On fundamental Fourier coefficients of Siegel cusp forms of degree 2

Let $F$ be a Siegel cusp form of degree 2, even weight $k \geq 2$ and odd squarefree level $N$. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of $F$ at fundamental matrices $S$ (i.e., with $-4 det(S)$ equal to a fundamental discriminant). We prove that as $S$ varies along the equivalence classes of fundamental matrices with $det(S) \asymp X$, the sequence $a(F,S)$ has at least $X^{1-\epsilon}$ sign changes, and takes at least $X^{1-\epsilon}$"large values". Furthermore, assuming the Generalized Riemann Hypothesis as well as the refined Gan--Gross--Prasad conjecture, we prove the bound $|a(F,S)| \ll_{F, \epsilon} \frac{\det(S)^{\frac{k}2 - \frac{1}{2}}}{ (\log |\det(S)|)^{\frac18 - \epsilon}}$ for fundamental matrices $S$.

For S = a b/2 b/2 c ∈ Λ 2 , we define disc(S) = −4 det(S) = b 2 −4ac. If d = disc(S) is a fundamental discriminant 1 , then S is called fundamental. A fundamental S is automatically primitive (i.e., gcd(a, b, c) = 1). Observe that if d is odd, then S is fundamental if and only if d is squarefree. The fundamental Fourier coefficients of Siegel cusp forms are deep and highly interesting objects. These are the basic building blocks, in the sense that one cannot use the theory of Hecke operators to relate the Fourier coefficients a(F, S) at these matrices to those at simpler matrices. Furthermore, fundamental Fourier coefficients are closely related to central L-values.
In [Sah13,SS13] it was proved that if k > 2 is even and N is squarefree, then elements of S k (Γ (2) 0 (N )) (under some mild assumptions) are uniquely determined by their fundamental Fourier coefficients. More precisely, it was proved there that for k, N as above, if F ∈ S k (Γ (2) 0 (N )) is non-zero and an eigenfunction of the U (p) operators for p|N , then a(F, S) = 0 for infinitely many matrices S such that disc(S) is odd and squarefree. This non-vanishing result is crucial for the existence of good Bessel models [PSS14, Lemma 5.1.1] and consequently was needed for removing a key assumption from theorems due to Furusawa [Fur93], Pitale-Schmidt [PS09a] and the thirdnamed author [Sah09,Sah10,PSS14] on the degree 8 L-function on GSp 4 × GL 2 . Furthermore, there is a remarkable identity, explained in more detail in Section 1.3 below, relating squares of 2010 Mathematics Subject Classification. Primary 11F30; Secondary 11F37, 11F46, 11F67. 1 Recall that an integer n is a fundamental discriminant if either n is a squarefree integer congruent to 1 modulo 4 or n = 4m where m is a squarefree integer congruent to 2 or 3 modulo 4. 1 (weighted averages of) fundamental Fourier coefficients and central values of dihedral twists of GSp 4 and GL 2 L-functions. Indeed, the fundamental Fourier coefficients are unipotent periods whose weighted averages are Bessel periods whose absolute squares are essentially central L-values of degree 8 L-functions, via the refined Gan-Gross-Prasad conjectures [Liu16].
Motivated by these connections, the objective of this paper is to understand better the nature of the fundamental Fourier coefficients. In particular, we investigate the following questions: • Are there many sign changes among the fundamental Fourier coefficients?
• How large (in the sense of both lower and upper bounds) are the fundamental Fourier coefficients? We emphasize that while these kinds of questions have been previously studied for the full sequence a(F, S) (S ∈ Λ 2 ) of Fourier coefficients attached to F (see [CGK15,GS17,HZ18] for results on sign-changes and [Das20] for an Omega-result) there appears to be virtually no previous work in the more subtle setting where one restricts to fundamental Fourier coefficients. There has also been a fair bit of work on sign changes of Hecke eigenvalues of Siegel cusp forms [Koh07,PS08,RSW16,DK18] which can be combined with the Hecke relations [And74] to deduce sign changes among the a(F, S) with disc(S) = dm 2 where d is a fixed fundamental discriminant and m varies. This should make it clear that the problem of obtaining sign changes or growth asymptotics for Fourier coefficients not associated to fundamental discriminants is of a different flavor (and relatively easier). Our focus in this paper is on the subsequence of Fourier coefficients a(F, S) with S restricted to matrices of fundamental discriminant, where these questions are more difficult.
1.1. Main results. Let k > 2 be an even integer and N be an odd squarefree integer. Fix F ∈ S k (Γ (2) 0 (N )). If N > 1, assume that F is an eigenform for the U (p) Hecke operator (see (48)) for the finitely many primes p|N ; we make no assumptions concerning whether F is a Hecke eigenform at primes not dividing the level N . Our main result on sign changes is as follows.
Theorem A. (see Theorem 5.2) For F as above with real Fourier coefficients one can fix M such that given ε > 0 and sufficiently large X, there are ≥ X 1−ε distinct odd squarefree integers n i ∈ [X, M X] and associated fundamental matrices S i ∈ Λ 2 with |disc(S i )| = n i , such that with the n i ordered in increasing manner we have a(F, S i )a(F, S i+1 ) < 0.
Thus, Theorem A asserts that there are at least X 1−ε (strict) sign changes among the fundamental Fourier coefficients of discriminant ≍ X. Interestingly, this also improves the exponent of the non-vanishing results of [Sah13,SS13] mentioned earlier, where it was proved that there are ≫ ε X 5/8−ε non-vanishing fundamental Fourier coefficients of discriminant up to X.
Another question left unanswered in all previous works is that of lower bounds for |a(F, S)| with S fundamental. Let F be as above and fixed. A famous (and very deep) conjecture of Resnikoff and Saldana [RS74] predicts that for S a fundamental 2 matrix in Λ 2 , (3) |a(F, S)| ≪ F,ε |disc(S)| k 2 − 3 4 +ε . We prove a lower bound for many fundamental Fourier coefficients with an exponent of the same strength.
Theorem B. (see Theorem 5.3) For F as above, ε > 0 fixed, and all sufficiently large X there are ≥ X 1−ε distinct odd squarefree integers n ∈ [X, 2X], with associated fundamental matrices S n such that |disc(S n )| = n and |a(F, S n )| ≥ n 2 The conjecture extends to non-fundamental matrices but then it needs to be modified slightly by excluding the Saito-Kurokawa lifts.
Theorem B tells us that there are at least X 1−ε fundamental Fourier coefficients of discriminant ≍ X whose sizes are "large". Incidentally, just like Theorem A, Theorem B also improves upon the exponent of the set of non-vanishing fundamental coefficients obtained in [SS13] from 5/8 to 1.
Next, we investigate upper bounds for the Fourier coefficients |a(F, S)| for fundamental S. The best currently known bound is due to Kohnen [Koh93] who proved that |a(F, S)| ≪ F,ε |disc(S)| k 2 − 13 36 +ε . This bound is quite far from the conjectured true bound (3). In fact, even if one were to assume the Generalized Lindelöf hypothesis, one only obtains the upper bound ≪ F,ε |disc(S)| k 2 − 1 2 +ε (as explained further below). Thus, the exponent k 2 − 1 2 appears to be a natural barrier. By employing probabilistic methods and assuming the Generalized Riemann Hypothesis (GRH) for several L-functions, we are able to go beyond this barrier for the first time.
Theorem C. (see Theorem 5.14) Let k > 2 be an even integer and N be an odd squarefree integer. Fix F ∈ S k (Γ (2) 0 (N )). Assume that the refined Gan-Gross-Prasad conjecture [Liu16, (1.1)] holds 3 for Bessel periods of holomorphic cusp forms on SO 5 (A). Assume that GRH holds for L-functions in the Selberg 4 class. Then we have for fundamental matrices S. We note that a bound similar to that obtained in Theorem C has been recently proved in the special case where F is a Yoshida lift by Blomer and Brumley [BB20, Corollary 4].
