Multiplicity one for L -functions and applications
aa r X i v : . [ m a t h . N T ] M a y Multiplicity one for L -functions and applications David W. FarmerAmerican Institute of [email protected] Ameya PitaleUniversity of [email protected] C. RyanBucknell [email protected] Ralf SchmidtUniversity of [email protected] 4, 2018
Abstract
We give conditions for when two Euler products are the same giventhat they satisfy a functional equation and their coefficients satisfy a par-tial Ramanujan bound and do not differ by too much. Additionally, weprove a number of multiplicity one type results for the number-theoreticobjects attached to L -functions. These results follow from our main resultabout L -functions. An L -function is a Dirichlet series that converges absolutely in some right halfplane, has a meromorphic continuation to a function of order , with finitelymany poles, satisfies a functional equation, and admits an Euler product. Forexample, the (incomplete) L -functions attached to tempered, cuspidal automor-phic representations, or the Hasse-Weil L -functions attached to non-singular,projective, algebraic varieties defined over a number field, conjecturally satisfythese conditions.In this paper, using standard techniques from analytic number theory, weprove a strong multiplicity one result for such L -functions (without reference toany underlying automorphic or geometric object). We closely follow the work[10] and we redo their arguments for two reasons. First, our results are moregeneral in that they have slightly weaker hypotheses. Second, we think that thetechniques should be better known, especially to those who study L -functionsautomorphically.One of the defining axioms for the class of L -functions we consider is theexistence of an Euler product. There exists a number d , called the degree ofthe L -function, such that the local Euler factor is of the form Q p ( p − s ) − , where Q p ( X ) is a polynomial satisfying Q p (0) = 1 , and Q p ( X ) has degree d for almost1ll primes. We say that a given L -function satisfies the Ramanujan conjecture ,if the roots of Q p are of absolute value at least , for all p .The multiplicity one results we discuss in this paper are statements whichassert that if two L -functions are sufficiently close, then they must be equal. Amodel example is: Theorem 1.1.
Suppose L ( s ) = P a ( n ) n − s and L ( s ) = P a ( n ) n − s areDirichlet series which continue to meromorphic functions of order satisfyingappropriate functional equations and having appropriate Euler products. As-sume that L ( s ) and L ( s ) satisfy the Ramanujan conjecture. Assume also that a ( p ) = a ( p ) for almost all p . Then L ( s ) = L ( s ) . The precise conditions on the functional equation and Euler product aredescribed in Section 2.1. A weaker version of Theorem 1.1, requiring equality ofthe local Euler factors instead of the p th Dirichlet coefficients, is given in [20].Theorem 1.1 is also a consequence of the main result in [10]. The result we willactually prove, Theorem 2.2, is stronger. First, instead of requiring equalityof the p th Dirichlet series coefficients, we only require that they are close onaverage. 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 representa-tion π factors as π = ⊗ π p , where π p is an irreducible admissible representationof GL( n, Q p ) (we mean Q p = R for p = ∞ ). Attached to π is an automor-phic L -function L ( s, π ) , whose finite part is L fin ( s, π ) = Q p< ∞ L ( s, π p ) . Thecompletion of L fin ( s, π ) is known to be “nice”, and hence L fin ( s, π ) is the kindof function to which Theorem 1.1 applies. At almost all primes p we have L ( s, π p ) = det(1 − A ( π p ) p − s ) − , where A ( π p ) = diag( α ,p , . . . , α n,p ) is a di-agonal matrix whose entries are the Satake parameters at p . The Ramanujanconjecture is the assertion that each π p is tempered, which in this context im-plies that | α j,p | = 1 . In particular note that L ( s, π p ) is a polynomial in p − s andthis polynomial has all its roots on the unit circle.An easy consequence of Theorem 1.1 is the following. Theorem 1.2.
Suppose that π and π ′ are (unitary) cuspidal automorphic rep-resentations of GL( n, A Q ) satisfying tr( A ( π p )) = tr( A ( π ′ p )) for almost all p .Assume that both L fin ( s, π ) and L fin ( s, π ′ ) satisfy the Ramanujan conjecture.Then π = π ′ . Most statements of strong multiplicity one in the literature are phrased interms of A ( π p ) and A ( π ′ p ) being conjugate, instead of the much weaker conditionof the equality of their traces. Using the stronger version of Theorem 1.1, wewill in fact prove a stronger result which only requires that the traces are closeenough on average; see Theorem 3.1 for the precise statement.Our second application is to Siegel modular forms of degree . For such mod-ular forms Weissauer [27] has proved the Ramanujan conjecture. The Dirichletcoefficients a i ( p ) appearing in Theorem 1.1 are essentially the Hecke eigenvaluesfor the Hecke operator T ( p ) . We therefore have the following:2 heorem 1.3. Suppose F j , for j = 1 , , are Siegel Hecke eigenforms of weight k j for Sp(4 , Z ) , with Hecke eigenvalues µ j ( n ) . If p / − k µ ( p ) = p / − k µ ( p ) for all but finitely many p , then k = k and F , F have the same eigenvaluesfor the Hecke operator T ( n ) for all n . The remarkable fact here is that the Hecke operator T ( p ) alone does notgenerate the local Hecke algebra at p . This Hecke algebra is generated by T ( p ) and T ( p ) . The fact that the coincidence of the eigenvalues for T ( p ) is enough isof course a global phenomenon. Using the result of [21], we see that if Böcherer’sconjecture is true then not only are the Hecke eigenvalues of F , F in Theorem1.3 equal but we get F = F . Again, using the averaged version of Theorem2.2, we can prove a stronger result; see Theorem 3.2.Our third application concerns the Hasse-Weil zeta function of hyperellipticcurves; see Proposition 3.3. This Proposition is in a similar spirit to thosementioned above. Assuming the L -functions satisfy a functional equation of aform they are expected to satisfy, we can apply our analytic theorems to provea result about the underlying (in this case) geometric object. Notation
