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Section 4Selberg orthogonality and strong multiplicity one for $\GL(n)$

The proof of Theorem 2.1.2 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 (3.1.1).

Using the fact that $1.4^2 < 2$ and the consequence of the prime number theorem

\begin{equation} \sum_{p\le X}\frac{1}{p} \sim \log\log(X),\tag{4.2} \end{equation}

we see that condition (4.1) holds if $\left| \tr A(\pi_p) - \tr A(\pi_p')\right| < 1.4$ for all but finitely many $p$. 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 (4.2) implies a version of a result of Ramakrishnan [Ram]: if $ \tr A(\pi_p) = \tr A(\pi_p')$ for $\frac78+\varepsilon$ of all primes $p$, then $\pi=\pi'$. This result was extended by Rajan [Raj].

The proof of Theorem 4.1 is a straightforward application of recent results toward the Selberg orthonormality conjecture [LWY, AvSm], which make use of progress on Rudnick and Sarnak's "Hypothesis H" [RudSar, K]. Suppose

\begin{equation}L_1(s)=\sum\frac{a(n)}{n^s},\qquad L_2(s)=\sum\frac{b(n)}{n^s}\tag{4.3} \end{equation}

are $L$-functions, meaning that they have a functional equation and Euler product as described in Section 2.1.

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 [Sel]. Recall that an $L$-function is primitive if it cannot be written nontrivially as a product of $L$-functions.

Proof of Theorem 4.1. Rudnick and Sarnak's Hypothesis H is the assertion

\begin{equation} \sum_p \frac{a(p^k)^2\log^2(p)}{p^k} < \infty \end{equation}

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

For the standard $L$-functions of cuspidal automorphic representations on $\GL(n)$, Rudnick and Sarnak [RudSar] proved Selberg's orthonormality conjecture under the assumption of Hypothesis H, and they proved Hypothesis H for $n=2$, $3$. The case of $n=4$ for Hypothesis H was proven by Kim [K]. Thus, under the conditions in Theorem 4.1, the Selberg orthonormality conjecture is true.

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 (4.4)

\begin{align} \sum_{p\le X} \frac{1}{p} |a(p)-b(p)|^2 =\mathstrut&\sum_{p\le X} \frac{1}{p} \bigl( |a(p|^2 + |b(p)|^2 - 2 \Re(a(p)\overline{b(p)})\bigr) \cr =\mathstrut& 2\log\log(X) - 2 \delta_{L_1, L_2} \log\log(X)+ O(1) \cr =& \begin{cases} O(1) & \text{ if } L_1= L_2 \cr 2\log\log(X) + O(1) & \text{ if } L_1\not = L_2. \cr \end{cases}\tag{4.5} \end{align}

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 (4.5) implies that $L_1(s) = L_2(s)$. This gives us $\pi = \pi'$. \qed

\vspace{2ex} Recently, the transfer of full level Siegel modular forms to $\GL(4)$ was obtained in [PSS]. Hence, we can apply Theorem 4.1 to the transfer to $\GL(4)$ of a Siegel modular form of full level and thus obtain a stronger version of Theorem 3.2.1.

\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.

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