1.2. The reduction of Theorems A and B to half-integral weight forms. The proofs of Theorems A and B rely on reducing these questions to corresponding ones about cusp forms of weight k − 1 2 on the upper half-plane, exploiting the Fourier-Jacobi expansion of F and the relation between Jacobi forms and classical cusp forms of half-integral weight. More precisely, using [Iwa74] it follows (see Section 5.2) that the set of primes p such that the p'th Fourier-Jacobi coefficient of F is non-zero has positive density in the set of all primes; fix any p in this set coprime to N . Using a classical construction going back to Eichler and Zagier [EZ85, Thm 5.6] in the case N = 1 and due to Manickam-Ramakrishnan [MR00] for squarefree N , we can now construct a non-zero cusp form h of level 4N p and weight k − 1 2 whose Fourier coefficients a(h, n) essentially equal some a(F, S) with |disc(S)| = n.
From the above construction, Theorem A will follow if we can demonstrate X 1−ε sign changes among the coefficients a(h, n) of the half-integral weight form h for odd squarefree n ≍ X, which is exactly what we prove in Theorem 3.1, a result which builds upon works of Matomäki and Radziwi l l [MR15] and [LR21] and may be of independent interest. A point worth noting here is that h is not typically a Hecke eigenform (even when F is a Hecke eigenform) as the passage from Siegel cusp forms to Jacobi forms described above is not a functorial correspondence. The main ingredient for our proof of Theorem 3.1 is the demonstration of cancellation in sums of a(h, n) over almost all short intervals together with bounds on their moments thereby providing a lower bound on sums of |a(h, n)| over almost all short intervals. Combining the two results shows that over many short intervals the absolute value of the average of a(h, n) is strictly smaller than that of |a(h, n)|. Consequently, a sign change of a(h, n) occurs in many short intervals.
Likewise, Theorem B follows provided we can demonstrate suitable large values for |a(h, n)|. This is done in Section 4. The main result of that section, Theorem 4.1, says that there are at least log n log log n . Theorem 4.1 generalizes recent work of Gun-Kohnen-Soundararajan [GKS20] which dealt with the case of h of level 4. The proof of Theorem 4.1 follows the "resonance method"-strategy of [Sou08,GKS20]; however there are additional complications coming from the level which we need to overcome. The starting point of the proof is to use Kohnen's basis for S + k+ 1 2 (4N ) consisting of newforms and an explicit form of Waldspurger's formula to reduce the problem to showing large values for (a weighted average of) a particular central L-value, while controlling sizes of certain other central L-values (see Proposition 4.2 and the discussion after it, in particular estimates (31) and (32)). This is achieved by the resonance-method as in [GKS20]. A key technical input for this method is the evaluation of the first moment of twisted central L-values (Proposition 4.3), which is obtained following the method of [SY10]. Complications arising from the level show up here in a form of extra congruence and coprimality conditions, and these are dealt with as in [RS15].
Theorem C, unlike Theorems A and B, does not involve a reduction to half-integral weight forms. We explain the main ideas behind its proof in Section 1.4 further below.
Finally, we remark that a variant of the Fourier-Jacobi expansion trick sketched at the beginning of this subsection has been recently developed by Böcherer and Das to prove non-vanishing of fundamental Fourier coefficients of Siegel modular forms of degree n [BD21]. By using their variant, it seems plausible that the methods of this paper may allow one to extend Theorems A and B above to Siegel cusp forms of higher degree. We do not pursue this extension here.
0 (N )) with k > 2 even and N odd and squarefree. Using the defining relation for Siegel cusp forms, we see that thus showing that a(F, S) only depends on the SL 2 (Z)-equivalence class of the matrix S. Let d < 0 be a fundamental discriminant, let Cl K denote the ideal class group of K = Q( √ d), and let w(K) ∈ {2, 4, 6} be the number of roots of unity in K. It is well-known that the SL 2 (Z)-equivalence classes of matrices in Λ 2 of discriminant d are in natural bijective correspondence with the elements of Cl K . So, for any character Λ of the finite group Cl K , we can define which may be viewed as a Bessel period [DPSS20, Prop. 3.5]. The space S k (Γ (2) 0 (N )) has a natural subspace S k (Γ (2) 0 (N )) CAP spanned by the Saito-Kurokawa lifts. If F is a Saito-Kurokawa lift then a(F, S) (for fundamental S) depends only on d = disc(S) and is fairly well-understood. In particular, for F ∈ S k (Γ (2) 0 (N )) CAP , the Bessel period B(F, Λ) vanishes whenever Λ = 1 K , where 1 K denotes the trivial character of Cl K . Now suppose that F is not a Saito-Kurokawa lift. Let φ be the adelization of F , and suppose that φ generates an irreducible automorphic representation π of GSp 4 (A). Böcherer [Böc86] made the remarkable conjecture that where χ d is the quadratic character associated to K/Q and A F is a constant depending only on F .
More generally, let AI(Λ) be the automorphic representation of GL(2, A) given by the automorphic induction of Λ from K; it is generated by (the adelization of) the dihedral modular form θ Λ (z) = 0 =a⊂O K Λ(a)e 2πiN (a)z of weight 1. It is easy to check that L(s, π ⊗ AI(Λ)) = L(s, π)L(s, π ⊗ χ d ). Now, assume that the refined Gan-Gross-Prasad conjecture (see [DPSS20, Conjecture 1.12] and [Liu16, (1.1)]) for the pair (φ, Λ) holds true. In fact, this conjecture for Λ = 1 K is now known thanks to work of Furusawa and Morimoto [FM21] (which combined with [DPSS20] completes the proof of Böcherer's conjecture) and the proof for general Λ has been recently announced by the same authors. Then Theorem 1.13 of [DPSS20] implies that under some mild assumptions, where c F is an explicit non-zero constant depending only on F and L(s, π × AI(Λ)) is the tensor product L-function of the spin (degree 4) L-function of π and the standard (degree 2) L-function of AI(Λ). We show in Proposition 5.9 that a variant of (6), where the equality is replaced by an inequality, holds in a more general setup (assuming the refined Gan-Gross-Prasad conjecture).
The identities (5) and (6) demonstrate that the fundamental Fourier coefficients of Hecke eigenforms in S k (Γ (2) 0 (N )) are intimately connected with central L-values of the degree 8 L-function L(s, π × AI(Λ)) as Λ varies over the ideal class characters of K. By inverting (5), we can write which expresses each fundamental a(F, S) as a weighted average of the Bessel periods B(F, Λ). Now, combining (6) and (7) with Theorem B, we obtain the following corollaries.
Corollary 1.1. Let π be a cuspidal automorphic representation of GSp 4 (A) that is not of Saito-Kurokawa type, such that π arises from a form in S k (Γ (2) 0 (N )) with k > 2 even and N odd and squarefree. Fix ε > 0. Assume the refined Gan-Gross-Prasad conjecture [DPSS20, Conjecture 1.12]. Then for all sufficiently large X, there are ≥ X 1−ε negative fundamental discriminants d with |d| ≍ X such that for K = Q( √ d), By specializing further to the case of Yoshida lifts, we obtain the following application which is purely about central L-values of dihedral twists of classical newforms.
Corollary 1.2. Let k > 2 be an even integer. Let N 1 , N 2 be two positive, squarefree integers such that M = gcd(N 1 , N 2 ) > 1. Let f be a holomorphic newform of weight 2k − 2 on Γ 0 (N 1 ) and g be a holomorphic newform of weight 2 on Γ 0 (N 2 ). Assume that for all primes p dividing M the Atkin-Lehner eigenvalues of f and g coincide. Fix ε > 0. Then for all sufficiently large X, there are ≥ X 1−ε negative fundamental discriminants d with |d| ≍ X with the property that there exists an ideal class group character Λ of K = Q( √ d) such that Corollary 1.2 strengthens the main theorem of [SS13] which showed the existence of Λ with (simultaneous) non-vanishing for L( 1 2 , f × AI(Λ)) and L( 1 2 , g × AI(Λ)) and remarked: "while our method gives a lower bound on the number of non-vanishing twists, it does not give a lower bound on the size of the non-vanishing L-value itself." Corollary 1.2 successfully achieves this.
1.4. Fractional moments of L-values. Combining (6) and (7), we can write Therefore, in order to prove Theorem C, we need to go beyond the bound obtained by a naive application of the Generalized Lindelöf hypothesis. We do this by using Soundararajan's method [Sou09] for bounding moments of L-functions. Assuming GRH, we prove the following bound (Theorem 6.1) which, thanks to (8), implies Theorem C: The main contribution to the moments of L( 1 2 , π × AI(Λ)) will come from its large values and we expect that these should be approximated by the large values of exp( p n <|d| b π×AI(Λ) (p n ) p n/2 ), where b π×AI(Λ) (n) is the n'th coefficient of the Dirichlet series of log L(s, π ×AI(Λ)). For ease of discussion let us assume here that d is prime, N = 1, and π transfers to a cuspidal representation of GL 4 5 . Separately analyzing the primes, squares of primes and higher prime powers we show under GRH where b π (p), b AI(Λ) (p) respectively denote the p'th coefficient of the Dirichlet series of log L(s, π) and log L(s, AI(Λ)). For primes with ( d p ) = 1 so that pO K = pp, as Λ varies over Cl K , we expect that b AI(Λ) (p) = Λ(p) + Λ(p) −1 behaves like the random variable X p + X −1 p where {X p } p are iid random variables uniformly distributed on the unit circle (if ( d p ) = −1, b AI(Λ) (p) = 0). Consequently the sum on the r.h.s. above is modelled by the random variable p<|d| , which can be shown to have a normal limiting distribution as d → ∞ with mean 0 and variance 2 p<|d| bπ(p) 2 p 1 ( d p )=1 ∼ log log |d|, which we prove under GRH. The preceding discussion suggests where in the last step we have used that the moment generating function of a normal random variable X with mean 0 and variance σ 2 is given by E(e zX ) = e 1 2 z 2 σ 2 . Remarkably, Soundararajan's method allows us to make this heuristic argument rigorous for the upper bound, up to the loss of a factor (log |d|) ε , which occurs due to a sub-optimal treatment of the large primes.
1.5. Notations. We use the notation A ≪ x,y,z B to signify that there exists a positive constant C, depending at most upon x, y, z, so that |A| ≤ C|B|. The symbol ε will denote a small positive quantity. We write A(x) = O y (B(x)) if there exists a positive real number M (depending on y) and a real number For a positive integer n with prime factorization n = k i=1 p α i i , we define ω(n) = k, Ω(n) = k i=1 α i . We let µ(n) denote the Möbius function, i.e., µ(n) = (−1) ω(n) if ω(n) = Ω(n), and µ(n) = 0 otherwise. We say that n is squarefree if µ(n) = 0. We let (a, b) or gcd(a, b) denote the greatest common divisor of a and b.
We say that d is a fundamental discriminant if d is the discriminant of the field Q( √ d). For a fundamental discriminant d, we let χ d be the associated quadratic Dirichlet character. Given any representation π of a group, we letπ denote the contragredient, and V π denote the representation space. We use A to denote the ring of adeles over Q and we use A F to denote the ring of adeles 5 If this is not the case (e.g., if π corresponds to a Yoshida lift) the estimates below will be slightly different and the resulting bound for the moment predicted by the heuristic will also differ. 6 over F for a general number field F . If G is a reductive group such that the local Langlands correspondence is known for each G(F v ) and π is an automorphic representation of G(A F ), then we formally (as an Euler product over finite places) define the L-function L(s, ρ(π)) := L(s, π, ρ) for each finite dimensional representation ρ of the dual group. All L-functions in this paper will denote the finite part of the L-function (i.e., without the archimedean factors), so that for a number field F and an automorphic representation π of GL n (F ), we have L(s, π) = v<∞ L(s, π v ). All L-functions will be normalized to take s → 1 − s. For an integer N we denote L N (s, π) = v∤N L(s, π v ). Given a reductive group G and two irreducible automorphic representations π = ⊗ v π v and σ = ⊗ v σ v of G(A F ), we say that π and σ are nearly equivalent if π v ≃ σ v for all but finite many places v of F . 1.6. Acknowledgements. We thank Ralf Schmidt for helpful discussions concerning the material in Section 5.4. We thank Valentin Blomer and Farrell Brumley for forwarding us their preprint [BB20]. We thank the anonymous referee for useful comments and corrections which improved this paper. This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/T028343/1].

Preliminaries on half integral weight forms
The goal of this section is to set up some notation and lay out some key properties concerning cusp forms of half-integral weight on the complex upper half-plane. (4N ) denote the space of holomorphic cusp forms of weight k + 1 2 for the group Γ 0 (4N ). In other words, a function f : We let c(f, n) denote the "normalized" Fourier coefficients, defined by For f, g ∈ S k+ 1 2 (4N ), we define the Petersson inner product f, g by 2.2. The Kohnen plus space and decomposition into old and newspaces. Fix positive integers k, N such that N is odd and squarefree. We recall the definition of the Kohnen plus space S + for which a(f, n) = 0 whenever n ≡ (−1) k+1 or 2 mod 4. According to the results of [Koh82], there exists a canonically defined subspace S +,new where we define According to the Shimura lifting [Shi73], as refined by Kohnen in [Koh82], there is an isomorphism as Hecke modules, where S new 2k (N ) is the orthogonal complement of the space of cuspidal oldforms of weight 2k for Γ 0 (N ) as defined by Atkin-Lehner [AL70]. The Shimura lifting (12) takes each newform in S +,new In view of (9) and the fact that the operators U (p) with p|N commute with T (p 2 ), p ∤ N , a basis of S + k+ 1 2 (4N ) consisting of eigenforms for T (p 2 ), p ∤ N , is given by (4ℓ) consisting of newforms. Note however that all members of B k+ 1 2 ,4N are not necessarily orthogonal to each other. The following result will be useful for us; recall the definitions of Ω(n) and ω(n) from Section 1.5.
Lemma 2.1. Let r, ℓ be positive, odd, squarefree integers with (r, ℓ) = 1 and let f ∈ S +,new k+ 1 2 (4ℓ) be a newform. Then for any odd squarefree integer n, putting d = (−1) k n, we have Additionally, for any odd integer r ≥ 1 with (r, ℓ) = 1 we have Proof. The first statement follows from [Shi73, Corollary 1.8 (i)]. Using that |λ f (p)| ≤ 2 and applying [Shi73, Corollary 1.8 (ii)] we will establish the second claim by the following simple induction argument. It suffices to show for each p ∤ 2ℓ that The case m = 0 is trivial and m = 1 follows from the first claim of the lemma. By [Shi73, Corollary 1.8 (ii)] we have for any m ≥ 1 Hence, for m ≥ 1 we get that

An explicit version of Waldspurger's formula.
A well-known formula of Waldspurger [Wal85] that was refined and made explicit in special cases by Kohnen [Koh85], expresses the squares of Fourier coefficients of half-integral weight eigenforms in terms of central L-values. We state a version of it below for elements of the basis (13).
Proposition 2.2. Let r, ℓ be positive, odd, squarefree integers with (r, ℓ) = 1. Let f be a newform in S +,new k+ 1 2 (4ℓ) and let g ∈ S new 2k (ℓ) be the Shimura lift of f . Then, for any squarefree positive integer n with (n, 4ℓ) = 1, and d = (−1) k n, we have where w p is the eigenvalue for the Atkin-Lehner operator at p acting on g. If either of the two conditions above are not met, then c(f |U (r 2 ), n) = 0.

2.5.
Estimates on moments of Fourier coefficients.
where N is odd and squarefree. Then there exists M ≥ 2 such that for all sufficiently large X, and for any ε > 0 Proof. We first prove (14). For the upper bound we use that y k+1/2 |f (z)| is bounded on H and hence we have For the lower bound, we use a result obtained in the proof of [Sah13, Proposition 3.7], which gives for any M ≥ 1 that Using (16) along with partial summation we can bound the tail end of the sum as follows Combining the two bounds above we have for M sufficiently large that Finally, we note that by (16) the contribution from terms to the l.h.s. above with n ≤ X is O f (X), which completes the proof of the lower bound in (14). For the proof of (15), we use (13) to reduce to the case f = f 1 |U (r 2 ) where f 1 ∈ S +,new newform with rℓ|N . Using Proposition 2.2, it now suffices to prove that where g is the Shimura lift of f 1 and the sum is over fundamental discriminants d. This follows from the approximate functional equation and Heath-Brown's quadratic large sieve [HB95], using a straightforward modification of the proof of [HB95, Theorem 2] (see also [SY10, Corollary 2.5]).

