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''Dirichlet L-series'' | '''Dirichlet L-series''' |
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=== Series Coefficients === | '''Series Coefficients''' |
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'''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 $s\in\mathbb{C}$. |
Tutorial Outline!
Introduction
Definition (Amy and Cassie)
- - Dirichlet L-series and zeta functions (Amy) - for elliptic curves (Cassie) - for modular forms (Cassie)
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:
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\CC. 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\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
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 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 (Lola)
An Euler product is an infinite product expansion of a Dirichlet series, indexed by the primes. For a Dirichlet series of the form
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