We review some notation from analytic number theory for completeness. Giventwo functions f ( x ) , g ( x ) , • we write f ( x ) ∼ g ( x ) as x → ∞ if lim x →∞ f ( x ) /g ( x ) = 1 ; • we write f ( x ) ≪ g ( x ) as x → ∞ if there exists C > and x > so thatif x > x then | f ( x ) | ≤ C | g ( x ) | ; this is also written as f ( x ) = O ( g ( x )) as x → ∞ .In this paper we drop the phrase “as x → ∞ ” when using the above notation. Acknowledgements
We thank Farrell Brumley and Abhishek Saha for carefully reading an earlierversion of this paper and for providing useful feedback on it. This work wassupported by an AIM SQuaRE. Ameya Pitale and Ralf Schmidt are supportedby National Science Foundation grant DMS 1100541. L -functions In this section we describe the L -functions for which we will prove a multiplicityone result. As in other approaches to L -functions viewed from a classical per-spective, such as that initiated by Selberg [23], we consider Dirichlet series witha functional equation and an Euler product. However, in contrast to Selberg, westrive to make all our axioms as specific as possible. Presumably (as conjecturedby Selberg) these different axiomatic approaches all describe the same objects: L ( s, π ) where π is a cuspidal automorphic representation of GL( n ) .3 .1 L -function background Before getting to L -functions, we recall two bits of terminology that will be usedin the following discussion. An entire function f : C → C is said to have orderat most α if for all ǫ > : f ( s ) = O (exp( | s | α + ǫ )) . Moreover, we say f has order equal to α if f has order at most α , and f doesnot have order at most γ for any γ < α . The notion of order is relevant becausefunctions of finite order admit a factorization as described by the HadamardFactorization Theorem and the Γ -function and L -functions are all of order 1.In order to ease notation, we use the normalized Γ -functions defined by: Γ R ( s ) := π − s/ Γ( s/ and Γ C ( s ) := 2(2 π ) − s Γ( s ) . An L -function is a Dirichlet series L ( s ) = ∞ X n =1 a ( n ) n s , (2.1)where s = σ + it is a complex variable. We assume that L ( s ) converges absolutelyin the half-plane σ > and has a meromorphic continuation to all of C . Theresulting function is of order , admitting at most finitely many poles, all ofwhich are located on the line σ = 1 . Finally, L ( s ) must have an Euler productand satisfy a functional equation as described below.The functional equation involves the following parameters: a positive integer N , complex numbers µ , . . . , µ J and ν , . . . , ν K , and a complex number ε . Thecompleted L -function Λ( s ) := N s/ J Y j =1 Γ R ( s + µ j ) K Y k =1 Γ C ( s + ν k ) · L ( s ) (2.2)is a meromorphic function of finite order, having the same poles as L ( s ) in σ > , and satisfying the functional equation Λ( s ) = ε Λ(1 − s ) . (2.3)The number d = J + 2 K is called the degree of the L -function.We require some conditions on the parameters µ j and ν j . The temperednesscondition is the assertion that ℜ ( µ j ) ∈ { , } and ℜ ( ν j ) a positive integer or half-integer. With those restrictions, there is only one way to write the parametersin the functional equation, as proved in Proposition 2.1. This restriction is notknown to be a theorem for most automorphic L -functions. In order to statetheorems which apply in those cases, we will make use of a “partial Selbergbound,” which is the assertion that ℜ ( µ j ) , ℜ ( ν j ) > − .4he Euler product is a factorization of the L -function into a product overthe primes: L ( s ) = Y p F p ( p − s ) − , (2.4)where F p is a polynomial of degree at most d : F p ( z ) = (1 − α ,p z ) · · · (1 − α d,p z ) . (2.5)If p | N then p is a bad prime and the degree of F p is strictly less than d , inother words, α j,p = 0 for at least one j . Otherwise, p is a good prime, in whichcase the α j,p are called the Satake parameters at p . The Ramanujan bound isthe assertion that at a good prime | α j,p | = 1 , and at a bad prime | α j,p | ≤ .The Ramanujan bound has been proven in very few cases, the most promi-nent of which are holomorphic forms on GL(2) and
GSp(4) . See [22] for a surveyof what progress is known towards proving the Ramanujan bound. Also see [6].We write | α j,p | ≤ p θ , for some θ < , to indicate progress toward theRamanujan bound, referring to this as a “partial Ramanujan bound.”We will need to use symmetric and exterior power L -functions associated toa L -function L ( s ) . Let S be the finite set of bad primes p of L ( s ) . The partialsymmetric and exterior square L -functions are defined as follows. L S ( s, sym n ) = Y p S Y i + ... + i d = n (1 − α i ,p . . . α i d d,p p − s ) − (2.6) L S ( s, ext n ) = Y p S Y ≤ i <...
Suppose that L j ( s ) = Q p F p,j ( p − s ) − , for j = 1 , , are L -functions which satisfy a partial Ramanujan bound for some θ < . If F p, = F p, for all but finitely many p , then F p, = F p, for all p , and L and L havethe same functional equation data.
5n particular, the proposition shows that the functional equation data ofan L -function is well defined. There are no ambiguities arising, say, from theduplication formula of the Γ -function. Also, we remark that the partial Ra-manujan bound is essential. One can easily construct counterexamples to theabove proposition using Saito-Kurokawa lifts, which do not satisfy the partialRamanujan bound. Proof.
Let Λ j ( s ) be the completed L -function of L j ( s ) and consider λ ( s ) = Λ ( s )Λ ( s )= (cid:16) N N (cid:17) s/ Q j Γ R ( s + µ j, ) Q k Γ C ( s + ν k, ) Q j Γ R ( s + µ j, ) Q k Γ C ( s + ν k, ) Y p F p, ( p − s ) − F p, ( p − s ) − . (2.8)By the assumption on F p,j , the product over p is really a finite product. Thus,(2.8) is a valid expression for λ ( s ) for all s .By the partial Ramanujan bound and the assumptions on µ j and ν j , we seethat λ ( s ) has no zeros or poles in the half-plane ℜ ( s ) > θ . But by the functionalequations for L and L we have λ ( s ) = ( ε /ε ) λ (1 − s ) . Thus, λ ( s ) also hasno zeros or poles in the half-plane ℜ ( s ) < − θ . Since θ < , we conclude that λ ( s ) has no zeros or poles in the entire complex plane.If the product over p in (2.8) were not empty, then the fact that { log( p ) } is linearly independent over the rationals implies that λ ( s ) has infinitely manyzeros or poles on some vertical line. Thus, F p, = F p, for all p .The Γ -factors must also cancel identically, because the right-most pole of Γ R ( s + µ ) is at − µ , and the right-most pole of Γ C ( s + ν ) is at − ν . This leavespossible remaining factors of the form Γ C ( s + 1) / Γ R ( s + 1) , but that also haspoles because the Γ R factor cancels the first pole of the Γ C factor, but not thesecond pole. Note that the restriction ℜ ( µ ) ∈ { , } is a critical ingredient inthis argument.This leaves the possibility that λ ( s ) = ( N /N ) s/ , but such a functioncannot satisfy the functional equation λ ( s ) = ( ε /ε ) λ (1 − s ) unless N = N and ε = ε . L -functions In this section we state a version of strong multiplicity one for L -functions whichis stronger than Proposition 2.1 because it only requires the Dirichlet coefficients a ( p ) and a ( p ) to be reasonably close. This is a significantly weaker conditionthan equality of the local factor.Although the main ideas behind the proof appear in Kaczorowski-Perelli [10]and Soundararajan [26], we give a slightly stronger version which assumes apartial Ramanujan bound θ < , plus an additional condition, instead ofthe full Ramanujan conjecture. We provide a self-contained account becausewe also wish to bring awareness of these techniques to people with a morerepresentation-theoretic approach to L -functions.6 heorem 2.2. Suppose L ( s ) , L ( s ) are Dirichlet series with Dirichlet coef-ficients a ( n ) , a ( n ) , respectively, which continue to meromorphic functions oforder 1 satisfying functional equations of the form (2.2) - (2.3) with a partial Sel-berg bound ℜ ( µ j ) , ℜ ( ν j ) > − for both functions, and having Euler productssatisfying (2.4) - (2.5) . Assume a partial Ramanujan bound for some θ < holdsfor both functions, and that the Dirichlet coefficients at the primes are close toeach other in the sense that X p ≤ X p log( p ) | a ( p ) − a ( p ) | ≪ X. (2.9) We have L ( s ) = L ( s ) if either of the following two conditions are satisfied1) X p ≤ X | a ( p ) − a ( p ) | log p ≪ X .2) For each of L ( s ) and L ( s ) , separately, any one of the following holds:a) The Ramanujan bound θ = 0 .b) The partial symmetric square (2.6) of the function has a meromorphiccontinuation past the σ = 1 line, and only finitely many zeros or polesin σ ≥ .c) The partial exterior square (2.7) of the function has a meromorphiccontinuation past the σ = 1 line, and only finitely many zeros or polesin σ ≥ . Note that condition (2.9) is satisfied if | a ( p ) − a ( p ) | ≪ / p p , in particular,if a ( p ) = a ( p ) for all but finitely many p , or more generally if a ( p ) = a ( p ) forall but a sufficiently thin set of primes. In particular, a ( p ) and a ( p ) can differat infinitely many primes. Also, by the prime number theorem [2, Theorem 4.4]in the form X p
Let L ( s ) = Y p ∞ X j =0 a ( p j ) p − js (2.22) and suppose there exists M , M ≥ and θ < so that | a ( p j ) | ≪ p θj and X p ≤ X | a ( p ) | log p ≤ M X + o ( X ) , (2.23) X p ≤ X p − | a ( p ) | log p ≤ M X + o ( X ) . (2.24) Then L ( s ) is a nonvanishing analytic function in the half-plane σ > . Further-more, if L ( s ) has a meromorphic continuation to a neighborhood of σ ≥ , then L ( s ) has at most ( M + 2 M ) zeros or poles on the σ = 1 line. Note that, by the prime number theorem (2.10), the condition (2.23) on a ( p ) is satisfied if | a ( p ) | ≤ M . Also, if θ < then condition (2.24) on a ( p ) holdswith M = 1 .The proof of Lemma 2.5 is in Section 2.5.10 .4 Proof of Theorem 2.2 Now we have the ingredients to prove Theorem 2.2. The proof begins the sameas that of Proposition 2.1, by considering the ratio of completed L -funtions: λ ( s ) := Λ ( s )Λ ( s ) , (2.25)which is a meromorphic function of order 1 and satisfies the functional equation λ ( s ) = ελ (1 − s ) , where ε = ε /ε . Lemma 2.6. λ ( s ) has only finitely many zeros or poles in the half-plane σ ≥ . Assuming the lemma, we complete the proof of Theorem 2.2 as follows. Bythe functional equation, λ ( s ) has only finitely many zeros or poles, so by theHadamard factorization theorem λ ( s ) = e As r ( s ) (2.26)where r ( s ) is a rational function.By (2.26), as σ → ∞ , λ ( σ ) = C σ m e Aσ (cid:0) C σ − + O ( σ − ) (cid:1) , (2.27)for some C = 0 and m ∈ Z . On the other hand, if b ( n ) is the first non-zeroDirichlet coefficient (with n > ) of L ( s ) /L ( s ) , then by (2.25) and Stirling’sformula, as σ → ∞ , λ ( σ ) = (cid:0) B σ B e B σ log σ + B σ (1 + o (1)) (cid:1)(cid:0) b ( n ) n − σ + O (( n + 1) − σ ) . (2.28)Comparing those two asymptotic formulas, the leading terms must be equal,so B = C , B = m , B = 0 , and B = A . Comparing second terms, wehave polynomial decay equal to exponential decay, which is impossible unless b ( n ) = 0 and C = 0 . But b ( n ) was the first nonzero coefficient of L ( s ) /L ( s ) ,so we conclude that L ( s ) = L ( s ) , as claimed.The rest of this section is devoted to the proof of Lemma 2.6. By (2.8) andthe partial Selberg bound assumed on µ and ν , only the product P ( s ) = Y p F p, ( p − s ) − F p, ( p − s ) − = Y p a ( p ) p − s + a ( p ) p − s + · · · a ( p ) p − s + a ( p ) p − s + · · · could contribute any zeros or poles to λ ( s ) in the half-plane σ ≥ .By the first line in equation (2.16) of Lemma 2.4 we have P ( s ) = Y p a ( p ) p − s a ( p ) p − s · Y p a ( p ) p − s a ( p ) p − s · H ( s )= A ( s ) H ( s ) , (2.29)say, where H ( s ) is regular and nonvanishing for σ > + θ .11 emma 2.7. Assuming θ < , bound (2.9) , and condition 1) in Theorem 2.2,with A ( s ) as defined in (2.29) we have A ( s ) = Y p (1 + ( a ( p ) − a ( p )) p − s ) · Y p (1 + ( a ( p ) − a ( p )) p − s ) · H ( s ) , (2.30) where H ( s ) is regular and nonvanishing for σ > . We finish the proof of Lemma 2.6 and then conclude with the proof ofLemma 2.7.Using the notation of Lemma 2.7, write (2.30) as A ( s ) = A ( s ) H ( s ) . Since A ( s ) and H ( s ) are meromorphic in a neighborhood of σ ≥ , so is A ( s ) .Changing variables s s + , which divides the n th Dirichlet coefficient by / √ n , we can apply Lemma 2.5, using the estimate (2.9) and condition 1) toconclude that A ( s ) has only finitely many zeros or poles in σ ≥ . Since thesame is true of H ( s ) and H ( s ) , we have shown that P ( s ) has only finitely manyzeros or poles in σ ≥ . This completes the proof for conditions 2a) and 1).In the other cases, the proof is almost the same, using Lemma 2.4 to rewriteequation (2.29) in terms of L Sj ( s, sym ) or L Sj ( s, ext ) , and using Lemma 2.5 forthe factors that remain. This concludes the proof of Lemma 2.6. Proof of Lemma 2.7. Using the identities ax bx = 1 + ( a − b ) x − b ( a − b ) x bx (2.31)and ax + bx = (1 + ax ) (cid:18) bx ax (cid:19) (2.32)we have ax bx = (1 + ( a − b ) x ) (cid:18) − b ( a − b ) x (1 + ( a − b ) x )(1 + bx ) (cid:19) . (2.33)Thus Y p a ( p ) p − s a ( p ) p − s = Y p (cid:0) a ( p ) − a ( p )) p − s (cid:1) × Y p (cid:18) − a ( p )( a ( p ) − a ( p )) p − s (1 + ( a ( p ) − a ( p )) p − s )(1 + a ( p ) p − s ) (cid:19) = Y p (cid:0) a ( p ) − a ( p )) p − s (cid:1) · h ( s ) (2.34)say. We wish to apply Lemma 2.5 to show that h ( s ) is regular and nonvanishingfor σ > σ for some σ < . Since θ < , if σ ≥ and p > P where P depends12nly on θ , then | a ( p ) p − σ | ≥ and | a ( p ) − a ( p )) p − σ | ≥ . Usingthose inequalities and | a ( p ) | ≪ p θ we have X P ≤ p ≤ X (cid:12)(cid:12)(cid:12)(cid:12) a ( p )( a ( p ) − a ( p ))(1 + ( a ( p ) − a ( p )) p − σ )(1 + a ( p ) p − σ ) (cid:12)(cid:12)(cid:12)(cid:12) log p ≤ X P ≤ p ≤ X | a ( p )( a ( p ) − a ( p )) | log p ≪ X θ X P ≤ p ≤ X | ( a ( p ) − a ( p )) | log p ≪ X +2 θ . (2.35)Changing variables s → s − and applying Lemma 2.5, we see that h ( s ) isregular and nonvanishing for σ > .Applying the same reasoning to the second factor in (2.30) completes theproof. Two basic results which are used in this section are:
Lemma 2.8. If P n ≤ X | a ( n ) | ≪ X ǫ for every ǫ > , then ∞ X n =1 a ( n ) n s convergesabsolutely for all σ > . Lemma 2.9. If P n ≤ X | a ( n ) | ≤ CX as X → ∞ , then ∞ X n =1 a ( n ) n σ ≤ Cσ − O (1) as σ → + . Both of those results follow by partial summation.We first state and prove a simplified version of Lemma 2.5.
Lemma 2.10.
Let L ( s ) = Y p ∞ X j =0 a ( p j ) p − js (2.36) and suppose there exists M ≥ and θ < so that | a ( p j ) | ≪ p jθ and X p ≤ X | a ( p ) | log p ≤ (1 + o (1)) M X. (2.37) Then L ( s ) is a nonvanishing analytic function in the half-plane σ > . Further-more, if L ( s ) has a meromorphic continuation to a neighborhood of σ ≥ , then L ( s ) has at most M zeros or poles on the σ = 1 line. a ( p ) issatisfied if | a ( p ) | ≤ M . Proof.
We have L ( s ) = Y p ∞ X j =0 a ( p j ) p − js = Y p (cid:18) a ( p ) p − s + X j ≥ a ( p j ) p − js (cid:19) = Y p (cid:0) a ( p ) p − s (cid:1) × Y p (cid:0) a ( p ) p − s + ( a ( p ) − a ( p ) a ( p )) p − s + ( a ( p ) − a ( p ) a ( p ) + a ( p ) a ( p )) p − s + · · · (cid:1) = Y p (cid:0) a ( p ) p − s (cid:1) Y p (cid:18) ∞ X j =2 b ( p j ) p − js (cid:19) , (2.38)say, where b ( p j ) ≪ jM j p jθ ≪ p j ( θ + ǫ ) for any ǫ > .Writing (2.38) as L ( s ) = f ( s ) g ( s ) we have log g ( s ) = X p log(1 + Y ) = X p (cid:0) Y + O ( Y ) (cid:1) , (2.39)where Y = ∞ X j =2 b ( p j ) p − js . Now, | Y | ≤ ∞ X j =2 | b ( p j ) | p − jσ ≪ ∞ X j =2 p j ( θ − σ + ǫ ) = p θ − σ + ǫ ) − p θ − σ + ǫ . (2.40)If σ > + θ we have | Y | ≪ /p δ for some δ > . Therefore the series (2.39) for log( g ( s )) converges absolutely for σ > + θ , so g ( s ) is a nonvanishing analyticfunction in that region. By (2.37), Cauchy’s inequality, and Lemma 2.8, f ( s ) isa nonvanishing analytic function for σ > , so the same is true for L ( s ) . Thisestablishes the first assertion in the lemma.Now we consider the zeros of L ( s ) on σ = 1 . Since θ < , the zeros or polesof L ( s ) on the σ = 1 line are the zeros or poles of f ( s ) . Furthermore, by (2.38)and the properties of g ( s ) , for σ > we have L ′ L ( s ) = X p − a ( p ) log( p ) p s + h ( s ) , (2.41)14here h ( s ) is bounded in σ > + θ + ǫ for any ǫ > . Suppose s , . . . , s J arezeros or poles of L ( s ) , with s j = 1 + it j having multiplicity m j . We have L ′ L ( σ + it j ) ∼ m j σ − , as σ → + , (2.42)therefore X p − a ( p ) log( p ) p σ + it j ∼ m j σ − , as σ → + . (2.43)Now write k ( s ) = J X j =1 m j X p − a ( p ) log( p ) p s + it j . (2.44)By (2.43) we have k ( σ ) ∼ P Jj =1 m j σ − , as σ → + . (2.45)On the other hand, for σ > we have | k ( σ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X p a ( p ) log( p ) p σ J X j =1 m j p − it j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X p | a ( p ) | log( p ) p σ ! X p log pp σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J X j =1 m j p − it j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 + o (1)) (cid:18) M σ − (cid:19) J X j =1 J X ℓ =1 m j m ℓ X p log pp σ + i ( t j − t ℓ ) ∼ (cid:18) M σ − (cid:19) J X j =1 m j σ − as σ → + . (2.46)On the first line we used the Cauchy-Schwartz inequality, on the next-to-last linewe wrote the sum over a ( p ) as a Stieltjes integral and used (2.37) and Lemma 2.9,and on the last line we used the fact that the Riemann zeta function has a simplepole at s = 1 and no other zeros or poles on the σ = 1 line.Combining (2.43) and (2.46) we have P Jj =1 m j ≤ M . Since m j ≥ , we seethat J ≤ M , as claimed.The proof of Lemma 2.5 is similar to Lemma 2.10.15 roof of Lemma 2.5. We have L ( s ) = Y p ∞ X j =0 a ( p j ) p − js = Y p (cid:18) a ( p ) p − s + a ( p ) p − s + X j ≥ a ( p j ) p − js (cid:19) = Y p (cid:0) a ( p ) p − s (cid:1) (cid:0) a ( p ) p − s (cid:1) × (cid:0) a ( p ) − a ( p ) a ( p )) p − s + ( a ( p ) − a ( p ) a ( p ) + a ( p ) a ( p )) p − s + · · · (cid:1) = Y p (cid:0) a ( p ) p − s (cid:1) (cid:0) a ( p ) p − s (cid:1) Y p (cid:18) ∞ X j =3 c ( p j ) p − js (cid:19) = f ( s ) g ( s ) , (2.47)say.We have c ( p j ) ≪ jM j p jθ ≪ p j ( θ + ǫ ) for any ǫ > . We use this to show that g ( s ) is a nonvanishing analytic function in σ > + θ . Writing g ( s ) = Q p (1 + Y ) we have log g ( s ) = X p log(1 + Y ) = X p (cid:0) Y + O ( Y ) (cid:1) , (2.48)where Y = ∞ X j =3 b ( p j ) p − js . Now, | Y | ≤ ∞ X j =3 | b ( p j ) | p − jσ ≪ ∞ X j =3 p j ( θ − σ + ǫ ) = p θ − σ + ǫ ) − p θ − σ + ǫ . (2.49)If σ > + θ we have | Y | ≪ /p δ for some δ > . Therefore by Lemma 2.8the series (2.48) for log( g ( s )) converges absolutely for σ > + θ , so g ( s ) is anonvanishing analytic function in that region. By the same argument, using(2.23) and (2.24), f ( s ) is a nonvanishing analytic function for σ > , so thesame is true for L ( s ) . This establishes the first assertion in the lemma.Now we consider the zeros of L ( s ) on σ = 1 . Since θ < , the zeros or polesof L ( s ) on the σ = 1 line are the zeros or poles of f ( s ) . Taking the logarithmicderivative of (2.47) and using the same argument as above for the lower order16erms, we have L ′ L ( s ) = X p − a ( p ) log( p ) p s + 2 a ( p ) log( p ) p s − a ( p ) log( p ) p s + h ( s )= X p − a ( p ) log( p ) p s − a ( p ) log( p ) p s + h ( s ) , (2.50)where h j ( s ) is bounded in σ > + θ + ǫ for any ǫ > . By (2.23) and Lemma 2.8,the middle term in the sum over primes in (2.50) converges absolutely for σ > ,so it was incorporated into h ( s ) .Suppose s , . . . , s J are zeros or poles of L ( s ) , with s j = 1 + it j havingmultiplicity m j . We have L ′ L ( σ + it j ) ∼ m j σ − , as σ → + , (2.51)therefore X p (cid:18) − a ( p ) log( p ) p σ + it j − a ( p ) log( p ) p σ + it j ) (cid:19) ∼ m j σ − , as σ → + . (2.52)Now write k ( s ) = J X j =1 m j X p (cid:18) − a ( p ) log( p ) p s + it j − a ( p ) log( p ) p s + it j ) (cid:19) (2.53)By (2.52) we have k ( σ ) ∼ P Jj =1 m j σ − , as σ → + . (2.54)We will manipulate (2.53) so that so that we can use (2.23) and (2.24) to givea bound on P m j in terms of M and M .17y Cauchy’s inequality and Lemma 2.9 we have | k ( σ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X p a ( p ) log( p ) p σ J X j =1 m j p it j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X p p − σ a ( p ) log( p ) p σ J X j =1 m j p it j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X p | a ( p ) | log( p ) p σ ! X p log pp σ (cid:12)(cid:12)(cid:12)(cid:12) J X j =1 m j p − it j (cid:12)(cid:12)(cid:12)(cid:12) + 2 X p p − σ | a ( p ) | log( p ) p σ ! X p log pp σ (cid:12)(cid:12)(cid:12)(cid:12) J X j =1 m j p − it j (cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 + o (1)) (cid:18) M σ − (cid:19) (cid:18) J X j =1 J X ℓ =1 m j m ℓ X p log pp σ + i ( t j − t ℓ ) (cid:19) + 2 (cid:18) M σ − (cid:19) (cid:18) J X j =1 J X ℓ =1 m j m ℓ X p log pp σ +2 i ( t j − t ℓ ) (cid:19) ! ∼ M + 2 M ( σ − J X j =1 m j σ − as σ → + . (2.55)In the last step we used the fact that the Riemann zeta function has a simplepole at 1 and no other zeros or poles on the -line.Combining (2.54) and (2.55) we have J X j =1 m j ≤ ( M + 2 M ) . Since m j ≥ ,the proof is complete. GL( n ) Let π = ⊗ π p and π ′ = ⊗ π ′ p be cuspidal automorphic representations of thegroup GL( n, A Q ) . For a finite prime p for which π p and π ′ p are both unramified,let A ( π p ) (resp. A ( π ′ p ) ) represent the semisimple conjugacy class in GL( n, C ) corresponding to π p (resp. π ′ p ). The strong multiplicity one theorem for GL( n ) states that if A ( π p ) = A ( π ′ p ) for almost all p , then π = π ′ . The followingresult implies, in particular, that the equality of traces tr( A ( π p )) = tr( A ( π ′ p )) for almost all p is sufficient to reach the same conclusion. The traces could evenbe different at every prime, if those differences decreased sufficiently rapidly asa function of p . Theorem 3.1.
Suppose that π and π ′ are (unitary) cuspidal automorphic rep- esentations of GL( n, A Q ) satisfying X p ≤ X p log( p ) (cid:12)(cid:12) tr A ( π p ) − tr A ( π ′ p ) (cid:12)(cid:12) ≪ X. (3.1) Assume a partial Ramanujan bound for some θ < holds for both incomplete L -functions L fin ( s, π ) and L fin ( s, π ′ ) . Then π = π ′ .Proof. We apply Theorem 2.2 to L ( s ) = L fin ( s, π ) and L ( s ) = L fin ( s, π ′ ) .The condition on the spectral parameters ℜ ( µ j ) , ℜ ( ν j ) > − is satisfied byProposition 2.1 of [4]. By [7], the partial symmetric square L -function for GL( n ) has meromorphic continuation to all of C and only finitely many poles in σ ≥ .Using the fact that the partial Rankin-Selberg L -function of a representation of GL( n ) with itself has no zeros in σ ≥ (see [25]) and that the partial exteriorsquare L -function of GL( n ) has only finitely many poles (see [5]), we see thatpartial symmetric square L -function for GL( n ) has only finitely many zerosin σ ≥ . This gives us condition 2b) of Theorem 2.2. The conclusion ofTheorem 2.2 is that L ( s ) = L ( s ) . By the familiar strong multiplicity onetheorem for GL( n ) , this implies π = π . In this section we prove Theorem 1.3. We start by giving some background onSiegel modular forms for
Sp(4 , Z ) . Let the symplectic group of similitudes ofgenus be defined by GSp(4) := { g ∈ GL(4) : t gJg = λ ( g ) J, λ ( g ) ∈ GL(1) } where J = (cid:20) I − I (cid:21) . Let
Sp(4) be the subgroup with λ ( g ) = 1 . The group GSp + (4 , R ) := { g ∈ GSp(4 , R ) : λ ( g ) > } acts on the Siegel upper half space H := { Z ∈ M ( C ) : t Z = Z, Im( Z ) > } by g h Z i := ( AZ + B )( CZ + D ) − where g = (cid:20) A BC D (cid:21) ∈ GSp + (4 , R ) , Z ∈ H . (3.2)Let us define the slash operator | k for a positive integer k acting on holomorphicfunctions F on H by ( F | k g )( Z ) := λ ( g ) k det( CZ + D ) − k F ( g h Z i ) ,g = (cid:20) A BC D (cid:21) ∈ GSp + (4 , R ) , Z ∈ H . (3.3)The slash operator is defined in such a way that the center of GSp + (4 , R ) actstrivially. Let S (2) k be the space of holomorphic Siegel cusp forms of weight k ,19enus with respect to Γ (2) := Sp(4 , Z ) . Then F ∈ S (2) k satisfies F | k γ = F forall γ ∈ Γ (2) .Let us now describe the Hecke operators acting on S (2) k . For a matrix M ∈ GSp + (4 , R ) ∩ M ( Z ) , we have a finite disjoint decomposition Γ (2) M Γ (2) = G i Γ (2) M i . (3.4)For F ∈ S (2) k , define T k (Γ (2) M Γ (2) ) F := det( M ) k − X i F | k M i . (3.5)Note that this operator agrees with the one defined in [1]. Let F ∈ S (2) k be asimultaneous eigenfunction for all the T k (Γ (2) M Γ (2) ) , M ∈ GSp + (4 , R ) ∩ M ( Z ) ,with corresponding eigenvalue µ F (Γ (2) M Γ (2) ) . For any prime number p , it isknown that there are three complex numbers α F ( p ) , α F ( p ) , α F ( p ) such that, forany M with λ ( M ) = p r , we have µ F (Γ (2) M Γ (2) ) = α F ( p ) r X i Y j =1 ( α Fi ( p ) p − j ) d ij , (3.6)where Γ (2) M Γ (2) = F i Γ (2) M i , with M i = (cid:20) A i B i D i (cid:21) and D i = (cid:20) p d i ∗ p d i (cid:21) . (3.7)Henceforth, if there is no confusion, we will omit the F and p in describingthe α Fi ( p ) to simplify the notations. The α , α , α are the classical Satake p -parameters of the eigenform F . It is known that they satisfy α α α = p k − . (3.8)For any n > , define the Hecke operators T k ( n ) by T k ( n ) = X λ ( M )= n T k (Γ (2) M Γ (2) ) . Let the eigenvalues of F corresponding to T k ( n ) be denoted by µ F ( n ) . Set α p = p − ( k − / α and β p = p − ( k − / α α . Then formulas for the Hecke eigenvalues µ F ( p ) and µ F ( p ) in terms of α p and β p are µ F ( p ) = p k − / (cid:0) α p + α − p + β p + β − p (cid:1) , (3.9) µ F ( p ) = p k − (cid:0) α p + α − p + ( α p + α − p )( β p + β − p ) + β p + β − p + 2 − p (cid:1) . (3.10)20he Ramanujan bound in this context is | α p | = | β p | = 1 . (3.11)This is closely related to our use of that term for L -functions, as can be seenfrom the spin L -function of F : L ( s, F, spin) = Y p F p ( p − s , spin) − , where F p ( X, spin) = (1 − α p X )(1 − α − p X )(1 − β p X )(1 − β − p X ) . It satisfies thefunctional equation Λ( s, F, spin) := Γ C ( s + )Γ C ( s + k − ) L ( s, F, spin)= ε Λ( s, F , spin) , (3.12)where ε = ( − k .Let a ( p ) be the p th Dirichlet coefficient of L ( s, F, spin) . We will use the factthat each F falls into one of two classes.i) a ( p ) = p / + p − / + β p + β − p , where β p is the Satake p -parameter of aholomorphic cusp form on GL(2) of weight k − . In this case F is a Saito-Kurokawa lifting ; for more details on Saito-Kurokawa liftings we refer to[9]. Note that | β p | = 1 , so that a ( p ) = p / + O (1) in the Saito-Kurokawacase.ii) a ( p ) = O (1) . This is the Ramanujan conjecture for non-Saito-Kurokawaliftings, which has been proven in [27].Theorem 1.3 is now a consequence of the following stronger result. Theorem 3.2.
Suppose F j , for j = 1 , , are Siegel Hecke eigenforms of weight k j for Sp(4 , Z ) , with Hecke eigenvalues µ j ( n ) . If X p ≤ X p log( p ) (cid:12)(cid:12)(cid:12) p / − k µ ( p ) − p / − k µ ( p ) (cid:12)(cid:12)(cid:12) ≪ X (3.13) as X → ∞ , then k = k and F and F have the same eigenvalues for theHecke operator T ( n ) for all n .Proof. For i = 1 , let a i ( p ) be the p th Dirichlet coefficient of L ( s, F i , spin) .Then a i ( p ) = α i,p + α − i,p + β i,p + β − i,p , where α i,p , β i,p are the Satake p -parametersof F i , as explained after (3.8). By (3.9), µ i ( p ) = p k i − / (cid:0) α i,p + α − i,p + β i,p + β − i,p (cid:1) . Hence, condition (3.13) translates into X p ≤ X p log( p ) | a ( p ) − a ( p ) | ≪ X. (3.14)21rom the remarks made before the theorem, we see that either F , F are bothSaito-Kurokawa lifts, or none of them is a Saito-Kurokawa lift.Assume first that F , F are both Saito-Kurokawa lifts. Then, for i = 1 , ,there exist modular forms f i of weight k i − and with Satake parameters β i,p such that a i ( p ) = p / + p − / + β i,p + β − i,p . From (3.14) we obtain X p ≤ X p log( p ) | b ( p ) − b ( p ) | ≪ X, (3.15)where b i,p = β i,p + β − i,p . Note that b i,p is the p th Dirichlet coefficient of(the analytically normalized L -function) L ( s, f i ) . Since the Ramanujan con-jecture is known for elliptic modular forms, Theorem 2.2 applies. We conclude k − k − and L ( s, f ) = L ( s, f ) . Hence k = k and L ( s, F , spin) = L ( s, F , spin) . The equality of spin L -functions implies µ ( p ) = µ ( p ) and µ ( p ) = µ ( p ) for all p . Since T ( p ) and T ( p ) generate the p -componentof the Hecke algebra, it follows that µ ( n ) = µ ( n ) for all n .Now assume that F and F are both not Saito-Kurokawa lifts. Then, usingthe fact that the Ramanujan conjecture is known for F and F , Theorem 2.2applies to L ( s ) = L ( s, F , spin) and L ( s ) = L ( s, F , spin) . We conclude that k = k and that the two spin L -functions are identical. As above, this implies µ ( n ) = µ ( n ) for all n . Let X/ Q be an elliptic or hyperelliptic curve, X : y = f ( x ) , where f ∈ Z [ x ] , and let N X ( p ) be the number of points on X mod p . In Serre’srecent book [24], the title of Section 6.3 is “About N X ( p ) − N Y ( p ) ,” in whichhe gives a description of what can happen if N X ( p ) − N Y ( p ) is bounded. ForSerre, X and Y are much more general than hyperelliptic cuves, but we use thehyperelliptic curve case to illustrate an application of multiplicity one resultsfor L -functions.Recall that the Hasse-Weil L -function of X , L ( X, s ) = ∞ X n =1 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 generalcase by multiplicativity. The L -function (conjecturally if g X ≥ ) satisfies thefunctional equation Λ( X, s ) = N s/ X Γ C ( s ) g X L ( X, s ) = ± Λ( X, − s ) , (3.16)where N X is the conductor and g X = ⌊ (deg( f ) − / ⌋ is the genus of X .22 roposition 3.3. 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 conjecturedfunctional equation (3.16) , 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 hypothesisof functional equation. But Serre’s proof involves more machinery than we usehere.