Sign changes for coefficients of half-integral weight forms
3.1. Statement of main result. Throughout this section let k ≥ 2 be an integer and N ≥ 1 be odd and squarefree. The main theorem to be proved in this section is be a fixed cusp form whose Fourier coefficients c(f, n) are all real. Then there exists M ≥ 2 such that given any ε > 0, the sequence has at least ≫ f,M,ε X 1−ε sign changes.
The main novelty here is that this result holds for all cusp forms f ∈ S + k+1/2 (4N ), not just Hecke eigenforms, and this is crucial for our later application. Previously it was not apparently even known that there are infinitely many sign changes of c(f, n) as n ranges over squarefree integers for f ∈ S + k+ 1 2 (4N ).
Our proof builds upon the methods developed in [MR15,LR21] and relies upon the following two propositions. The first of which shows that the size of |c(f, n)| is relatively well-behaved for most short intervals [x, x + y].
Proposition 3.2. Let f ∈ S + k+ 1 2 (4N ). There exists M ≥ 2 such that given any ε > 0 and Our other main proposition shows that we can obtain square-root cancellation in sums of c(f, n) over almost all short intervals [x, x + y].
We will now prove Theorem 3.1 using Propositions 3.2 and 3.3. The proof of Propositions 3.2 and 3.3 will be given in Sections 3.2 and 3.3, respectively.
Proof of Theorem 3.1. Observe that if the Fourier coefficients c(f, n) are real and then the interval [x, x + y] must contain a sign change of c(f, n) where n ∈ [x, x + y] ranges over odd squarefree integers that are coprime to N . We will show that for most integers X ≤ x ≤ M X that the above inequality holds for intervals of length y = X 6ε .
By Chebyshev's inequality, the number of integers X ≤ x ≤ M X for which where we have used Proposition 3.3 in the last inequality. By Proposition 3.2 we have that (17) and (18) hold. Therefore, we obtain at least ≫ f,M,ε X 1−3ε/2 y = X 1−15ε/2 sign changes of c(f, n) along integers X ≤ n ≤ M X, that are odd, squarefree, and coprime to N .
3.2. Proof of Proposition 3.2. We first prove the following result, which is an easy consequence of Proposition 2.3.
Then there exists M ≥ 2 such that given any ε > 0 Proof. Applying Hölder's inequality gives Hence, using Proposition 2.3 we conclude that Combining this with (20) completes the proof.
where the second inequality above follows since every term in the sum on the l.h.s. is counted ⌊y⌋+1 times on the r.h.s..
Hence we must have that 3.3. Proof of Proposition 3.3. Throughout this section we write The proof of the proposition proceeds directly, beginning with an application of Cauchy-Schwarz. This leads naturally to a shifted convolution sum of Fourier coefficients of f over squarefree integers and to bound this sum we require the following fairly standard result.
and v ∈ Z with (v, r) = 1 we have for any given ε > 0 that Proof. This is an extension of Proposition 6.1 of [LR21] to the case of general level and below we will describe how to adapt the arguments given there to this case. The initial step is to use the Fourier expansion of f to express the l.h.s. of (22) as We now use the circle method following Jutila [Jut92], as in [Har03,Proposition 2]. An important feature in Jutila's version of the circle method is that we have freedom over our choice of moduli, which we choose as follows Notably, to estimate the error term we use that y k 2 + 1 4 |f (z)| is bounded on H, since f is a cusp form. Since we have chosen our moduli q ∈ Q so that 4N |q we are able to use the modularity of f by applying [LR21, Lemma 6.1], which extends to general level in straightforward way, then once again use the Fourier expansion of f . Consequently, we have transformed the original sum on the l.h.s. of (22), which is effectively over n ≤ X 1+ε , to dual sums which are effectively over m, n ≤ X ε Q 2 /X. The summands in the dual sums include the Fourier coefficients of f twisted by additive characters and factors from the half-integral weight multiplier system along with a Kloosterman sum S(⋆, −h; p), where the first argument, ⋆, depends on N, p, m, n, v, r. An important observation is that since p is a prime with 0 < |h| < p, the Weil bound gives |S(⋆, −h; p)| ≤ 2 √ p for any ⋆ ∈ Z. Using the Weil bound and estimating the dual sums over m, n by applying Cauchy-Schwarz and (16) to handle the Fourier coefficients of f we can show that (23) is bounded by Recalling our earlier error term of O(X 1−η/8+ε ), which arose from applying Jutila's circle method we now take η = 4/51 to complete the proof.
To sum over squarefree integers we will sieve out integers that have a square divisor and require the following estimate for sums of Fourier coefficients.
Using this bound then applying Cauchy-Schwarz and (16) we get that Proof of Proposition 3.3. To handle the condition that 2n is squarefree we first recall that the indicator function of squarefree numbers is µ 2 (n) = d 2 |n µ(d). We then treat the cases of divisors d ≤ Y and d > Y separately, and let First we consider the large divisors and get that Using the definition of µ 2 >Y (n) and applying Lemma 3.6 we see that the r.h.s. above is For Y ≥ √ yX ε this is ≪ X √ y, as needed.
Next, we will consider the contribution from the small divisors d ≤ Y .
Let C(f, n) = c(f, n)µ 2 ≤Y (n)1 (n,2N )=1 . Applying Cauchy-Schwarz and using that W (u) 2 ≫ 1, for any u ∈ [1, 2] we get X≤x≤2X x≤n≤x+y Assume y ≤ X 1/4 . We use the convention that c(f, n) = 0 if n / ∈ N. To estimate the inner sums on the r.h.s. we expand the square, combine appropriate terms, use that W is a smooth function, and apply (16) to get that the r.h.s. above equals = 0≤h 1 ,h 2 ≤y n≥1 Using (16) once again we get that the term with h = 0 in the sum on the r.h.s. above contributes 14 We will next estimate the contribution from the terms in (25) with h = 0. Recalling the definition of µ 2 ≤Y and using that 1 (n,2N )=1 = d|(n,2N ) it follows that the contribution to the r.h.s. of (25) from the terms with h = 0 is For n with d 2 1 |n, d 2 2 |n + h, d 3 |n, d 4 |n + h we have n ≡ a (mod r) for some a, r ∈ Z with r ≤ 16N 2 Y 4 . Using additive characters to detect this congruence, we get that the inner sum above is For v = 0 write v/r = v ′ /r ′ with (r ′ , v ′ ) = 1 and if v = 0 we set r ′ = 1. Applying Proposition 3.5, we get that the sum above is ≪ f,ε X 1− 1 102 +ε provided that r ′ ≤ X 1 102 and 0 < |h| < X 1 2 . Hence, by this along with (26) we conclude that for 16N 2 Y 4 ≤ X 1 102 that the r.h.s. of (25) is ≪ f,ε yX + y 2 Y 2 X 1− 1 102 +ε + y 3 X ε , which is ≪ yX, as needed, provided that y ≤ X 1/4 and (28) yY 2 X ε ≪ X 1 102 .
It remains to optimize our parameters. Recall that to handle the contribution of the small divisors we required Y ≥ √ yX ε . We now choose Y = √ yX ε . Taking the constraint (28) into account the largest we can choose y is y = X 1 204 − 3 2 ε . We conclude by noting that with these choices we have 16N 2 Y 4 ≪ N X 1 102 −3ε ≤ X 1 102 as required for the application of Proposition 3.5.

Large values for coefficients of half-integral weight forms
The main result of this section, Theorem 4.1 below, generalizes [GKS20, Theorem 1] (which treated the case N = 1).

