet L-series and zeta functions (Amy) - for elliptic curves (Cassie) - for modular forms (Cassie)

\noindent\Large{The Dedekind \zeta-function} \normalsize

If K is a number field over \mathbb{Q} and s\in\mathbb{C} such that \mathfrak{R}(s)>1 then we can create \zeta_K(s), the Dedekind \zeta-function of K:

\zeta_K(s)=\sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbb{Q}} (I))^s} = \sum_{n\geq1} \frac{a_n}{n^s}.

In the first sum, I runs through the nonzero ideals I of \mathcal{O}_K, the ring of integers of K, and a_n is the number of ideals in \mathcal{O}_K of norm n. These \zeta-functions are a generalization of the Riemann \zeta-function, which can be thought of as the Dedekind \zeta-function for K=\mathbb{Q}. The Dedekind \zeta-function of K also has an Euler product expansion and an analytic continuation to the entire complex plane with a simple pole at s=1, as well as a functional equation. Any \zeta_K(s) can be decomposed as a product of L-series of Dirichlet characters in the character group of K:

\zeta_K(s)=\prod_{\chi} L(s,\chi).

\noindent\Large{L-series of Elliptic Curves} \normalsize

Let E be an elliptic curve over \mathbb{Q} and let p be prime. Let N_p be the number of points on the reduction of E mod p and set a_p=p+1-N_p when E has good reduction mod p. Then the L-series of E, L(s,E), is defined to be

L(s,E)=\prod_p \frac{1}{L_p(p^{-s})}=\prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \left(1 - a_p p^{-s}\right)^{-1}

where $ L_p(T) = \begin{cases} 1-a_pT+pT^2 \text{, if E has good reduction at p$}, \\

1-T \text{, if E has split multiplicative reduction at p},\\ 1+T \text{, if E has non-split multiplicative reduction at p},\\ 1 \text{, if E has additive reduction at p} \end{cases} $$

and a_p \in \set{0,1,-1} if E has bad reduction mod p. (All of these definitions can be rewritten if you have an elliptic curve defined over a number field K; see Silverman's \emph{The Arithmetic of Elliptic Curves}, Appendix C \S16.) Notice in particular that although one can certainly rewrite L(s,E) as a sum over the natural numbers, the sequence of numerators no longer has an easily interpretable meaning in terms of the elliptic curve itself.

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}.

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:

\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 (over \mathbb{Q}):

L(E, s) = \sum_{n = 1}^\infty \frac{a_n}{n^s} = \prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \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

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+\noindent\Large{The Dedekind $\zeta$-function}
\normalsize

If $K$ is a number field over $\mathbb{Q}$ and $s\in\mathbb{C}$ such that $\mathfrak{R}(s)>1$ then we can create $\zeta_K(s)$, the Dedekind $\zeta$-function of $K$:
$$\zeta_K(s)=\sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbb{Q}} (I))^s} = \sum_{n\geq1} \frac{a_n}{n^s}. $$
In the first sum, $I$ runs through the nonzero ideals $I$ of $\mathcal{O}_K$, the ring of integers of $K$, and  $a_n$ is the number of ideals in $\mathcal{O}_K$ of norm $n$.  These $\zeta$-functions are a generalization of the Riemann $\zeta$-function, which can be thought of as the Dedekind $\zeta$-function for $K=\mathbb{Q}$.  The Dedekind $\zeta$-function of $K$ also has an Euler product expansion and an analytic continuation to the entire complex plane with a simple pole at $s=1$, as well as a functional equation.  Any $\zeta_K(s)$ can be decomposed as a product of $L$-series of Dirichlet characters in the character group of $K$:
$$\zeta_K(s)=\prod_{\chi} L(s,\chi).$$

\noindent\Large{$L$-series of Elliptic Curves}
\normalsize

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $p$ be prime.  Let $N_p$ be the number of points on the reduction of $E$ mod $p$ and set $a_p=p+1-N_p$ when $E$ has good reduction mod $p$.  Then the $L$-series of $E$, $L(s,E)$, is defined to be
$$L(s,E)=\prod_p \frac{1}{L_p(p^{-s})}=\prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \left(1 - a_p p^{-s}\right)^{-1} $$
where
$$ L_p(T) = \begin{cases} 1-a_pT+pT^2  \text{, if $E$ has good reduction at $p$}, \\
 1-T  \text{, if $E$ has split multiplicative reduction at $p$},\\
 1+T  \text{, if $E$ has non-split multiplicative reduction at $p$},\\
 1  \text{, if $E$ has additive reduction at $p$} \end{cases} $$
and $a_p \in \set{0,1,-1}$ if $E$ has bad reduction mod $p$.  (All of these definitions can be rewritten if you have an elliptic curve defined over a number field $K$; see Silverman's \emph{The Arithmetic of Elliptic Curves}, Appendix C $\S$16.)  Notice in particular that although one can certainly rewrite $L(s,E)$ as a sum over the natural numbers, the sequence of numerators no longer has an easily interpretable meaning in terms of the elliptic curve itself.
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