8
$\begingroup$

Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion

$$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$

and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke operators $T_p$, not just those with $(p,N) = 1$, and for the normalized L-function

$$L^{S_{\infty}}(f,s) = \sum\limits_{n=1}^{\infty} \frac{n^{(1-k)/2}a_n}{n^s}$$

one can add in an archimedean factor to make a completed L-function $L(f,s)$ which is an Euler product

$$L(f,s) = \prod\limits_{p \leq \infty} L_p(f,s)$$

and, for a suitable "contragredient" $g \in S_k(\Gamma_0(N))$ and epsilon root number $\epsilon(f,s)$, satisfies the functional equation

$$L(f,s) = \epsilon(f,s) L(g,1-s).$$

See for example Kowalski's article "Classical Automorphic Forms" in the book "An Introduction to the Langlands Program."

Now, $f$ can be identified in various ways with a cuspidal automorphic form on $\operatorname{GL}_2(\mathbb Q) \backslash \operatorname{GL}_2(\mathbb A_{\mathbb Q})$. There is a cuspidal automorphic representation $\pi = \otimes \pi_p$, unique up to isomorphism, which contains $f$. The normalized newform $f$ is conversely uniquely determined by $\pi$. This is one way to state the "Multiplicity One" theorem.

Also, $\pi$ itself has an L-function $L(\pi,s) = \prod\limits_{p \leq \infty} L(\pi_p,s)$ satisfying its own functional equation $L(\pi,s) = \epsilon(\pi,s) L(\tilde{\pi},1-s)$.

At the primes $p$ not dividing $N$, the representation $\pi_p$ is spherical, and the L-function $L(\pi_p,s)$ arises from the local spherical Hecke algebra of $\operatorname{GL}_2(\mathbb Q_p)$. Here things can be normalized so that

$$L(\pi_p,s) = L_p(f,s). \tag{1} $$ What about the primes dividing $N$? Here $L_p(f,s)$ arises from a Hecke operator $T_p$, but the representation $\pi_p$ could be supercuspidal, or it could have an Iwahori-fixed vector but be non-spherical. With the proper normalizations, can we have equation (1) holding for all primes $p$ (and consequently $L(\pi,s) = L(f,s)$)? Note that when $\pi_p$ is supercuspidal or is induced from a ramified character, $L(\pi_p,s) = 1$.

I expect it's possible to normalize things so that the local factors of $L(f,s)$ and $L(\pi,s)$ agree at all places, but I have never seen any reference which does this.

$\endgroup$
  • $\begingroup$ You know the exact functional equation satisfied by $L(f,s)$ from the Fricke involution, you want $L(s,\pi)$ to satisfy a similar one, which implies they are equal (since changing finitely primes adds some Euler factors in the functional equation), morally being left invariant under $GL_2(\Bbb{Q})$ is really the functional equation. $\endgroup$ – reuns Dec 26 '19 at 16:00
9
$\begingroup$

$L(\pi,s)$ agrees with $L(f,s)$ if $f\in\pi$ is a newform, and this is even true for $\mathrm{GL}_n$. Of course, things are complicated by the fact that there are many ways to define $L(\pi,s)$ and $L(f,s)$. I found the following papers very useful to check various consistencies: Schmidt and Kondo-Yasuda.

Here is a summary based on Kondo-Yasuda. Let $G:=\mathrm{GL}_n$. Let $(\pi,V_\pi)$ be a cuspidal automorphic representation of $G$ over $\mathbb{Q}$ of unitary central character $\omega$. Let $K_p(c)$ denote the subgroup of elements of $G(\mathbb{Z}_p)$ whose bottom row is congruent to $\begin{pmatrix}0 & \cdots & 0 & 1\end{pmatrix}$ modulo $c$. Let $K(c):=\prod_p K_p(c)$. The conductor $c_\pi$ is the smallest $c$ such that $V_\pi^{K(c)}\neq\{0\}$. An element of $V_\pi^{K(c_\pi)}$ is called a global newform: it is an eigenfunction of the convolution by any element of $C_c(K_p(c_\pi)\backslash G(\mathbb{Q}_p)/K_p(c_\pi))$. Let $c_\omega$ be the conductor of $\omega$, and consider the corresponding primitive Dirichlet character modulo $c_\omega$: $$\chi_\omega(m):=\omega(1,\underbrace{1,\dotsc,1}_{v\mid c_\omega},\underbrace{m,m,\dotsc}_{v\nmid c_\omega})= \omega(1,\underbrace{m^{-1},\dotsc,m^{-1}}_{v\mid c_\omega},\underbrace{1,1,\dotsc}_{v\nmid c_\omega}),$$ for $m>0$ and $(m,c_\omega)=1$. Note that $c_\omega\mid c_\pi$. Let $\chi_\pi$ denote the Dirichlet character modulo $c_\pi$ induced by $\chi_\omega$. For $k\in\{1,\dotsc,n-1\}$, we define the $k$-th Hecke operator at $p$ as the characteristic function of $$H_k(p):=K_p(c_\pi)\,\mathrm{diag}(\underbrace{p,\dotsc,p}_\text{$k$ entries},\underbrace{1,\dotsc, 1}_\text{$n-k$ entries})\,K_p(c_\pi).$$ Accordingly, let $\lambda_{\pi,k}(p)$ be the $k$-th normalized Hecke eigenvalue at $p$: $$\int_{H_k(p)}f(xg)\,dg=\lambda_{\pi,k}(p)p^{\frac{k(n-k)}{2}}\mathrm{vol}(K_p)f(x),\qquad f\in V_\pi^{K(c_\pi)}.$$ Further, let $\lambda_{\pi,0}(p):=1$ and $\lambda_{\pi,n}(p):=\chi_\pi(p)$.

Theorem (Tamagawa 1963, Satake 1963, Kondo-Yasuda 2010). $$L(\pi,s)=\prod_p\left(\sum_{k=0}^n(-1)^k\lambda_{\pi,k}(p)p^{-ks}\right)^{-1}.$$

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Thanks very much for your answer. I assume the archimedean equality $L_{\infty}(f,s) = L(\pi_{\infty},s)$ is easy to check ($\pi_{\infty}$ for a newform should be like a discrete series representation) $\endgroup$ – D_S Dec 25 '19 at 21:49
  • 1
    $\begingroup$ @D_S: Yes. As a rule of thumb, as long as the $L$-function "looks good", i.e., it has an Euler product decomposition and a functional equation of the usual shape, "it is good". If the definition is not OK, it will show up as a problem in the analytic properties. $\endgroup$ – GH from MO Dec 25 '19 at 22:44
  • $\begingroup$ For $L(\pi_\infty,s)$ I recommend Tate's article "Number theoretic background" in Volume 2 of Borel-Casselman (eds.): Automorphic forms, representations, and $L$-functions. $\endgroup$ – GH from MO Dec 25 '19 at 22:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.