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:
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 ap=p+1−Np when E has good reduction mod p. Then the L-series of E, L(s,E), is defined to be
L(s,E)=∏p1Lp(p−s)=∏p good reduction(1−app−s+p1−2s)−1∏p bad reduction(1−app−s)−1
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 Functions (Amy)
- - not everything, but hit the highlights
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)=∑∞n=1nsan,
the corresponding Euler product (if it exists) has the form
F(s)=∏p(1−psap)−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
ζ(s)=∑∞n=11ns=∏p(1−p−s)−1
2. Dirichlet L-function
L(s,χ)=∑∞n=1nsχ(n)=∏p(1−psχ(p))−1
3. L-function of an Elliptic Curve (over Q)
L(E,s)=∑∞n=1nsan=∏p good reduction(1−app−s+p1−2s)−1∏p bad reduction(1−app−s)−1
Not all L-series have an associated Euler product, however. For example, the Epstein Zeta Functions, defined by
ζQ(s)=∑(u,v)/=(0,0)(au2+buv+cv2)−s,
where Q(u,v)=au2+buv+cv2 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
