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| Our L-function tutorial goes here. | 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 $$F(s) = \prod_p \left(1 - \frac{a_p}{p^s}\right)^{-1}.$$ 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 Taylor Series Zeros and Poles Analytic Rank Precision Issues Advanced Topics: - creating a new L-series class 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)
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
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