Hecke L-functions
introduction
- In 1920, Hecke 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 1928, Artin proved his celebrated reciprocity law that every abelian L-function is in fact a Hecke L-function
- In 1936, Chevalley introduced the concept of ideles and the idele group of an algebraic number field and reinterpreted Hecke's grossencharakter as characters of the idele class groups
- In 1945, Artin and whaples defined the adele ring of an algebraic number field
- In 1950, Tate carried out the suggestion of Artin to use harmonic analysis of adele groups to prove Hecke's theorems abour L-functions attached to grossencharacters
- for example, analytic continuation of classical $\zeta$-functions and Dirichlet $L$-functions
- the following is taken 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
overview
- A Hecke character (or Größencharakter) of a number field $K$ is defined to be a quasicharacter of the idèle class group of $K$
- Robert Langlands interpreted Hecke characters as automorphic forms on the reductive algebraic group GL(1) over the ring of adeles of $K$
- Let $E⁄K$ be an abelian Galois extension with Galois group $G$
- Then for any character $\sigma:G \to \mathbb{C}^{\times}$ (i.e. one-dimensional complex representation of the group $G$), there exists a Hecke character $\chi$ of $K$ such that
\[L_{E/K}^{\mathrm{Artin}}(\sigma, s) = L_{K}^{\mathrm{Hecke}}(\chi, s)\] where the left hand side is the Artin L-function associated to the extension with character $\sigma$ and the right hand side is the Hecke $L$-function associated with $\chi$
Riemann zeta function
analytic continuation
- 자코비 세타함수를 이용하여, 리만제타함수를 복소평면 전체로 확장할 수 있음.\[\theta(\tau)= \sum_{n=-\infty}^\infty e^{\pi i n^2 \tau}\]
- 감마함수\[\Gamma(s) = \int_0^\infty e^{-t} t^{s} \frac{dt}{t}\] 를 이용하면, \[\int_0^\infty e^{-\pi n^2t} t^{\frac{s}{2}} \frac{dt}{t} = {\pi}^{-\frac{s}{2}}\Gamma(\frac{s}{2})\frac{1}{n^s}\]
- 형식적으로는 다음과 같은 적분에 의해, 리만제타함수를 얻을 수 있음.
\[\xi(s) : = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)= \int_0^\infty (\frac{\theta(it)-1}{2})t^{\frac{s}{2}} \frac{dt}{t}\]
- 그러나 위의 적분은 모든 s에 대하여 수렴하지 않음. 따라서 다음과 같이 수정하여, 적분이 모든 s에 대하여 정의되도록 함.
\[\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s) = \frac{1}{s-1}-\frac{1}{s} +\frac{1}{2}\int_0^1 (\theta(it)-\frac{1}{\sqrt{t}})t^{\frac{s}{2}} \frac{dt}{t} +\frac{1}{2}\int_1^\infty (\theta(it)-1)t^{\frac{s}{2}} \frac{dt}{t}\]
여기서는 자코비 세타함수의 성질 \[\theta(iy)=\frac{1}{\sqrt{y}}\theta(\frac{i}{y})\] 이 사용됨.
함수방정식
- 리만제타함수는 \(s=\frac{1}{2}\) 에 대하여 대칭성을 가지고, 그에 따른 함수방정식을 만족시킴.\[\xi(s) = \xi(1 - s)\] 즉,\[\pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)=\pi^{-(1-s)/2}\ \Gamma\left(\frac{1-s}{2}\right)\ \zeta(1-s)\]
- 증명
자코비 세타함수의 모듈라 성질을 사용하면, \[\int_0^1 (\theta(it)-\frac{1}{\sqrt{t}})t^{\frac{s}{2}} \frac{dt}{t}= \int_1^\infty (\theta(it)-1)t^{\frac{1-s}{2}} \frac{dt}{t}\]
이므로, \(\xi(s)\) 의 정의를 이용하면, \[\xi(s) = \frac{1}{s-1}-\frac{1}{s} +\frac{1}{2}\int_1^\infty (\theta(it)-1)t^{\frac{1-s}{2}} \frac{dt}{t}+\frac{1}{2}\int_1^\infty (\theta(it)-1)t^{\frac{s}{2}} \frac{dt}{t}\]
를 얻는다.
이 식에서 \(s \leftrightarrow 1-s\) 는 우변을 변화시키지 않음므로 함수방정식 \(\xi(s) = \xi(1 - s)\)을 얻는다. ■
Dirichlet L-functions
global 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 $$ \zeta(f,\chi,s)=\zeta(\widehat{f},\overline{\chi},1-s) $$
memo
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.