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Tutorial Outline! |
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=== Definition (Amy and Cassie) === - Dirichlet L-series and zeta functions (Amy) - for elliptic curves (Cassie) - for modular forms (Cassie) ''The Dedekind $\zeta$-function'' |
Authors: Amy Feaver, Lola Thompson, Cassie Williams === Definitions === '''The Dedekind $\zeta$-function''' |
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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'' |
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''' |
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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 |
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=== Basic Functions (Amy) === - not everything, but hit the highlights === Euler Product (Lola) === |
''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 === |
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=== Taylor Series === |
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The analytic rank of an $L$-series is the order of vanishing of the zero at the central critical point (half the weight of the $L$-series). In Sage, you can compute the analytic rank of any $L$-series $L$ by using the command: sage: L.analytic_rank() WARNING: It is important to note that the analytic rank is computed using numerical methods, and is not provably correct. The analytic rank of an L-series of an elliptic curve plays an important role in the Birch and Swinnerton-Dyer conjecture. Specifically, part of the BSD conjecture claims that the rank of the group or points on an elliptic curve $E$ over a number field $K$ is the order of vanishing of the $L$-function $L(s,E)$ at $s=1$. Generalizations of the BSD conjecture exist for abelian varieties of higher dimension as well. |
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- creating a new L-series class |
Introduction
Authors: Amy Feaver, Lola Thompson, Cassie Williams
Definitions
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:
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
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
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 y2 + y = x3 - 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 y2 + y = x3 + (-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
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
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
Zeros and Poles
Analytic Rank
The analytic rank of an L-series is the order of vanishing of the zero at the central critical point (half the weight of the L-series). In Sage, you can compute the analytic rank of any L-series L by using the command:
sage: L.analytic_rank()
WARNING: It is important to note that the analytic rank is computed using numerical methods, and is not provably correct.
The analytic rank of an L-series of an elliptic curve plays an important role in the Birch and Swinnerton-Dyer conjecture. Specifically, part of the BSD conjecture claims that the rank of the group or points on an elliptic curve E over a number field K is the order of vanishing of the L-function L(s,E) at s=1. Generalizations of the BSD conjecture exist for abelian varieties of higher dimension as well.
Precision Issues
Advanced Topics:
- - finding L-series from incomplete information