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Tutorial Outline!

Introduction

Definition (Amy and Cassie)

The Dedekind ζ-function

If K is a number field over Q and s∈C such that Re(s)>1 then we can create ζK(s), the Dedekind ζ-function of K:

ζK(s)=∑I⊆OK1(NK/Q(I))s=∑n≥1nsan. 
In the first sum, I runs through the nonzero ideals I of OK, the ring of integers of K, and an is the number of ideals in OK of norm n. These ζ-functions are a generalization of the Riemann ζ-function, which can be thought of as the Dedekind ζ-function for K=Q. The Dedekind ζ-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 ζK(s) can be decomposed as a product of L-series of Dirichlet characters in the character group of K:
ζK(s)=∏χL(s,χ). 

L-series of Elliptic Curves

Let E be an elliptic curve over Q and let p be prime. Let Np 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) = 1-a_pT+pT^2 if E has good reduction at p, and L_p(T)= 1-a_p T with a_p \in \{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 The Arithmetic of Elliptic Curves, Appendix C, Section 16.) If Re(s)>3/2 then L(s,E) is analytic, and it is conjectured that these L-series have analytic continuations to the complex plane and functional equations.

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)

\noindent \begin{Large}\textbf{Basic Sage Functions for L-series}\end{Large}

\noindent \begin{large}\textbf{Series Coefficients}\end{large}

\noindent The command L.anlist(n) will return a list V of n+1 numbers; 0, followed by the first n coefficients of the L-series L. The zero is included simply as a place holder, so that the kth L-series coefficient a_k will correspond to the kth entry V[k] of the list.

\vspace{0.1in}

\noindent For example: \newline sage: K.\langle a\rangle = NumberField(x^3 + 29) \newline sage: L = LSeries(K) \newline sage: L.anlist(5) \newline will return [0,1,1,1,2,1], which is [0,a_1,a_2,a_3,a_4,a_5] for this L-series.

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\noindent To access the value of an individual coefficient, you can use the function an. For example, for the series used above:

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\noindent sage: L.an(3)\newline will return 1 (the value of a_3), and\newline sage: L.an(4) \newline returns 2.

Euler Product (Lola)

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 (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.

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

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

Note: In order to use this class, the authors created a derived class that implements a method _local_factor(P), which takes as input a prime ideal P of K=base\_field, and returns a polynomial that is typically the reversed characteristic polynomial of Frobenius at P of Gal(\overline{K}/K) acting on the maximal unramified quotient of some Galois representation. This class automatically computes the Dirichlet series coefficients a_n from the local factors of the L-function.

Functional Equation

Taylor Series

Zeros and Poles

Analytic Rank

Precision Issues

Advanced Topics: