Diff for "days33/lfunction/tutorial"

Introduction

Authors: Amy Feaver, Lola Thompson, Cassie Williams

Definition

The Dedekind \zeta-function

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

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.

In Sage it is simple to construct the L-series for a number field K. For example,

• sage: K.<a>=NumberField(x^2-x+1)

  sage: L=LSeries(K);L

returns the Dedekind \zeta-function associated to this quadratic imaginary field. The command

• sage: LSeries('zeta')

will return the Riemann \zeta-function. One function that has interesting functionality for Dedekind \zeta-functions is the residues command, which computes the residues at each pole. If you ask for the residues of a Dedekind \zeta-function, Sage will return 'automatic':

• sage: K.<a>=NumberField(x^2-x+1)

  sage: L=LSeries(K)

  sage: L.residues()

  • 'automatic'

but if you ask for the residues to a given precision you will get more information.

• sage: L.residues(prec=53)

  • [-0.590817950301839]

  sage: L.residues(prec=100)

  • [-0.59081795030183867576605582778]

Remember that the coefficients count the number of ideals of a given norm:

• sage: K.<a>=NumberField(x^2+1)

  sage: L=LSeries(K)

  sage: L.anlist(10)

  • [0, 1, 1, 0, 1, 2, 0, 0, 1, 1, 2]

implying that there is no ideal of norm 3 in \mathbb{Q}[i].

Dirichlet L-series

Dirichlet L-series are defined in terms of a Dirichlet characters. A Dirichlet character \chi mod k, for some positive integer k, is a homomorphism (\mathbb{Z}/k\mathbb{Z})^*\rightarrow\mathbb{C}. The series is given by

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\mapsto e^{2\pi i/10}).

L-series of Elliptic Curves

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

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.

To construct L(s,E) in Sage, first define an elliptic curve over some number field.

• sage: E=EllipticCurve('37a')

  sage: L=LSeries(E);L

  • L-series of Elliptic Curve defined by y^{2 + y = x}3 - x over Rational Field

  sage: K.<a>=NumberField(x^2-x+1)

  sage: E2 = EllipticCurve(K, [0, 0,1,-1,0])

  sage: LSeries(E2)

  • L-series of Elliptic Curve defined by y^{2 + y = x}3 + (-1)*x over Number Field in a with defining polynomial x^2 - x + 1

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.

• sage: L.anlist(10)

  • [0, 1, -2, -3, 2, -2, 6, -1, 0, 6, 4]

L-series of Modular Forms

If f is a modular form of weight k, it has a Fourier expansion f(z)=\sum_{n\geq0} a_n (e^{2\pi i z})^n. Then the L-series of f is

which does converge on some half-plane. These L-series have an analytic continuation and functional equation, but not necessarily an Euler product formula.

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

For example:

• sage: K.\langle a\rangle = NumberField(x^3 + 29) sage: L = LSeries(K) sage: L.anlist(5)

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.

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 a_3), 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\in\mathbb{C}.

Euler Product

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 \mathbb{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