An \(L\)-function is a Dirichlet series that converges absolutely in some right half plane, has a meromorphic continuation to a function of order \(1\), with finitely many poles, satisfies a functional equation, and admits an Euler product. For example, the (incomplete) \(L\)-functions attached to tempered, cuspidal automorphic representations, or the Hasse-Weil \(L\)-functions attached to non-singular, projective, algebraic varieties defined over a number field, conjecturally satisfy these conditions.
In this paper, using standard techniques from analytic number theory, we prove a strong multiplicity one result for such \(L\)-functions (without reference to any underlying automorphic or geometric object). We closely follow the work of Kaczorowski and Perelli [12] and we redo their arguments for two reasons. First, our results are more general in that they have slightly weaker hypotheses. Second, we think that the techniques should be better known, especially to those who study \(L\)-functions automorphically.
One of the defining axioms for the class of \(L\)-functions we consider is the existence of an Euler product. There exists a number \(d\), called the degree of the \(L\)-function, such that the local Euler factor is of the form \(Q_p(p^{-s})^{-1}\), where \(Q_p(X)\) is a polynomial satisfying \(Q_p(0)=1\), and \(Q_p(X)\) has degree \(d\) for almost all primes. We say that a given \(L\)-function satisfies the Ramanujan conjecture, if the roots of \(Q_p\) are of absolute value at least \(1\), for all \(p\).
The multiplicity one results we discuss in this paper are statements which assert that if two \(L\)-functions are sufficiently close, then they must be equal. A model example is:
Suppose \(L_1(s)=\sum a_1(n)n^{-s}\) and \(L_2(s)=\sum a_2(n)n^{-s}\) are Dirichlet series which continue to meromorphic functions of order \(1\) satisfying appropriate functional equations and having appropriate Euler products. Assume that \(L_1(s)\) and \(L_2(s)\) satisfy the Ramanujan conjecture. Assume also that \(a_1(p)=a_2(p)\) for almost all \(p\). Then \(L_1(s)=L_2(s)\).
The precise conditions on the functional equation and Euler product are described in Section 2.1. A weaker version of Theorem 1.1, requiring equality of the local Euler factors instead of the \(p\)th Dirichlet coefficients, is given in [24]. Theorem 1.1 is also a consequence of the main result in [12]. The result we will actually prove, Theorem 2.2.1, is stronger. First, instead of requiring equality of the \(p\)th Dirichlet series coefficients, we only require that they are close on average. Second, the Ramanujan hypothesis can be slightly relaxed.
We will present three applications of strong multiplicity one for \(L\)-functions. The first application is to cuspidal automorphic representations of \(\GL(n,\A_\Q)\), where \(\A_\Q\) denotes the ring of adeles of the number field \(\Q\). Any such representation \(\pi\) factors as \(\pi=\otimes\pi_p\), where \(\pi_p\) is an irreducible admissible representation of \(\GL(n,\Q_p)\) (we mean \(\Q_p=\R\) for \(p=\infty\)). Attached to \(\pi\) is an automorphic \(L\)-function \(L(s,\pi)\), whose finite part is \(L_{{\rm fin}}(s,\pi) = \prod_{p < \infty} L(s, \pi_p)\). The completion of \(L_{{\rm fin}}(s,\pi)\) is known to be "nice", and hence \(L_{{\rm fin}}(s,\pi)\) is the kind of function to which Theorem 1.1 applies. At almost all primes \(p\) we have \(L(s,\pi_p)=\det(1-A(\pi_p)p^{-s})^{-1}\), where \(A(\pi_p)={\rm diag}(\alpha_{1,p},\ldots,\alpha_{n,p})\) is a diagonal matrix whose entries are the Satake parameters at \(p\). The Ramanujan conjecture is the assertion that each \(\pi_p\) is tempered, which in this context implies that \(|\alpha_{j,p}|=1\). In particular note that \(L(s,\pi_p)\) is a polynomial in \(p^{-s}\) and this polynomial has all its roots on the unit circle.
An easy consequence of Theorem 1.1 is the following.
