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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:

\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.

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

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

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.

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

L(s,f)=\sum_{n\geq1} \frac{a_n}{n^s}

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}.

Taylor Series for L-functions

This function will return the Taylor series of an L-function L. If the user does not enter any arguments, the center of the series will default to weight/2. For example, if L is the Riemann zeta function,

sage: L.taylor_series()

will output the Taylor series centered at weight/2=0.5. You can also specify degree, variable and precision. Entering

sage: L.taylor_series(center=2, degree=4, variable='t', prec=30)

will give you the Taylor series with the properties you would expect. Note that degree=4 actually means you will compute the first 4 terms of the series, giving you a degree 3 polynomial. The output of the above line therefore will be the Taylor polynomial 1.6449341 - 0.93754825t + 0.99464012t^{2} - 1.0000243t^{3} + O(t^{4}).

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

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 6 as of 2011-09-21 18:15:30 → 
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+   ← Revision 48 as of 2012-01-10 19:17:42 → ⇥
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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)


Basic Functions (Amy)
  - not everything, but hit the highlights


Euler Product (Lola)
  - translating between Euler product and Dirichlet series

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: $$L(E, s) = \sum_{n = 1}^\infty \frac{a_n}{n^s} = \prod_{\substack{p \\ E \ \mathrm{has good reduction at} \ p}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{\substack{p \\ E \ mathrm{does not have good reduction at} \ p}} \left(1 - a_p p^{-s}\right)^{-1}$$
+=== 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 $K$:
$$\zeta_K(s)=\sum_{I \subseteq \mathcal{O}_K} \frac{1}{(N_{K/\mathbb{Q}} (I))^s} = \sum_{n\geq1} \frac{a_n}{n^s}. $$
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.

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
$$L(s,\chi)=\sum_{n\in\mathbb{N}}\frac{\chi(n)}{n^s},\ s\in\mathbb{C}, \text{Re}(s)>1.$$
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
$$L(s,E)=\prod_p \frac{1}{L_p(p^{-s})}=\prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \left(1 - a_p p^{-s}\right)^{-1} $$
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.

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 
$$L(s,f)=\sum_{n\geq1} \frac{a_n}{n^s}$$ 
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 $k$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.


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 $s\in\mathbb{C}$. 


'''Taylor Series for L-functions'''

This function will return the Taylor series of an L-function $L$. If the user does not enter any arguments, the center of the 
series will default to weight/2. For example, if $L$ is the Riemann zeta function, 

sage: L.taylor_series()

will output the Taylor series centered at weight/2=0.5. You can also specify degree, variable and precision. Entering

sage: L.taylor_series(center=2, degree=4, variable='t', prec=30)

will give you the Taylor series with the properties you would expect. Note that degree=4 actually means you will compute the first 4 terms of the series, giving you a degree 3 polynomial. The output of the above line therefore will be the Taylor polynomial
$1.6449341 - 0.93754825t + 0.99464012t^{2} - 1.0000243t^{3} + O(t^{4})$.


=== 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 $$F(s) = \sum_{n = 1}^\infty \frac{a_n}{n^s},$$ 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 $\mathbb{Q}$)''' $$L(E, s) = \sum_{n = 1}^\infty \frac{a_n}{n^s} = \prod_{p \ \mathrm{good \ reduction}} \left(1 - a_p p^{-s} + p^{1-2s}\right)^{-1} \prod_{p \ \mathrm{bad \ reduction}} \left(1 - a_p p^{-s}\right)^{-1}$$
-Line 29:
+Line 172:
-$$\zeta_Q(s) = sum_{(u,v) \neq (0,0)} (au^2 + buv + cv^2)^{-s},$$
+$$\zeta_Q(s) = \sum_{(u,v) \neq (0,0)} (au^2 + buv + cv^2)^{-s},$$
-Line 33:
+Line 176:
-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
+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 ===


=== Zeros and Poles ===


=== Analytic Rank ===


=== Precision Issues ===


=== Advanced Topics: ===
  - finding L-series from incomplete information