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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:

\zeta_K(s)=\sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbb{Q}} (I))^s} = \sum_{n\geq1} \frac{a_n}{n^s}.

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. Any \zeta_K(s) can be decomposed as a product of L-series of Dirichlet characters in the character group of K:

\zeta_K(s)=\prod_{\chi} L(s,\chi).

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

L(s,E)=\prod_p \frac{1}{L_p(p^{-s})}=\prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \left(1 - a_p p^{-s}\right)^{-1}

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.

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.

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) = \sum_{n = 1}^\infty \frac{a_n}{n^s},

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

F(s) = \prod_p \left(1 - \frac{a_p}{p^s}\right)^{-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

\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_p \left(1 - p^{-s}\right)^{-1}

2. Dirichlet L-function

L(s, \chi) = \sum_{n = 1}^\infty \frac{\chi(n)}{n^s} = \prod_p \left(1 - \frac{\chi(p)}{p^s}\right)^{-1}

3. L-function of an Elliptic Curve (over \mathbb{Q})

L(E, s) = \sum_{n = 1}^\infty \frac{a_n}{n^s} = \prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \left(1 - a_p p^{-s}\right)^{-1}

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

\zeta_Q(s) = \sum_{(u,v) \neq (0,0)} (au^2 + buv + cv^2)^{-s},

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.

-  ⇤ ← Revision 3 as of 2011-09-21 03:47:43 → 
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+   ← Revision 33 as of 2011-09-21 21:22:32 → ⇥
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-Deletions are marked like this.
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-Introduction
+=== Introduction ===
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-Definition (Amy and Cassie)
+=== Definition (Amy and Cassie) ===
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+''The Dedekind $\zeta$-function''
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-Basic Functions (Amy)
  - not everything, but hit the highlights
+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 $K$:
$$\zeta_K(s)=\sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbb{Q}} (I))^s} = \sum_{n\geq1} \frac{a_n}{n^s}. $$
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.  Any $\zeta_K(s)$ can be decomposed as a product of $L$-series of Dirichlet characters in the character group of $K$:
$$\zeta_K(s)=\prod_{\chi} L(s,\chi).$$

''$L$-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
$$L(s,E)=\prod_p \frac{1}{L_p(p^{-s})}=\prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \left(1 - a_p p^{-s}\right)^{-1} $$
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.

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 $k$th L-series coefficient $a_k$ will correspond to the $k$th 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.
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+Line 42:
-Euler Product (Lola)
  - translating between Euler product and Dirichlet 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.
-Line 20:
+Line 53:
-Functional Equation
+=== 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 $$F(s) = \sum_{n = 1}^\infty \frac{a_n}{n^s},$$ the corresponding Euler product (if it exists) has the form $$F(s) = \prod_p \left(1 - \frac{a_p}{p^s}\right)^{-1}.$$ 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''' $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_p \left(1 - p^{-s}\right)^{-1}$$

2. '''Dirichlet L-function''' $$L(s, \chi) = \sum_{n = 1}^\infty \frac{\chi(n)}{n^s} = \prod_p \left(1 - \frac{\chi(p)}{p^s}\right)^{-1}$$

3. '''L-function of an Elliptic Curve (over $\mathbb{Q}$)''' $$L(E, s) = \sum_{n = 1}^\infty \frac{a_n}{n^s} = \prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \left(1 - a_p p^{-s}\right)^{-1}$$

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

$$\zeta_Q(s) = \sum_{(u,v) \neq (0,0)} (au^2 + buv + cv^2)^{-s},$$

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.
-Line 23:
+Line 70:
-Taylor Series
+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 ===
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-Zeros and Poles
+=== Taylor Series ===
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-Analytic Rank
+=== Zeros and Poles ===
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-Precision Issues
+=== Analytic Rank ===
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-Advanced Topics:
  - creating a new L-series class
+=== Precision Issues ===
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+Line 95:
-(Finding L-series from incomplete info)
+=== Advanced Topics: ===
  - creating a new L-series class
  
  - finding L-series from incomplete information