Siegel modular forms
In this section we prove Theorem. 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
\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}{}.
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
g \langle Z \rangle := (AZ+B)(CZ+D)^{-1} \qquad \mbox{ where } g =
\mat{A}{B}{C}{D} \in \GSp^+(4,\R), Z \in \HH_2.
Let us define the slash operator |_k for a positive integer k
acting on holomorphic functions F on \HH_2 by
(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.
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
\Gamma^{(2)} M \Gamma^{(2)} = \bigsqcup\limits_i \Gamma^{(2)}
M_i.
For F \in S_k^{(2)}, define
T_k(\Gamma^{(2)} M \Gamma^{(2)})F :=
\det(M)^{\frac{k-3}2}\sum\limits_i F|_kM_i.
Note that this operator agrees with the one defined in.
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
\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}},
where \Gamma^{(2)} M \Gamma^{(2)} = \bigsqcup_i \Gamma^{(2)}
M_i, with
M_i = \mat{A_i}{B_i}{0}{D_i} \quad \mbox{ and } \quad D_i =
\begin{bmatrix}p^{d_{i1}}\amp \ast\\0\amp p^{d_{i2}}
\end{bmatrix} .
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
\alpha_0^2 \alpha_1 \alpha_2 = p^{2k-3}.
For any n > 0, define the Hecke operators T_k(n) by
T_k(n) = \sum\limits_{\lambda(M)=n} T_k(\Gamma^{(2)} M \Gamma^{(2)}).
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
\mu_F(p)\amp = p^{k-3/2} \big(\alpha_p + \alpha_p^{-1} + \beta_p + \beta_p^{-1}\big),
\mu_F(p^2)\amp =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) .
The Ramanujan bound in this context is
|\alpha_p| = |\beta_p| = 1.
This is closely related to our use of that term for L-functions, as can be seen from the spin L-function ofF:
L(s,F,\spin)=\prod_p F_p(p^{-s},\spin)^{-1},
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
\Lambda(s,F,\spin) :=\mathstrut\amp \Gamma_\C(s+\tfrac12)\Gamma_\C(s+k-\tfrac32) L(s,F,\spin)
=\mathstrut\amp \varepsilon \Lambda(s,\overline{F},\spin),
where \varepsilon=(-1)^k.
Let a(p) be the pth Dirichlet coefficient of L(s,F,\spin). We will use the fact that each F falls into one of two classes.
- 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. Note that |\beta_p|=1, so that a(p)=p^{1/2}+O(1) in the Saito-Kurokawa case.
- a(p)=O(1). This is the Ramanujan conjecture for non-Saito-Kurokawa liftings, which has been proven in.
Theorem is now a consequence of the following stronger result.
Suppose F_j, for j=1,2, are Siegel Hecke eigenforms of weight k_j for \Sp(4,\Z), with Hecke eigenvalues \mu_j(n). If
\sum_{p\le X} p\,\log(p) \left|p^{3/2-k_1}\mu_1(p)-p^{3/2-k_2}\mu_2(p)\right|^2\ll X
as X\to\infty, then k_1=k_2 and F_1 and F_2 have the same eigenvalues for the Hecke operator T(n) for alln.
For i=1,2 let a_i(p) be the pth 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 . By ,
\mu_i(p)= p^{k_i-3/2} \big(\alpha_{i,p} + \alpha_{i,p}^{-1} + \beta_{i,p} + \beta_{i,p}^{-1}\big).
Hence, condition translates into
\sum_{p\le X} p\,\log(p) \left|a_1(p)-a_2(p)\right|^2\ll X.
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 we obtain
\sum_{p\le X} p\,\log(p) \left|b_1(p)-b_2(p)\right|^2\ll X,
where b_{i,p}=\beta_{i,p}+\beta_{i,p}^{-1}. Note that b_{i,p}
is the pth Dirichlet coefficient of (the analytically normalized
L-function) L(s,f_i). Since the Ramanujan conjecture is known
for elliptic modular forms, Theorem 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 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.
Hyperelliptic curves
Let X/\Q be an elliptic or hyperelliptic curve,
X:\ y^2 = f(x),
where f\in \Z[x], and let N_X(p) be the number of points
on X modp. In Serre's recent book, the title
of Section6.3 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,
L(X,s) = \sum_{n=1}^\infty \frac{a_X(n)}{n^s},
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
\Lambda(X,s) = N_X^{s/2}\,\Gamma_\C(s)^{g_X} L(X,s) = \pm \Lambda(X,2-s),
where N_X is the conductor and
g_X= \lfloor ({\rm deg}(f)-1)/2 \rfloor is the genus ofX.
Suppose X and Y are hyperelliptic curves and N_X(p)-N_Y(p)
is bounded. If the Hasse-Weil L-functions of X and Y satisfy
their conjectured functional equation,
then X and Y have the same conductor and genus, and
N_X(p^e)=N_Y(p^e) for all p,e.
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.
To apply Theorem, we first form the
analytically normalized L-function
L(s,X)=L(X,s+\tfrac12) =\sum \frac{a_X(n)/\sqrt{n}}{n^{s}}
=\sum \frac{b_X(n)}{n^{s}},
say. Note that we have the functional equation
\Lambda(s,X) = N_X^{s/2} \Gamma_\C(s+\tfrac12)^{g_X} L(s,X) = \pm \Lambda(1-s,X).
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
|b_X(p) - b_Y(p)|\ll \frac{1}{\sqrt{\mathstrut p}},
which implies
\sum_{p\le T} p|b_X(p) - b_Y(p)|^2 \log(p) \ll
\sum_{p\le T} \log(p) \sim T,
by the prime number theorem.
Thus, Theorem 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 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 allp.
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