Statement of main result.
Theorem 4.1. Let k ≥ 2 be an integer, N ≥ 1 be odd and squarefree, and h ∈ S + k+ 1 2 (4N ) be a cusp form. Let ε > 0 be fixed. Then for all X sufficiently large, there exist at least X 1−ε odd squarefree integers n coprime to N such that X ≤ n ≤ 2X and |c(h, n)| ≥ exp 1 82 log n log log n .
We will prove this theorem by combining methods of [GKS20] and [RS15,LR21]. Our first job is to reduce the question to bounding central values of L-functions. This is done by using the explicit form of Waldspurger's formula due to Kohnen (Proposition 2.2 above).

4.2.
Reduction to bounds on L-values. Fix an integer k ≥ 2 and an odd squarefree integer N ≥ 1 throughout Section 4. Let h be as in Theorem 4.1. We use the basis (13) to write where the coefficients α r,ℓ,f depend only on r, ℓ, f, and h. For each odd squarefree n, we use Lemma 2.1 to get the following identity for the Fourier coefficients: We already know that c(h, n) = 0 for some odd squarefree n (this follows from [SS13] for example).
, and a reduced residue class η mod 4N such that Above, w p is the eigenvalue of the Atkin-Lehner operator W p acting on f 0 . For brevity, we denote for each f ∈ B new We will denote the Shimura lift of f ∈ B new k+ 1 2 ,4ℓ by g f ∈ S new 2k (ℓ) with Fourier coefficients m k− 1 2 λ g f (m) normalized so that λ g f (1) = 1. Also, write g 0 for g f 0 and m k− 1 2 λ 0 (m) for its Fourier coefficients. For each odd squarefree integer n such that d = (−1) k n ≡ η (mod 4N ), we use the triangle inequality, Cauchy-Schwarz, and Proposition 2.2 to obtain where A > 0 and B > 0 are independent of d. Now, Theorem 4.1 follows from the following auxiliary result.

Combining Proposition 4.2 and (29),
Let S be the subset of D N,η (X) for which the estimate (30) holds. Then certainly Suppose that the following estimates hold: where R = L 2 ≤p≤L 4 1 + r(p) 2 λ 0 (p) 2 .
Assuming (31)-(34) the proof of Proposition 4.2 can be finished as follows. We observe that by using (31), (32) and (33). Hence, On the other hand, r.h.s of the previous display can be estimated by Hölder's inequality and (34) as where, as before, the average of the squares of central L-values is estimated by using the quadratic large sieve of Heath-Brown [HB95] (here we can extend the sum to all fundamental discriminants ≤ X in magnitude by non-negativity). Combining this with (35) gives |S| ≫ k,N X 1−ε/2 ≥ X 1−ε , as desired.
So it suffices to establish estimates (31)-(34). Notice that (33) and (34) follow directly from Proposition 3 of [GKS20] by simply estimating The other two estimates are consequences of the following first moment result.

4.3.
A twisted first moment asymptotic. In this subsection, we sketch the proof of the proposition above. The starting point is the approximate functional equation (which follows by an easy modification of the proof of Lemma 5 in [RS15]) saying that where W g,η is a smooth weight function defined by, for c > 1/2, Notice that the value of L g,η (s) is the same for each d ≡ η (mod 4N ). The weight function satisfies W g,η (ξ) = L g,η (1/2) + O k,N,ε (ξ 1 2 −ε ) as ξ −→ 0 and W g,η (ξ) ≪ k,N,A |ξ| −A for any A ≥ 1 as ξ −→ ∞. Thus the sum we have to evaluate takes the shape Notice that by definition any d ∈ D N,η is squarefree and odd. We pick out this property by the identity Note that the above identity holds without the condition (α, 2N ) = 1, but this can be added as by construction (d, 2N ) = 1 for d ∈ D N,η . Inserting this to the above expression gives that the d-sum is given by We will evaluate the r-sum by applying a version of Poisson summation formula [RS15, Lemma 7]. The terms where mu is a square will contribute the main term in the zero-frequency term on the dual side and the rest will give the error term. By using standard arguments [LR21] the contribution of the latter terms can be bounded by ≪ k,N,Φ,ε X 7/8+ε u 3/8+ε .

Using (36) the zero-frequency contribution is given by
A simple computation shows that, for (m, 2N ) = 1, For mu a square and u = u 1 u 2 2 with u 1 squarefree, it follows that m = u 1 ℓ 2 for some ℓ ∈ Z. Hence the r.h.s of the previous display is From this it is easy to see, by using the Euler product expression of the symmetric square L-function, that for p ∤ 2N u the corresponding Euler factor is where {α p , β p } are the Satake parameters of the cusp form g at p.
To summarize, the r.h.s of (39) equals λ g (u 1 ) u s+1/2 1 L 2s + 1, Sym 2 g G(2s + 1; u), 20 where G(2s + 1; u) = p G p (2s + 1; u) is the Euler product locally given by By estimating trivially it follows that G(2s + 1; u) extends analytically to the domain Re(s) > −1/4 and is bounded there by Consequently the s-integrand in (38) extends to an analytic function in the above domain (apart from a simple pole at s = 0). Thus moving the line of integration in (38) to the line Re(s) = −1/4+ε shows that the expression equals where the main terms comes from the residue at s = 0 and the error term from the contour shift. It follows immediately from the definition of G(s; u) that G(1; ·) is multiplicative and that G(1; p k ) = 1 + O(1/p) at prime powers. This concludes the sketch of the proof of Proposition 4.3.
Write n = n 1 n 2 to express the double sum above as a single sum over n and note that by the fact that r(·) is supported only on squarefree integers coprime to N the only integers n which contribute to the sum over n satisfy p 3 ∤ n and (n, N ) = 1. Hence, by multiplicativity of r(·) we can express the sum over n as an Euler product and the expression on the r.h.s of the previous display equals where the last estimate follows from Deligne's bound |λ g (p)| ≤ 2, the fact that G(1; p k ) = 1+O(1/p), and the definition of r(n).
Let us now choose α = 1/(8 log L). Then by the prime number theorem and partial summation the above is by the choices L = 1 8 √ log M log log M and M = X 1/24 . From the above arguments we deduce that where the last step follows exactly as in [GKS20,§6].

Siegel cusp forms of degree 2
In this Section we first review various properties of Siegel cusp forms of degree 2 and then go on to prove our main results stated in the introduction. where I 2 is the identity matrix of size 2. Define the algebraic groups GSp 4 and Sp 4 over Z by GSp 4 (R) = {g ∈ GL 4 (R) | t gJg = µ 2 (g)J, µ 2 (g) ∈ R × }, Sp 4 (R) = {g ∈ GSp 4 (R) | µ 2 (g) = 1}, for any commutative ring R. The Siegel upper-half space H 2 of degree 2 is defined by We define 0 (N )) denote the space of Siegel cusp forms of weight k and level N ; precisely, they consist of the holomorphic functions F on H 2 which satisfy the relation for γ ∈ Γ (2) 0 (N ), Z ∈ H 2 , and vanish at all the cusps. Any F in S k (Γ for A ∈ GL 2 (Z). In particular, the Fourier coefficient a(F, T ) depends only on the SL 2 (Z)equivalence class of T . We say that a matrix S ∈ Λ 2 is fundamental if disc(S) = −4 det(S) is a fundamental discriminant. Given a fundamental discriminant d < 0 and K = Q( √ d), let Cl K denote the ideal class group of K. It is well-known that the SL 2 (Z)-equivalence classes of matrices in Λ 2 of discriminant d can be identified with Cl K ; so the expression S∈Cl K a(F, S)Λ(S) makes sense for each Λ ∈ Cl K . 5.2. Constructing half-integral weight forms. Each F in S k (Γ (2) 0 (N )) has a Fourier-Jacobi expansion F (Z) = m>0 φ m (τ, z)e 2πimτ ′ where we write Z = τ z z τ ′ and for each m > 0, a F, n r/2 r/2 m e 2πi(nτ +rz) ∈ J cusp k,m (N ).
