Diff for "days33/lfunction/tutorial"

Differences between revisions 20 and 21

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.

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.

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.

-  ⇤ ← Revision 20 as of 2011-09-21 19:26:30 → 
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-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  $F (s) = \prod_{p} {(1 - \frac{a_{p}}{p^{s}})}^{- 1} .$  

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:
+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  $F (s) = \prod_{p} {(1 - \frac{a_{p}}{p^{s}})}^{- 1} .$  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:
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+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.

Diff for "days33/lfunction/tutorial"

Introduction

Definition (Amy and Cassie)

Basic Functions (Amy)

Euler Product (Lola)

Functional Equation

Taylor Series

Zeros and Poles

Analytic Rank

Precision Issues

Advanced Topics: