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\emph{The Dedekind $\zeta$-function} | ''The Dedekind $\zeta$-function'' |
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\noindent\Large{$L$-series of Elliptic Curves} \normalsize |
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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. |
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.) 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. |
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:
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
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