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Introduction | === Introduction === |
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Definition (Amy and Cassie) | === Definition (Amy and Cassie) === |
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\emph{The Dedekind $\zeta$-function} | |
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Basic Functions (Amy) | 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 $K$: $$\zeta_K(s)=\prod_{\chi} L(s,\chi).$$ \noindent\Large{$L$-series of Elliptic Curves} \normalsize 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 $$ L_p(T) = \begin{cases} 1-a_pT+pT^2 \text{, if $E$ has good reduction at $p$}, \\ 1-T \text{, if $E$ has split multiplicative reduction at $p$},\\ 1+T \text{, if $E$ has non-split multiplicative reduction at $p$},\\ 1 \text{, if $E$ has additive reduction at $p$} \end{cases} $$ and $a_p \in \set{0,1,-1}$ 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 \emph{The Arithmetic of Elliptic Curves}, Appendix C $\S$16.) 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) === |
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Euler Product (Lola) - translating between Euler product and Dirichlet series |
=== 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 $$F(s) = \prod_p \left(1 - \frac{a_p}{p^s}\right)^{-1}.$$ 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''' $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} = \prod_p \left(1 - p^{-s}\right)^{-1}$$ 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. === Functional Equation === |
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Functional Equation | === Taylor Series === |
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Taylor Series | === Zeros and Poles === |
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Zeros and Poles | === Analytic Rank === |
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Analytic Rank | === Precision Issues === |
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Precision Issues Advanced Topics: |
=== Advanced Topics: === |
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(Finding L-series from incomplete info) |
- finding L-series from incomplete information |
Tutorial Outline!
Introduction
Definition (Amy and Cassie)
- - Dirichlet L-series and zeta functions (Amy) - for elliptic curves (Cassie) - for modular forms (Cassie)
\emph{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:
\noindent\Large{L-series of Elliptic Curves} \normalsize
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
1-T \text{, if E has split multiplicative reduction at p},\\ 1+T \text{, if E has non-split multiplicative reduction at p},\\ 1 \text{, if E has additive reduction at p} \end{cases} $$
and a_p \in \set{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 \emph{The Arithmetic of Elliptic Curves}, Appendix C \S16.) 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
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
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.
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