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Section 3Applications

Subsection 3.1Strong multiplicity one for $\GL(n)$

Let $\pi=\otimes\pi_p$ and $\pi'=\otimes\pi_p'$ be cuspidal automorphic representations of the group $\GL(n,\A_\Q)$. For a finite prime $p$ for which $\pi_p$ and $\pi_p'$ are both unramified, let $A(\pi_p)$ (resp. $A(\pi_p')$) represent the semisimple conjugacy class in $\GL(n,\C)$ corresponding to $\pi_p$ (resp. $\pi_p'$). The strong multiplicity one theorem for $\GL(n)$ states that if $A(\pi_p)=A(\pi_p')$ for almost all $p$, then $\pi=\pi'$. The following result implies, in particular, that the equality of traces ${\rm tr}(A(\pi_p))={\rm tr}(A(\pi_p'))$ for almost all $p$ is sufficient to reach the same conclusion. The traces could even be different at every prime, if those differences decreased sufficiently rapidly as a function of $p$.

Proof

We apply Theorem 2.1.2 to $ L_1(s)=L_{{\rm fin}}(s,\pi)$ and $L_2(s)=L_{{\rm fin}}(s,\pi')$. The condition on the spectral parameters $\Re(\mu_j), \Re(\nu_j)>-\frac12$ is satisfied by Proposition 2.1 of [BR]. By [BG], the partial symmetric square $L$-function for $\GL(n)$ has meromorphic continuation to all of $\C$ and only finitely many poles in $\sigma \geq 1$. Using the fact that the partial Rankin-Selberg $L$-function of a representation of $\GL(n)$ with itself has no zeros in $\sigma \geq 1$ (see [Sh]) and that the partial exterior square $L$-function of $\GL(n)$ has only finitely many poles (see [Be]), we see that partial symmetric square $L$-function for $\GL(n)$ has only finitely many zeros in $\sigma \geq 1$. This gives us condition 2.2 of Theorem 2.1.2. The conclusion of Theorem 2.1.2 is that $L_1(s)=L_2(s)$. By the familiar strong multiplicity one theorem for $\GL(n)$, this implies $\pi_1=\pi_2$.

Subsection 3.2Siegel modular forms

In this section we prove Theorem 1.3. We start by giving some background on Siegel modular forms for $\Sp(4,\Z)$. Let the symplectic group of similitudes of genus $2$ be defined by

\begin{multline}\GSp(4) := \{g \in \GL(4) : {}^{t}g J g = \lambda(g) J, \lambda(g) \in \GL(1) \}
\mbox{ where } J = \mat{}{I_2}{-I_2}{}. \end{multline}

Let $\SSp(4)$ be the subgroup with $\lambda(g)=1$. The group $\GSp^+(4,\R) := \{ g \in \GSp(4,\R) : \lambda(g) > 0 \}$ acts on the Siegel upper half space $\HH_2 := \{ Z \in M_2(\C) : {}^{t}Z = Z,\:{\rm Im}(Z) > 0 \}$ by

\begin{multline}g \langle Z \rangle := (AZ+B)(CZ+D)^{-1}
\mbox{ where } g = \mat{A}{B}{C}{D} \in \GSp^+(4,\R), Z \in \HH_2. \end{multline}

Let us define the slash operator $|_k$ for a positive integer $k$ acting on holomorphic functions $F$ on $\HH_2$ by

\begin{multline} (F|_kg)(Z) := \lambda(g)^k \det(CZ+D)^{-k} F(g \langle Z \rangle),
g = \mat{A}{B}{C}{D} \in \GSp^+(4,\R), Z \in \HH_2.\tag{3.2.1} \end{multline}

The slash operator is defined in such a way that the center of $\GSp^+(4,\R)$ acts trivially. Let $S_k^{(2)}$ be the space of holomorphic Siegel cusp forms of weight $k$, genus $2$ with respect to $\Gamma^{(2)} := \SSp(4,\Z)$. Then $F \in S_k^{(2)}$ satisfies $F |_k \gamma = F$ for all $\gamma \in \Gamma^{(2)}$.

Let us now describe the Hecke operators acting on $S_k^{(2)}$. For a matrix $M \in \GSp^+(4,\R) \cap M_{4}(\Z)$, we have a finite disjoint decomposition

\begin{equation}\Gamma^{(2)} M \Gamma^{(2)} = \bigsqcup\limits_i \Gamma^{(2)} M_i.\tag{3.2.2} \end{equation}

For $F \in S_k^{(2)}$, define

\begin{equation} T_k(\Gamma^{(2)} M \Gamma^{(2)})F := \det(M)^{\frac{k-3}2}\sum\limits_i F|_kM_i.\tag{3.2.3} \end{equation}

Note that this operator agrees with the one defined in [An]. Let $F \in S_k^{(2)}$ be a simultaneous eigenfunction for all the $T_k(\Gamma^{(2)} M \Gamma^{(2)}), M \in \GSp^+(4,\R) \cap M_{4}(\Z)$, with corresponding eigenvalue $\mu_F(\Gamma^{(2)} M \Gamma^{(2)})$. For any prime number $p$, it is known that there are three complex numbers $\alpha_0^F(p), \alpha_1^F(p), \alpha_2^F(p)$ such that, for any $M$ with $\lambda(M) = p^r$, we have

\begin{equation} \mu_F(\Gamma^{(2)} M \Gamma^{(2)}) =\alpha_0^F(p)^r \sum_i \prod\limits_{j=1}^2 (\alpha_i^F(p)p^{-j})^{d_{ij}},\tag{3.2.4} \end{equation}

where $\Gamma^{(2)} M \Gamma^{(2)} = \bigsqcup_i \Gamma^{(2)} M_i$, with

\begin{equation}M_i = \mat{A_i}{B_i}{0}{D_i} \quad \mbox{ and } \quad D_i = \begin{bmatrix}p^{d_{i1}}&\ast\\0&p^{d_{i2}}\end{bmatrix}.\tag{3.2.5} \end{equation}

Henceforth, if there is no confusion, we will omit the $F$ and $p$ in describing the $\alpha_i^F(p)$ to simplify the notations. The $\alpha_0, \alpha_1, \alpha_2$ are the classical Satake $p$-parameters of the eigenform $F$. It is known that they satisfy

\begin{equation} \alpha_0^2 \alpha_1 \alpha_2 = p^{2k-3}.\tag{3.2.6} \end{equation}

For any $n > 0$, define the Hecke operators $T_k(n)$ by

\begin{equation}T_k(n) = \sum\limits_{\lambda(M)=n} T_k(\Gamma^{(2)} M \Gamma^{(2)}). \end{equation}

Let the eigenvalues of $F$ corresponding to $T_k(n)$ be denoted by $\mu_F(n)$. Set $\alpha_p =p^{-(k-3/2)}\alpha_0$ and $\beta_p = p^{-(k-3/2)}\alpha_0 \alpha_1$. Then formulas for the Hecke eigenvalues $\mu_F(p)$ and $\mu_F(p^2)$ in terms of $\alpha_p$ and $\beta_p$ are

\begin{align}\mu_F(p)&= p^{k-3/2} \big(\alpha_p + \alpha_p^{-1} + \beta_p + \beta_p^{-1}\big),\\ \mu_F(p^2)&=p^{2k-3}\big(\alpha_p^2+\alpha_p^{-2}+(\alpha_p+\alpha_p^{-1})(\beta_p+\beta_p^{-1})+\beta_p^2+\beta_p^{-2}+2-\frac1p\big) .\tag{3.2.7} \end{align}

The Ramanujan bound in this context is

\begin{equation} |\alpha_p| = |\beta_p| = 1.\tag{3.2.8} \end{equation}

This is closely related to our use of that term for $L$-functions, as can be seen from the spin $L$-function of $F$:

\begin{equation} L(s,F,\spin)=\prod_p F_p(p^{-s},\spin)^{-1}, \end{equation}

where $F_p(X,\spin)=(1 - \alpha_p X)(1 - \alpha_p^{-1} X)(1 - \beta_p X)(1 - \beta_p^{-1} X)$. It satisfies the functional equation

\begin{align} \Lambda(s,F,\spin) :=\mathstrut&\Gamma_\C(s+\tfrac12)\Gamma_\C(s+k-\tfrac32) L(s,F,\spin)\cr=\mathstrut& \varepsilon \Lambda(s,\overline{F},\spin),\tag{3.2.9} \end{align}

where $\varepsilon=(-1)^k$.

Let $a(p)$ be the $p$th Dirichlet coefficient of $L(s,F,\spin)$. We will use the fact that each $F$ falls into one of two classes.

  1. $a(p)=p^{1/2}+p^{-1/2}+\beta_p+\beta_p^{-1}$, where $\beta_p$ is the Satake $p$-parameter of a holomorphic cusp form on $\GL(2)$ of weight $2k-2$. In this case $F$ is a Saito-Kurokawa lifting; for more details on Saito-Kurokawa liftings we refer to [EZ]. Note that $|\beta_p|=1$, so that $a(p)=p^{1/2}+O(1)$ in the Saito-Kurokawa case.
  2. $a(p)=O(1)$. This is the Ramanujan conjecture for non-Saito-Kurokawa liftings, which has been proven in [W].

Theorem 1.3 is now a consequence of the following stronger result.

Proof

For $i=1,2$ let $a_i(p)$ be the $p$th Dirichlet coefficient of $L(s,F_i,\spin)$. Then $a_i(p)=\alpha_{i,p}+\alpha_{i,p}^{-1}+\beta_{i,p}+\beta_{i,p}^{-1}$, where $\alpha_{i,p},\beta_{i,p}$ are the Satake $p$-parameters of $F_i$, as explained after (3.2.6). By (3.2.7),

\begin{equation} \mu_i(p)= p^{k_i-3/2} \big(\alpha_{i,p} + \alpha_{i,p}^{-1} + \beta_{i,p} + \beta_{i,p}^{-1}\big). \end{equation}

Hence, condition (3.2.10) translates into

\begin{equation} \sum_{p\le X} p\,\log(p) \left|a_1(p)-a_2(p)\right|^2\ll X.\tag{3.2.11} \end{equation}

From the remarks made before the theorem, we see that either $F_1,F_2$ are both Saito-Kurokawa lifts, or neither of them is a Saito-Kurokawa lift.

Assume first that $F_1,F_2$ are both Saito-Kurokawa lifts. Then, for $i=1,2$, there exist modular forms $f_i$ of weight $2k_i-2$ and with Satake parameters $\beta_{i,p}$ such that $a_i(p)=p^{1/2}+p^{-1/2}+\beta_{i,p}+\beta_{i,p}^{-1}$. From (3.2.11) we obtain

\begin{equation} \sum_{p\le X} p\,\log(p) \left|b_1(p)-b_2(p)\right|^2\ll X,\tag{3.2.12} \end{equation}

where $b_{i,p}=\beta_{i,p}+\beta_{i,p}^{-1}$. Note that $b_{i,p}$ is the $p$th Dirichlet coefficient of (the analytically normalized $L$-function) $L(s,f_i)$. Since the Ramanujan conjecture is known for elliptic modular forms, Theorem 2.1.2 applies. We conclude $2k_1-2=2k_2-2$ and $L(s,f_1)=L(s,f_2)$. Hence $k_1=k_2$ and $L(s,F_1,\spin)=L(s,F_2,\spin)$. The equality of spin $L$-functions implies $\mu_1(p)=\mu_2(p)$ and $\mu_1(p^2)=\mu_2(p^2)$ for all $p$. Since $T(p)$ and $T(p^2)$ generate the $p$-component of the Hecke algebra, it follows that $\mu_1(n)=\mu_2(n)$ for all $n$.