Suppose that \(\pi\) and \(\pi'\) are (unitary) cuspidal automorphic representations of \(\GL(n,\A_\Q)\) satisfying \(\tr(A(\pi_p))=\tr(A(\pi_p'))\) for almost all \(p\). Assume that both \(L_{{\rm fin}}(s,\pi)\) and \(L_{{\rm fin}}(s,\pi')\) satisfy the Ramanujan conjecture. Then \(\pi=\pi'\).
Most statements of strong multiplicity one in the literature are phrased in terms of \(A(\pi_p)\) and \(A(\pi_p')\) being conjugate, instead of the much weaker condition of the equality of their traces. Using the stronger version of Theorem 1.1, we will in fact prove a stronger result which only requires that the traces are close enough on average; see Theorem 3.1.1 for the precise statement.
Our second application is to Siegel modular forms of degree \(2\). For such modular forms Weissauer [31] has proved the Ramanujan conjecture. The Dirichlet coefficients \(a_i(p)\) appearing in Theorem 1.1 are essentially the Hecke eigenvalues for the Hecke operator \(T(p)\). We therefore have the following:
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 \(p^{3/2-k_1}\mu_1(p)=p^{3/2-k_2}\mu_2(p)\) for all but finitely many \(p\), then \(k_1 = k_2\) and \(F_1, F_2\) have the same eigenvalues for the Hecke operator \(T(n)\) for all \(n\).
The remarkable fact here is that the Hecke operator \(T(p)\) alone does not generate the local Hecke algebra at \(p\). This Hecke algebra is generated by \(T(p)\) and \(T(p^2)\). The fact that the coincidence of the eigenvalues for \(T(p)\) is enough is of course a global phenomenon. Using the result of [25], we see that if Böcherer's conjecture is true then not only are the Hecke eigenvalues of \(F_1, F_2\) in Theorem 1.3 equal but we get \(F_1 = F_2\). Again, using the averaged version of Theorem 2.2.1, we can prove a stronger result; see Theorem 3.2.1.
Our third application concerns the Hasse-Weil zeta function of hyperelliptic curves; see Proposition 3.4.1. This Proposition is in a similar spirit to those mentioned above. Assuming the \(L\)-functions satisfy a functional equation of a form they are expected to satisfy, we can apply our analytic theorems to prove a result about the underlying (in this case) geometric object.
Now that we have described the consequences of our main result, we briefly compare our result to other "analytic" strong multiplicity one results due to Murty and Murty [18], Murty [17], Kaczorowski and Perelli [12] and Kaczorowski [11]. Roughly speaking, these earlier versions of strong multiplicity one place more and more conditions on the two \(L\)-function. In [18], one can find the hypothesis that \(a_1(p)=a_2(p)\) and \(a_1(p^2)=a_2(p^2)\), in [17] it is assumed that \(a_1(p)=a_2(p)\) and some conditions on the twists of the \(L\)-functions. Later, in [12], the conditions on the twists of the \(L\)-functions is replaced with the assumption that the \(L\)-functions have Euler products of a form similar to what we assume.
We highlight a few notable differences between these earlier versions of strong multiplicity one and ours, making a more thorough comparison between the version by Kaczorowski [11] as it is the most recent and is closest to our version. There are a couple of ways in which our version is stronger. First, in [11] it is assumed that the \(L\)-functions satisfy the Ramanujan bound while we only assume a partial Ramanujan bound. Second, [11] requires the coefficients to be close in the sense \(|a_1(p)-a_2(p)|\ll p^{-\frac12 -\delta}\) for some \(\delta>0\), while we allow \(\delta=0\). In the proof, our result requires an analysis of possible zeros on the \(\sigma=\frac12\)-line, while this is avoided if \(\delta>0\). Finally, the main corollaries of our result do not follow if one requires \(\delta>0\). Despite these differences, the method of proof is similar, being closely based on [12].
The most salient difference between this paper and the papers with the earlier versions of strong multiplicity one is that we have applications in mind. We prove a number of, to us, very remarkable consequences of this analytic result, as indicated above. The applications are of broad interest, lie in different areas of number theory and we include the proofs of the analytic results for completeness and to expose these techniques to as broad an audience as possible.
We review some notation from analytic number theory for completeness. Given two functions \(f(x),\,g(x)\),
In this paper we drop the phrase "as \(x\to\infty\)" when using the above notation.