Here J cusp k,m (N ) denotes the space of Jacobi cusp forms of weight k, level N and index m. Given a primitive matrix S = a b/2 b/2 c ∈ Λ 2 (i.e., gcd(a, b, c) = 1) we let P(S) denote the set of primes of the form ax 2 + bxy + cy 2 . The set P(S) is infinite; indeed by [Iwa74, Theorem 1 (i)], For each prime p dividing N , define the operator U (p) acting on the space S k (Γ Lemma 5.1. Let k > 2 be even and N be squarefree. Let F ∈ S k (Γ (2) 0 (N )) be an eigenfunction of the U (p) operator for each prime p|N (note that if N = 1, this condition is trivially true).
i) Then there exists S 0 = a b/2 b/2 c ∈ Λ 2 such that a(F, S 0 ) = 0 and d 0 = b 2 − 4ac is odd and squarefree (and hence, S 0 is fundamental). ii) Let S 0 , d 0 be as above and let p ∤ 2N d 0 be a prime such that p ∈ P(S 0 ). Put Proof. The claim that there exists S 0 = a b/2 b/2 c ∈ Λ 2 such that a(F, S 0 ) = 0 and d 0 = b 2 − 4ac is odd and squarefree follows from Theorem 2.2 of [SS13]. Now let p ∈ P(S 0 ), p ∤ 2N d 0 . The fact that h p ∈ S k− 1 2 (4pN ) follows from φ p ∈ J cusp k,p (N ) and Theorem 4.8 of [MR00]; by definition h p belongs to the Kohnen plus space. Next, we prove (49). Let gcd(m, 4p) = 1. Observe that the sum is non-empty if and only if −m is a 24 square modulo 4p in which case the sum has exactly two terms. Indeed, we get that (50) a(m) = a F, , we obtain (49).

The proofs of Theorems A and B.
We are now ready to prove Theorems A and B of the introduction in a slightly stronger form.
Theorem 5.2. Let F ∈ S k (Γ (2) 0 (N )) with k > 2 even and N odd and squarefree. Assume that F is an eigenfunction of the U (p) operator for each prime p|N , and that F has real Fourier coefficients. Then there exists a set P of primes satisfying |{p ∈ P : p ≤ X}| ≫ F X log X such that given ε > 0 and p ∈ P, there exist M ≥ 2 (depending only on F and p) and X 0 ≥ 1 (depending on F , p and ε) so that for all X ≥ X 0 , there are r X ≥ X 1−ε matrices S 1 , S 2 , . . . , S r X ∈ Λ 2 satisfying the following: Proof. Using Lemma 5.1, we fix a fundamental matrix S 0 such that a(F, S 0 ) = 0. Take P = P(S 0 ); then the estimate |{p ∈ P : p ≤ X}| ≫ X log X follows from (47). Given any p ∈ P, let f = h p be as in Lemma 5.1, so that 0 = f ∈ S + k− 1 2 (4pN ) and the coefficients of f are all real since the coefficients of F are.
By Theorem 3.1, there exists M ≥ 2 such that for any ε > 0, the sequence {a(f, n)µ 2 (n)} X≤n≤M X (n,2N )=1 has ≥ C f,M,ε X 1−ε/2 sign changes for some constant C f,M,ε . Pick X 0 ≥ 1 so that for all X ≥ X 0 we have X ε/2 ≥ 1 C f,M,ε . Then for all X ≥ X 0 , there exists r X ≥ X 1−ε , and an increasing sequence (n i ) 1≤i≤r X of odd squarefree integers satisfying a(f, n i )a(f, n i+1 ) < 0. For each n i as above, (49) tells us that a(f, n i ) = 2a(F, S i ) for some S i = * * * p ∈ Λ 2 with |disc(S i )| = n i . This completes the proof.
0 (N )) with k > 2 even and N odd and squarefree. Assume that F is an eigenfunction of the U (p) operator for each prime p|N . Then there exists a set P of primes satisfying |{p ∈ P : p ≤ X}| ≫ F X log X such that given ε > 0 and p ∈ P, one can find X 0 ≥ 1 (depending on F , p and ε) so that for all X ≥ X 0 , there are r X ≥ X 1−ε matrices S 1 , S 2 , . . . , S r X ∈ Λ 2 satisfying the following: i) For each 1 ≤ i ≤ r X , S i = * * * p , and disc(S i ) is a squarefree integer coprime to 2N , ii) X ≤ |disc(S 1 )| < |disc(S 2 )| < . . . < |disc(S r X )| ≤ 2X, Proof. The proof is essentially identical to Theorem 5.2, except that we use Theorem 4.1 (rather than Theorem 3.1) on f = h p .
5.4. L-functions of Siegel cusp forms. For the rest of this section we assume that k > 2.
Given an irreducible cuspidal automorphic representation π of GSp 4 (A), we let L(s, π) denote the associated degree 4 L-function (known as the "spin" L-function). Furthermore, we let L(s, std(π)) denote the associated degree 5 L-function (the "standard" L-function) and L(s, ad(π)) denote the associated degree 10 L-function (the "adjoint" L-function). Each of these L-functions is defined as an Euler product with the local L-factor at each prime (including the ramified primes) constructed via the associated representation of the dual group GSp 4 (C) using the local Langlands correspondence (which is known for GSp 4 by the work of Gan-Takeda [GT11]). More precisely, for n = 4, 5, 10, let ρ n denote the irreducible n-dimensional representation of GSp 4 (C) given as follows: ρ 4 is the inclusion of GSp 4 (C) ֒→ GL 4 (C), ρ 5 is the map defined in [RS07,A.7], and ρ 10 is the adjoint representation of GSp 4 (C) on the Lie algebra of Sp 4 (C). Then L(s, π), L(s, std(π)), and L(s, ad(π)) correspond to the representations ρ 4 , ρ 5 and ρ 10 respectively.
We say that an element F of S k (Γ (2) 0 (N )) gives rise to an irreducible representation if its adelization (in the sense of [Sah15,§3]) generates an irreducible cuspidal automorphic representation of GSp 4 (A). The automorphic representation associated to any such F is of trivial central character and hence may be viewed as an automorphic representation of PGSp 4 (A) ≃ SO 5 (A). It can be checked [Sah15,Prop. 3.11] that if F gives rise to an irreducible representation then F is an eigenform of the local Hecke algebras at all primes not dividing N . In addition, we say that such an F is factorizable if its adelization corresponds to a factorizable vector in the representation generated.
We say that an irreducible cuspidal automorphic representation π of GSp 4 (A) arises from S k (Γ (2) 0 (N )) if there exists F ∈ S k (Γ (2) 0 (N )) whose adelization generates V π (in which case, by definition, F gives rise to the irreducible representation π, which therefore must be of trivial central character by the comments above). We say that an irreducible cuspidal automorphic representation π of GSp 4 (A) is CAP if it is nearly equivalent to a constituent of a global induced representation of a proper parabolic subgroup of GSp 4 (A). If a CAP π arises from S k (Γ (2) 0 (N )), then by Corollary 4.5 of [PS09b] it is associated to the Siegel parabolic (i.e., it is of Saito-Kurokawa type). Such a π is non-tempered at almost all primes (in particular, it violates the Ramanujan conjecture). On the other hand, if π arises from S k (Γ (2) 0 (N )) and is not CAP, then π satisfies the Ramanujan conjecture by a famous result of Weissauer [Wei09,Thm. 3.3].
Thus, the space S k (Γ (2) 0 (N )) has a natural decomposition into orthogonal subspaces 0 (N )) CAP is spanned by forms F which give rise to irreducible representations of Saito-Kurokawa type, and S k (Γ (2) 0 (N )) T is its orthogonal complement, spanned by forms F which give rise to irreducible representations that are not of Saito-Kurokawa type. Furthermore, one can get a basis of each of the spaces S k (Γ (2) 0 (N )) CAP and S k (Γ (2) 0 (N )) T in terms of factorizable forms. We refer the reader to [DPSS20, §3.1 and §3.2] for further comments related to the above discussion.
