Hecke L-functions
introduction
- http://math.stackexchange.com/questions/409200/functional-equation-for-hecke-l-series
- Tate's approach to analytic continuation of classical $\zeta$-functions and Dirichlet $L$-functions
- from [Leahy2010]
In the early 20th century, Erich Hecke attempted to find a further generalization of the Dirichlet L-series and the Dedekind zeta function. In 1920, he introduced the notion of a Grossencharakter, an ideal class character of a number field, and established the analytic continuation and functional equation of its associated L-series, the Hecke L-series. In 1950, John Tate, following the suggestion of his advisor, Emil Artin, recast Hecke's work. Tate provided a more elegant proof of the functional equation of the Hecke L-series by using Fourier analysis on the adeles and employing a reformulation of the Grossencharakter in terms of a character on the ideles. Tate's work now is generally understood as the GL(1) case of automorphic forms
Dirichlet L-functions
zeta integral
Riemann zeta function
- $f\in \mathcal{S}(\mathbb{A})$
- define
$$ \zeta(f,s)=\int_{\mathbb{A}^{\times}}f(x)|x|^s\, d^{\times}x $$
- thm
The integral converges locally uniformly for $\Re(s)>1$ and so it defines a holomorphic function in that range, which extends to an meromorphic function on $\mathbb{C}$. This function is holomorphic away from the points $s=0,1$, where it has at most simple poles of residue $-f(0)$ and $\hat{f}(0)$, respectively. The zeta integral satisfies the functional equation One has $$ \zeta(f,s)=\zeta(\widehat{f},1-s) $$
Dirichlet L-functions
- $f\in \mathcal{S}(\mathbb{A})$
- $\chi$ : character of $\mathbb{A}^{\times}/\mathbb{Q}^{\times}$ with finite image
- define
$$ \zeta(f,\chi,s)=\int_{\mathbb{A}^{\times}}f(x)\chi(x)|x|^s\, d^{\times}x $$
- thm
Let $\chi\neq 1$. The integral converges locally uniformly for $\Re(s)>1$ and so it defines a holomorphic function in that range, which extends to an entire function on $\mathbb{C}$. One has $$ \zeta(f,\chi,s)=\zeta(\widehat{f},\overline{\chi},1-s) $$
expositions
- Alayont, Adelic approach to Dirichlet L-function
- [Leahy2010] James-Michael Leahy, An introduction to Tate's Thesis
- Herz, Carl, Stephen William Drury, and Maruti Ram Murty. 1997. Harmonic Analysis and Number Theory: Papers in Honour of Carl S. Herz : Proceedings of a Conference on Harmonic Analysis and Number Theory, April 15-19, 1996, McGill University, Montréal, Canada. American Mathematical Soc.