Proof.
To apply Theorem 2.2, we first form the analytically normalized L -function L ( s, X ) = L ( X, s + ) = X a X ( n ) / √ nn s = X b X ( n ) n s , (3.17)say. Note that we have the functional equation Λ( s, X ) = N s/ X Γ C ( s + ) g X L ( s, X ) = ± Λ(1 − s, X ) . (3.18)The Hasse bound for a X ( n ) implies the Ramanujan bound for L ( s, X ) . Thecondition | N X ( p ) − N Y ( p ) | ≪ is equivalent to | b X ( p ) − b Y ( p ) | ≪ p p , (3.19)which implies X p ≤ T p | b X ( p ) − b Y ( p ) | log( p ) ≪ X p ≤ T log( p ) ∼ T, (3.20)by the prime number theorem. Thus, Theorem 2.2 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 A.1would apply, and a much weaker bound on | N X ( p ) − N Y ( p ) | would allow one toconclude that N X ( p e ) = N Y ( p e ) for all p, e . For example, if E, E ′ are ellipticcurves over Q , then | N E ( p ) − N E ′ ( p ) | ≤ . p p for all but finitely many p implies N E ( p ) = N E ′ ( p ) for all p . A Selberg orthogonality and strong multiplicityone for
GL( n ) The proof of Theorem 2.2 used only standard techniques from analytic numbertheory. Utilizing recent results concerning the Selberg orthonormality conjec-ture, and restrictng to the case of L -functions of cuspidal automorphic repre-sentations of GL( n ) , one obtains the following theorem, which is stronger thanTheorem 3.1. 23 heorem A.1. Suppose that π , π ′ are (unitary) cuspidal automorphic repre-sentations of GL( n, A F ) , and suppose X p ≤ X p (cid:12)(cid:12) tr A ( π p ) − tr A ( π ′ p ) (cid:12)(cid:12) ≤ (2 − ǫ ) log log( X ) (A.1) for some ǫ > as X → ∞ . If n ≤ , or if Hypothesis H holds for both L fin ( s, π ) and L fin ( s, π ′ ) (in particular if the partial Ramanujan conjecture θ < is truefor π and π ′ ), then π = π ′ . Using the fact that . < and the consequence of the prime numbertheorem X p ≤ X p ∼ log log( X ) , (A.2)we see that condition (A.1) holds if (cid:12)(cid:12) tr A ( π p ) − tr A ( π ′ p ) (cid:12)(cid:12) < . for all but finitelymany p . Thus, the strong multiplicity one theorem only requires consideringthe traces of π p , and futhermore those traces can differ at every prime, and byan amount which is bounded below.For GL (2 , A Q ) , the Ramanujan bound along with (A.2) implies a version ofa result of Ramakrishnan [17]: if tr A ( π p ) = tr A ( π ′ p ) for + ε of all primes p ,then π = π ′ . This result was extended by Rajan [18].The proof of Theorem A.1 is a straightforward application of recent resultstoward the Selberg orthonormality conjecture [13, 3], which make use of progresson Rudnick and Sarnak’s “Hypothesis H” [20, 11]. Suppose L ( s ) = X a ( n ) n s , L ( s ) = X b ( n ) n s (A.3)are L -functions, meaning that they have a functional equation and Euler productas described in Section 2.1.The point of the strong multiplicity one theorem is that two L -function musteither be equal, or else they must be far apart. The essential idea was elegantlydescribed by Selberg; see [23]. Recall that an L -function is primitive if it cannotbe written nontrivially as a product of L -functions. Conjecture A.2 (Selberg Orthonormality Conjecture) . Suppose that L and L are primitive L -functions with Dirichlet coefficients a ( p ) and b ( p ) . Then X p ≤ X a ( p ) b ( p ) p = δ ( L , L ) log log( X ) + O (1) , (A.4) where δ ( L , L ) = 1 if L = L , and otherwise.Proof of Theorem A.1. Rudnick and Sarnak’s Hypothesis H is the assertion X p a ( p k ) log ( p ) p k < ∞ k ≥ . For a given k , this follows from a partial Ramanujan bound θ < − k . Since k ≥ , Hypothesis H follows from the partial Ramanujanbound θ < .For the standard L -functions of cuspidal automorphic representations on GL( n ) , Rudnick and Sarnak [20] proved Selberg’s orthonormality conjectureunder the assumption of Hypothesis H, and they proved Hypothesis H for n = 2 , . The case of n = 4 for Hypothesis H was proven by Kim [11]. Thus, underthe conditions in Theorem A.1, the Selberg orthonormality conjecture is true.Since π and π ′ are cuspidal automorphic representations of GL( n, A F ) , the L -functions L ( s ) = L fin ( s, π ) and L ( s ) = L fin ( s, π ′ ) are primitive L -functions.Hence, by (A.4) X p ≤ X p | a ( p ) − b ( p ) | = X p ≤ X p (cid:0) | a ( p | + | b ( p ) | − ℜ ( a ( p ) b ( p )) (cid:1) = 2 log log( X ) − δ L ,L log log( X ) + O (1)= ( O (1) if L = L X ) + O (1) if L = L . (A.5)We have P p ≤ X p | a ( p ) − b ( p ) | ≤ (2 − ǫ ) log log( X ) for some ǫ > . Thisimplies that ǫ log log( X ) is unbounded, and hence (A.5) implies that L ( s ) = L ( s ) . This gives us π = π ′ .Recently, the transfer of full level Siegel modular forms to GL(4) was ob-tained in [16]. Hence, we can apply Theorem A.1 to the transfer to
GL(4) of aSiegel modular form of full level and thus obtain a stronger version of Theorem3.2.
Theorem A.3.
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