If π arises from S k (Γ (2) 0 (N )) and is of Saito-Kurokawa type, then there exists a representation π 0 of GL 2 (A) and a Dirichlet character χ 0 satisfying χ 2 0 = 1 such that L N (s, π) = L N (s, π 0 )L N (s + 1/2, χ 0 )L N (s − 1/2, χ 0 ). Additionally, if N is squarefree, then only χ 0 = 1 is possible by a wellknown result of Borel [Bor76] and so in this case we have L N (s, π) = L N (s, π 0 )ζ N (s + 1/2)ζ N (s − 1/2). There exists another typical situation where the L-function factors: we say that a π arising from S k (Γ (2) 0 (N )) is of Yoshida type if there are representations π 1 and π 2 of GL 2 (A) such that L(s, π) = L(s, π 1 )L(s, π 2 ); in this case one has (after possibly swapping π 1 and π 2 ) that π 1 arises from a classical holomorphic newform of weight 2 and π 2 arises from a classical holomorphic newform of weight 2k − 2 (see [Sah15,Sec. 4] for more details). We let S k (Γ (2) 0 (N )) Y denote the subspace of S k (Γ (2) 0 (N )) T spanned by forms which give rise to an irreducible representation of Yoshida type, and let S k (Γ (2) 0 (N )) G , which represents the general type, denote the orthogonal complement of 0 (N )) T . So we get the following key orthogonal decomposition into subspaces (51) 0 (N )) G . In the notation of [Sch18], the three subspaces on the right side above correspond to the global Arthur packets of type (P), (Y) and (G) respectively. In the sequel we will be mostly concerned with the space S k (Γ (2) 0 (N )) G because the other two spaces are easier to handle. The following proposition collects together some relevant facts about the associated L-functions that follow from the work of Arthur [Art13].
i) The representation π has a strong functorial lifting to an irreducible cuspidal automorphic representation Π 4 of GL 4 (A). In particular, if σ is any irreducible automorphic representation of GL n (A), then we have an equality of degree 4n Rankin-Selberg L-functions L(s, π × σ) = L(s, Π 4 × σ) and therefore L(s, π × σ) satisfies the usual properties 6 of an L-function. If n ≤ 3, then L(s, π × σ) is entire. ii) The representation π has a strong functorial lifting to an irreducible automorphic representation Π 5 of GL 5 (A). In particular, if σ is any irreducible automorphic representation of GL n (A), then we have an equality of degree 5n Rankin-Selberg L-functions L(s, std(π)×σ) = L(s, Π 5 × σ) and therefore L(s, std(π) × σ) satisfies the usual properties of an L-function.
If n ≤ 2, and σ has the property that the set of finite primes where it is ramified is either empty or contains at least one prime not dividing N , then L(s, std(π) × σ) is entire. iii) The degree 10 L-function L(s, ad(π)) satisfies the usual properties of an L-function, is entire, and has no zeroes on the line Re(s) = 1.
Proof. Since F corresponds to the "general" Arthur parameter, Arthur's work [Art13] (see also Section 1.1 of [Sch18]) shows that π has a strong lifting to a cuspidal automorphic representation Π 4 of GL 4 ; clearly Π 4 has trivial central character since π does. It is known from [Art13, Thm. 1.5.3] that Π 4 is self-dual and symplectic 7 . The fact that the lifting is "strong", i.e., corresponds to a local lift at all places, implicitly uses that the local parameters of Arthur coincide with the local Langlands parameters of Gan-Takeda [GT11]; this consistency of local parameters follows from [CG15]. The required properties of L(s, π × σ) now follow from Rankin-Selberg theory of GL n . For the second assertion, let ∧ 2 denotes the exterior square, and note that ∧ 2 ρ 4 = 1 + ρ 5 , hence (52) L(s, Π 4 , ∧ 2 ) = L(s, std(π))ζ(s).
Recall that L(s, Π 4 × Π 4 ) = L(s, Π 4 , Sym 2 )L(s, Π 4 , ∧ 2 ). Since Π 4 is symplectic, L(s, Π 4 , ∧ 2 ) has a simple pole at s = 1 and L(s, Π 4 , Sym 2 ) is an entire function by [BG92,Thm. 7.5]. It follows from (52) that L(s, std(π)) is holomorphic and non-zero at s = 1. Together with [Gri79, Theorem 2], we obtain that L(s, std(π)) has no poles on Re(s) = 1. On the other hand, by [Kim03, Thm. A] and [Hen09] L(s, Π 4 , ∧ 2 ) is the L-function of an automorphic representation of GL 6 (A) of the form 6 By this we mean that this L-function has meromorphic continuation to the entire complex plane, satisfies the standard functional equation taking s → 1 − s, has an Euler product, and is bounded in vertical strips (in particular, the L-function is in the extended Selberg class). 7 Recall that a self-dual representation Π of GLn(A) is said to be symplectic if L(s, Sym 2 Π) is holomorphic at s = 1.
Observe that each τ i is unitary, cuspidal, and unramified outside primes dividing N . By Rankin-Selberg theory, L(s, Π 5 × σ) = m−1 i=1 L(s, τ i × σ) satisfies the usual properties of an L-function. To complete the proof, it suffices to show that if σ is an irreducible cuspidal representation of GL 1 (A) or GL 2 (A) such that the set of ramification primes for σ is either empty or contains at least one prime outside N , then τ i ≃σ for each i. First, consider the case that σ is a character, in which case we are reduced to n i = 1. In this case, since τ i is unramified outside N , the assumption on σ means that the situation τ i ≃σ will force τ i to be unramified everywhere, in which case the right hand side of (53) would have a pole on Re(s) = 1. This contradicts the observation from above that L(s, std(π)) has no poles on Re(s) = 1. Next consider the case that σ is a cuspidal representation of GL 2 (A). An identical argument to the above reduces us to the case where τ i ≃σ is unramified at all finite primes. The easy relation ∧ 2 ρ 5 = Sym 2 ρ 4 = ρ 10 implies that (54) L(s, Π 5 , ∧ 2 ) = L(s, ad(π)) = L(s, Π 4 , Sym 2 ), which we know is an entire function. On the other hand, if some τ i in (53) has n i = 2 and is unramified at all finite primes, then L(s, Π 5 , ∧ 2 ) will contain a factor of L(s, ω τ i ) and so will have a pole on Re(s) = 1. This contradiction completes the proof that each L(s, τ i × σ) and hence L(s, Π 5 × σ) is entire. Finally, the assertion concerning the degree 10 L-function L(s, ad(π)) follows from the identity (54) and the fact that L(s, Π 4 , Sym 2 ) represents a holomorphic L-function. Since symmetric square L-functions are accessible via the Langlands-Shahidi method, the non-vanishing on Re(s) = 1 follows (see Sect. 5 of [Sha81] and [Sha97, Thm 1.1]).
For our future applications, we will also need that L(s, ad(π) × σ) has the properties of an Lfunction and is entire for certain special automorphic representations σ of GL 1 (A) or GL 2 (A). The next two lemmas achieve this for σ a quadratic character or σ of the form AI(Λ 2 ) where Λ is a character of A × K where K is a quadratic field and AI denotes automorphic induction. Lemma 5.5. Let K/Q be a quadratic field. Let F and π be as in Proposition 5.4 and assume that N is squarefree. Then the base change π K of π to GSp 4 (A K ) is cuspidal. Furthermore, the base change Π 4,K of Π 4 to GL 4 (A K ) is cuspidal and Π 4,K is the lifting of π K from GSp 4 (A K ) to GL 4 (A K ).