Now assume that $F_1$ and $F_2$ are both not Saito-Kurokawa lifts. Then, using the fact that the Ramanujan conjecture is known for $F_1$ and $F_2$, Theorem 2.1.2 applies to $L_1(s)=L(s,F_1,\spin)$ and $L_2(s)=L(s,F_2,\spin)$. We conclude that $k_1=k_2$ and that the two spin $L$-functions are identical. As above, this implies $\mu_1(n)=\mu_2(n)$ for all $n$.

Subsection 3.3Hyperelliptic curves

Let $X/\Q$ be an elliptic or hyperelliptic curve,

\begin{equation} X:\ y^2 = f(x), \end{equation}

where $f\in \Z[x]$, and let $N_X(p)$ be the number of points on $X$ mod $p$. In Serre's recent book [Ser], the title of Section 6.3 is "About $N_X(p)-N_Y(p)$," in which he gives a description of what can happen if $N_X(p)-N_Y(p)$ is bounded. For Serre, $X$ and $Y$ are much more general than hyperelliptic cuves, but we use the hyperelliptic curve case to illustrate an application of multiplicity one results for $L$-functions.

Recall that the Hasse-Weil $L$-function of $X$,

\begin{equation} L(X,s) = \sum_{n=1}^\infty \frac{a_X(n)}{n^s}, \end{equation}

has coefficients $a_X(p^n)=p^n+1-N_X(p^n)$ if $p$ is prime, which gives the general case by multiplicativity. The $L$-function (conjecturally if $g_X\ge 2$) satisfies the functional equation

\begin{equation} \Lambda(X,s) = N_X^{s/2}\,\Gamma_\C(s)^{g_X} L(X,s) = \pm \Lambda(X,2-s),\tag{3.3.1} \end{equation}

where $N_X$ is the conductor and $g_X= \lfloor ({\rm deg}(f)-1)/2 \rfloor$ is the genus of $X$.

Note that this result can be found in Serre's book without the hypothesis of functional equation. But Serre's proof involves more machinery than we use here.

Proof

To apply Theorem 2.1.2, we first form the analytically normalized $L$-function

\begin{equation}L(s,X)=L(X,s+\tfrac12) =\sum \frac{a_X(n)/\sqrt{n}}{n^{s}} =\sum \frac{b_X(n)}{n^{s}},\tag{3.3.2} \end{equation}

say. Note that we have the functional equation

\begin{equation} \Lambda(s,X) = N_X^{s/2} \Gamma_\C(s+\tfrac12)^{g_X} L(s,X) = \pm \Lambda(1-s,X).\tag{3.3.3} \end{equation}

The Hasse bound for $a_X(n)$ implies the Ramanujan bound for $L(s,X)$. The condition $|N_X(p)-N_Y(p)| \ll 1$ is equivalent to

\begin{equation}|b_X(p) - b_Y(p)|\ll \frac{1}{\sqrt{\mathstrut p}},\tag{3.3.4} \end{equation}

which implies

\begin{equation}\sum_{p\le T} p|b_X(p) - b_Y(p)|^2 \log(p) \ll \sum_{p\le T} \log(p) \sim T,\tag{3.3.5} \end{equation}

by the prime number theorem. Thus, Theorem 2.1.2 applies and we conclude that $L(X,s)=L(Y,s)$.

If one knew that $L(s,X)$ and $L(s,Y)$ were "automorphic", then Theorem 4.1 would apply, and a much weaker bound on $|N_X(p)-N_Y(p)|$ would allow one to conclude that $N_X(p^e)=N_Y(p^e)$ for all $p,e$. For example, if $E,E'$ are elliptic curves over $\Q$, then $|N_{E}(p)-N_{E'}(p)|\le 1.4 \sqrt{\mathstrut p}$ for all but finitely many $p$ implies $N_{E}(p)=N_{E'}(p)$ for all $p$.

\appendix

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