Selberg orthogonality and strong multiplicity one for \(\GL(n)\)

The proof of Theorem used only standard techniques from analytic number theory. Utilizing recent results concerning the Selberg orthonormality conjecture, and restrictng to the case of L-functions of cuspidal automorphic representations of \GL(n), one obtains the following theorem, which is stronger than Theorem.

Suppose that \pi, \pi' are (unitary) cuspidal automorphic representations of \GL(n,\A_F), and suppose

\sum_{p\le X} \frac{1}{p}\left| \tr A(\pi_p) - \tr A(\pi_p')\right|^2 \le (2-\epsilon) \log\log(X)

for some \epsilon>0 as X\to\infty. If n\le 4, or if HypothesisH holds for both L_\mathrm{fin}(s,\pi) and L_\mathrm{fin}(s,\pi') (in particular if the partial Ramanujan conjecture \theta\lt \frac14 is true for \pi and \pi'), then \pi=\pi'.

Using the fact that 1.4^2\lt 2 and the consequence of the prime number theorem

\sum_{p\le X}\frac{1}{p} \sim \log\log(X),

we see that condition holds if \left| \tr A(\pi_p) - \tr A(\pi_p')\right|\lt 1.4 for all but finitely manyp. Thus, the strong multiplicity one theorem only requires considering the traces of\pi_p, and futhermore those traces can differ at every prime, and by an amount which is bounded below.

For GL(2,\A_\Q), the Ramanujan bound along with implies a version of a result of Ramakrishnan: if \tr A(\pi_p) = \tr A(\pi_p') for \frac78+\varepsilon of all primesp, then \pi=\pi'. This result was extended by Rajan.

The proof of Theorem is a straightforward application of recent results toward the Selberg orthonormality conjecture[15, 3], which make use of progress on Rudnick and Sarnak's ``Hypothesis H''[24, 13]. Suppose

L_1(s)=\sum\frac{a(n)}{n^s},\qquad L_2(s)=\sum\frac{b(n)}{n^s}

are L-functions, meaning that they have a functional equation and Euler product as described in Section.

Rudnick and Sarnak's Hypothesis H is the assertion

\sum_p \frac{a(p^k)^2\log^2(p)}{p^k} \lt \infty

for all k\ge 2. For a given k, this follows from a partial Ramanujan bound \theta\lt \frac12 - \frac{1}{2k}. Since k\ge 2, HypothesisH follows from the partial Ramanujan bound \theta\lt \frac14.

For the standard L-functions of cuspidal automorphic representations on \GL(n), Rudnick and Sarnak proved Selberg's orthonormality conjecture for pi\cong \pi' under the assumption of HypothesisH, and they proved HypothesisH for n=2, 3. The case of n=4 for HypothesisH was proven by Kim. For \pi\not\cong \pi' the corresponding result was proved by independently by Avdispahi\'c-Smajlovi\'c and Liu-Wang-Ye. Thus, under the conditions in Theorem, the Selberg orthonormality conjecture is true.

The point of the strong multiplicity one theorem is that two L-functions must either be equal, or else they must be far apart. The essential idea was elegantly described by Selberg; see. Recall that an L-function is primitive if it cannot be written nontrivially as a product of L-functions.

Selberg Orthonormality Conjecture

Suppose that L_1 and L_2 are primitive L-functions with Dirichlet coefficients a(p) and b(p). Then

\sum_{p\le X} \frac{a(p)\overline{b(p)}}{p}=\delta({L_1, L_2}) \log\log(X)+O(1),

where \delta({L_1, L_2}) = 1 if L_1=L_2, and 0 otherwise.

Since \pi and \pi' are cuspidal automorphic representations of \GL(n,\A_F), the L-functions L_1(s) = L_\mathrm{fin}(s,\pi) and L_2(s) = L_\mathrm{fin}(s, \pi') are primitive L-functions. Hence, by

\sum_{p\le X} \frac{1}{p} |a(p)-b(p)|^2 =\mathstrut\amp \sum_{p\le X} \frac{1}{p} \bigl( |a(p|^2 + |b(p)|^2 - 2 \Re(a(p)\overline{b(p)})\bigr) =\mathstrut\amp 2\log\log(X) - 2 \delta_{L_1, L_2} \log\log(X)+ O(1) =\amp \begin{cases}O(1) \amp \text{ if } L_1= L_2 \cr 2\log\log(X) + O(1) \amp \text{ if } L_1\not = L_2. \cr \end{cases}

We have \sum_{p\le X} \frac{1}{p} |a(p)-b(p)|^2 \le (2-\epsilon)\log\log(X) for some \epsilon > 0. This implies that \epsilon \log\log(X) is unbounded, and hence implies that L_1(s) = L_2(s). This gives us \pi = \pi'.

Recently, the transfer of full level Siegel modular forms to \GL(4) was obtained in. Hence, we can apply Theorem to the transfer to \GL(4) of a Siegel modular form of full level and thus obtain a stronger version of Theorem.

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} \frac 1p \left|p^{3/2-k_1}\mu_1(p)-p^{3/2-k_2}\mu_2(p)\right|^2 \le (2-\epsilon) \log\log(X)

for some \epsilon > 0, 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.

\proof[Acknowledgements] We thank Farrell Brumley and Abhishek Saha for carefully reading an earlier version of this paper and for providing useful feedback on it. We are also grateful for the careful reading done by the referee who, among other things, proposed some new applications of our theorem and helped us better organize the paper.