Diff for "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)

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:

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:

Dirichlet L-series

Dirichlet L-series are defined in terms of a Dirichlet characters. A Dirichlet character χ mod k, for some positive integer k, is a homomorphism (Z/kZ)*→C. The series is given by $$L(s,\chi )=\sum _{n\in \mathbb{N}}\frac{\chi (n)}{n^s},s\in \mathbb{C},\text{Re}(s)>1.$$ Although these series can formally be defined for any Dirichlet character, it only makes (practical) sense to define these series in terms of primitive characters, because non-primitive characters will give rise to series which have missing factors in their Euler products and thus do not have an associated functional equation.

To define an L-series in Sage, you must first create a primitive character:

sage: G=DirichletGroup(11)

G is now the group of Dirichlet characters mod 11. We may then define the Dirichlet L-series over a single character from this group:

sage: L=LSeries(G.0)

gives the L-series for the character G.0 (the character which maps 2|→e2πi/10).

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 ap=p+1−Np when E has good reduction mod p. Then the L-series of E, L(s,E), is defined to be

where Lp(T)=1−apT+pT2 if E has good reduction at p, and Lp(T)=1−apT with ap∈{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 Sage Functions for L-series

Series Coefficients

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 ak will correspond to the kth entry V[k] of the list.

For example:

• sage: K.〈a〉 = NumberField(x3+29) sage: L = LSeries(K) sage: L.anlist(5)

will return [0,1,1,1,2,1], which is [0,a1,a2,a3,a4,a5] for this L-series.

To access the value of an individual coefficient, you can use the function an (WE ACTUALLY HAVE TO WRITE AN INTO SAGE FIRST...). For example, for the series used above:

sage: L.an(3)

will return 1 (the value of a3), and

sage: L.an(4)

returns 2.

Evaluation of L-functions at Values of s

For any L-function L, simply type

sage: L(s)

to get the value of the function evaluated at s∈C.

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

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

2. Dirichlet L-function

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

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

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,

• 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.

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:

• - creating a new L-series class - finding L-series from incomplete information