Proof. Let π K and Π 4,K be as in the statement of the lemma. Since Π 4 is the lifting of π, it follows from the definition of base change that Π 4,K is the lifting of π K . Let σ be an arbitrary cuspidal automorphic representation of GL 1 (A K ) or GL 2 (A K ). By the definition of lifting, we have L(s, π K × σ) = L(s, Π 4,K × σ) and so to prove that π K and Π 4,K are cuspidal it suffices to show that L(s, Π 4,K × σ) has no poles. By the adjointness formula [PR99, Prop 3.1] we have L(s, Π 4,K × σ) = L(s, Π 4 × AI(σ)). Note that AI(σ) is an automorphic representation of GL 2 (A) if σ is a character of A × K and AI(σ) is an automorphic representation of GL 4 (A) if σ is a cuspidal representation of GL 2 (A K ). By Proposition 5.4, L(s, Π 4 × AI(σ)) is entire when σ is a character. This reduces us to the case that σ is a cuspidal representation of GL 2 (A K ); in this case L(s, Π 4,K × σ) = L(s, Π 4 × AI(σ)) has a pole if and only if Π 4 ≃ AI(σ) (recall that Π 4 is self-dual). However, by looking at a prime p which ramifies in K, we see that this is impossible: at any such prime, the local Langlands parameter of Π 4,p is the local lifting of a representation of GSp 4 (Q p ) that has a vector fixed by the local Siegel congruence subgroup of level p (since N is squarefree), but an inspection of Table 1 of [JLR12] tells us that the local parameter of AI(σ p ) can never equal one of those. Hence we have completed the proof that L(s, Π 4,K × σ) has no poles.
Remark 5.6. The above proof crucially relies on the fact that F has squarefree level N . If N is allowed to be divisible by squares of primes, there indeed exists F whose adelization generates a representation π whose base change π K (for certain K) is non-cuspidal. Such F are constructed in [JLR12] for K real quadratic and [BDPŞ15] for K imaginary quadratic.
Lemma 5.7. Let K/Q be a quadratic field. Let F and π be as in Proposition 5.4 and assume that N is squarefree. Let χ K be the quadratic Dirichlet character associated to the extension K/Q. Let Λ be any idele class character of K × \A × K . Then the degree 10 L-function L(s, ad(π) × χ K ) and the degree 20 L-function L(s, ad(π) × AI(Λ 2 )) both satisfy the usual properties of an L-function, and are both entire.
Remark 5.8. Let F ∈ S k (Γ (2) 0 (N )) G give rise to an irreducible representation π and assume that N is squarefree. Let d be a fundamental discriminant that is divisible by at least one prime not dividing N , put K = Q( √ d), and let χ d be the quadratic Dirichlet character associated to K/Q and let Λ be a character of the ideal class group Cl K . Then combining Proposition 5.4 and Lemma 5.7, we see that the L-functions L(s, π × AI(Λ)), L(s, ad(π) × AI(Λ 2 )), L(s, ad(π) × χ d ), L(s, std(π) × AI(Λ 2 )), L(s, std(π) × χ d ) are all holomorphic everywhere in the complex plane. 5.5. A consequence of the refined Gan-Gross-Prasad identity. Let K = Q( √ d) where d < 0 is a fundamental discriminant. Recall from the introduction that for any character Λ of the finite group Cl K , we define B(F, Λ) = S∈Cl K a(F, S)Λ(S). The refined Gan-Gross-Prasad (GGP) conjecture in the context of Bessel periods of PGSp 4 ≃ SO 5 (and more generally, for Bessel periods of orthogonal groups) was formulated in [Liu16, (1.1)], and made more explicit in the context of Siegel cusp forms in [DPSS20]. This conjecture implies an identity expressing the square of |B(F, Λ)| as a ratio of L-values, up to some local integrals. For the purpose of this paper, we only need a relatively weak consequence of this identity, which we formulate explicitly as Hypothesis G below. In this subsection we do not assume that N is squarefree.
Suppose that F ∈ S k (Γ (2) 0 (N )) T gives rise to an irreducible representation π. Then F is said to satisfy Hypothesis G if there exists a constant C F such that for each imaginary quadratic field K = Q( √ d) (with d < 0 a fundamental discriminant) and each character Λ of Cl K we have (55) |B(F, Λ)| 2 ≤ C F |d| k−1 L( 1 2 , π × AI(Λ)).
Proposition 5.9. Let π be an irreducible representation that arises from S k (Γ (2) 0 (N )) T . Suppose that for each factorizable F ∈ S k (Γ (2) 0 (N )) T that gives rise to π, and each ideal class character Λ of an imaginary quadratic field K = Q( √ d) (with d a fundamental discriminant), the refined Gan-Gross-Prasad identity (in the form written down in Conjecture 1.3 of [DPSS20]) holds for (φ, Λ), where φ is the adelization of F . Then any F ∈ S k (Γ (2) 0 (N )) T that gives rise to π satisfies Hypothesis G.
Proof. Note that the subspace of S k (Γ (2) 0 (N )) T generated by forms that give rise to a fixed irreducible representation π, has a basis consisting of factorizable forms. So, for the purpose of verifying Hypothesis G, it suffices to prove (55) for factorizable F whose adelization φ = ⊗φ v generates π. Assuming the truth of Conjecture 1.2 of [DPSS20] for (φ, Λ), and combining it with the explicit calculations performed in [DPSS20, §2.2, §3.3-3.5], we see that where A F depends on F , N F |N is the smallest integer such that F ∈ S k (Γ (2) 0 (N F )) T , and the normalized local factors J p (φ p , Λ p ) (which depend on p, φ p , Λ p and d) are defined in [DPSS20,(30)]. To complete the proof, it suffices to show that J p (φ p , Λ p ) ≪ p,φp 1 (i.e., is bounded by some constant that depends on p, φ p but not on d or Λ p ). It suffices to show this for the unnormalized local factors J 0,p (φ p , Λ p ) defined in [DPSS20,(29)] since the normalized local L-factors J p (φ p , Λ p ) only differ from these by certain absolutely bounded L-factors appearing in [DPSS20,(30)].
To show the above, we move to a purely local setup. Let p be a prime dividing N F . Let F = Q p . We fix a set M of coset representatives of F × /(F × ) 2 such that all elements of M are taken from Z p , and each r ∈ M generates the discriminant ideal of F ( √ r)/F . We let r p equal the unique representative in M that corresponds to d. The assumptions imply that r p = du 2 p , for some u ∈ (Z × p ) 2 .
Let K equal F × F (the "split case") if r p ∈ F 2 and K = F ( √ r p ) = F ( √ d) (the "field case") if r p / ∈ F 2 . Fix the matrix S = S d as in [DPSS20,(75)], so that S d has discriminant d. Let T (F ) = T S (F ) ≃ K × be the associated subgroup of GSp 4 (F ), let N (F ) ⊂ GSp 4 (F ) denote the unipotent for the Siegel parabolic and θ S the character on N (F ) given by θ S ( I 2 X 0 I 2 ) = ψ(tr(SX)) where ψ is a fixed unramified additive character. Then we need to show that the integral π p (t p n p )φ p , φ p Λ −1 p (t p )θ −1 S (n p ) dn p dt p is bounded by some quantity that does not depend on d or Λ p . Note above that the superscript St denotes stable integral as in [Liu16]; this means that the integral can be replaced by any sufficiently large compact subgroup (as we will do below). Put S ′ = − 1 . In either case, A ′ ∈ GL 2 (Z p ) and S ′ = t ASA. To show that J p is bounded independently of d, we use a simple change of variables (n p → An p t A, t p → At ′ p A −1 ) to see that the integral J 0,p (φ p , Λ p ) remains unchanged when the matrix S is replaced by S ′ . This shows that J 0,p (φ p , Λ p ) does not depend on the actual value of d but only on the class of d in F × /(F × ) 2 together with d modulo 4, of which there are only finitely many possibilities. To show that the resulting integral is absolutely bounded independent of Λ, we replace the integral Remark 5.10. Furusawa and Morimoto [FM21] have proved the refined GGP identity in the form required in Prop. 5.9 for Λ = 1 K , and they have announced a proof of this identity for general Λ.
Remark 5.11. Under certain assumptions (namely N odd and squarefree, F a newform, and d p = −1 for all primes p dividing N ) the relevant local integrals were explicitly computed in [DPSS20], and so under these assumptions Proposition 5.9 follows from Theorem 1.13 of [DPSS20].
Corollary 5.12. Suppose that F ∈ S k (Γ (2) 0 (N )) Y gives rise to an irreducible representation. Then F satisfies Hypothesis G.