On conjectures of Sato-Tate and Bruinier-Kohnen

arXiv:1305.5443v3 [math.NT] 19 Nov 2013 On conjectures of Sato-Tate and Bruinier-Kohnen Sara Arias-de-Reyna∗ , Ilker Inam†and Gabor Wiese‡ 20th November 2013 Abstract This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of SatoTate equidistribution for CM modular forms with an error term similar to that in the prime number theorem. MSC 2010: 11F37 (primary), 11F30, 11F80, 11F11. Keywords: Half-integral weight modular forms, Shimura lift, Sato-Tate equidistribution, Fourier coefficients of modular forms, density of sets of primes. 1 Introduction A very significant recent result in pure mathematics is the proof of the Sato-Tate conjecture for nonCM modular eigenforms (even for Hilbert eigenforms) [2]. It asserts that for a normalised (A(1) = 1) cuspidal eigenform f=P∞ n=1 A(n)qn(with q=e2πiz) of weight k≥2on Γ0(N)(some N) the normalised coefficients A(p) 2p(k−1)/2∈[−1,1] are equidistributed with respect to the so-called Sato-Tate measure, when pruns through the set of primes. The corresponding result for CM forms has been known for a long time and in fact is quite a simple corollary of the equidistribution of the values of Hecke characters. In Section 3 of this article we include a proof of a form of this result that additionally provides an error bound like the one in the prime number theorem (see Theorem 3.1.1). It relies on an error bound for the equidistribution of the values of Hecke characters given in [15]. ∗Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg, [email protected] †Uludag University, Deparment of Mathematics, Faculty of Arts and Sciences, 16059 Gorukle, Bursa, Turkey, [email protected] or ilker.i[email protected] ‡Université du Luxembourg, Faculté des Sciences, de la Technologie et de la Communication, 6, rue Richard Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg, gabor[email protected] 1 A special case of Sato-Tate equidistribution for non-CM eigenforms shows that the sets of primes {pprime :A(p)>0}and {pprime :A(p)<0} both have natural density equal to 1/2. A conjecture of Bruinier and Kohnen ([3] and [9]) asserts that something similar should hold for certain half-integral weight modular forms f=P∞ n=1 a(n)qn; namely they conjecture that the sets {n∈N:a(n)>0}and {n∈N:a(n)<0} have the same natural density, namely, half of the natural density of {n∈N:a(n)6= 0}. The interest in the distribution of signs is explained by a famous theorem of Waldspurger relating the squares (a(t))2for squarefree tto the critical values of the Hecke L-function of the Shimura lift Fttwisted by an explicit quadratic character (see [23]); this precisely leaves the sign of a(t)undetermined. The Bruinier-Kohnen conjecture appears to be quite hard. The main contribution of the previous work [7] is the observation that the Shimura lift Ftallows one to utilise Sato-Tate equidistribution for the coefficients of the integral weight eigenform Ftin order to compute the densities of the sets of primes {pprime :a(tp2)>0}and {pprime :a(tp2)<0}. If the Shimura lift Ftis non-CM, in [7] it is proved that the densities of these two sets are equal. In this paper we extend this computation to the CM case, see Theorem 4.1.1. It turns out that in the CM case the densities can either be both 1/2or they can be 1/4and 3/4(see Example 4.1.2). In order to study the set of natural numbers {n∈N:a(tn2)>0}(and similarly for ‘<0’) we set up some general theory, that grew out of analysing the rather ad hoc methods of [7]. We now describe this. Let χ:N→ {−1,0,+1}be a multiplicative arithmetic function and define S±={pprime :χ(p) = ±1}and A±={n∈N:χ(n) = ±1}. Motivated by the Bruinier-Kohnen conjecture (take χ(n)to be the sign of a(tn2)supposing a(t)>0), westudy therelation betweenthe densities ofS±and A±. We were unable to prove any results without the assumption of some error term in the convergence of the natural density of S±. If there is a rather weak error term, then the sets of primes S±are weakly regular; if the error term is strong (often implied by variations of the Riemann Hypothesis), then we obtain regular sets (see Definition 2.2.1). Our main results in this abstract context are Propositions 2.2.2, 2.3.1, and 2.5.2. In this introduction we do not repeat their precise assertions, but we explain what they imply for the Bruinier-Kohnen conjecture. In the case that the Shimura lift Fthas CM, we use the error bound from Theorem 3.1.1 in order to obtain the weak regularity of the set {pprime :a(tp2)>0}(and similarly for ‘<0’ and ‘= 0’) and to deduce that {n∈N:a(tn2)>0}and {n∈N:a(tn2)<0} 2 have the same Dedekind-Dirichlet density (see Definition 2.1.3), which is equal to half the DedekindDirichlet density of {n∈N:a(tn2)6= 0}. Maybe at first sight astonishingly, one obtains this result even in the situation when the densities of the corresponding sets of primes are not equal. Under the assumption of a similar error bound in the case that Fthas no CM, one obtains the same result. This had already been established in [7] under the assumption of a stronger error bound. See Remark 3.1.3 for some relation of this error bound and the Generalised Riemann Hypothesis. If we assume this stronger error bound (whether Ftis CM or not), then one can use a result of Delange to derive that the previous statement even holds in terms of natural density. The study of the densities of S±and A±is done in Section 2. Our aim there is to give a coherent treatment so that we also recall the relevant definitions. Section 3 is devoted to proving Sato-Tate equidistribution for CM modular forms (in fact we show slightly more) with an error term as in the prime number theorem. In the final Section 4 the results towards the Bruinier-Kohnen conjecture are derived from the techniques provided in the other sections. Acknowledgements I.I. is supported by The Scientific and Technological Research Council of Turkey (TUBITAK) and Uludag University Research Project No: UAP(F) 2012/15. G.W. acknowledges partial support by the priority program 1489 of the Deutsche Forschungsgemeinschaft (DFG). S.A. is partially supported by the project MTM2012-33830 of the Ministerio de Economía y Competitividad of Spain. I.I. would like to thank the University of Luxembourg for its hospitality. The authors would like to thank Juan Arias de Reyna for his remarks. They also thank Jeremy Rouse for explanations concerning [18]. I.I. and G.W. are grateful to Winfried Kohnen for interesting discussions. Thanks are also due to the anonymous referee for helpful suggestions concerning the presentation of the paper. 2 Densities and sets of primes In this section we are concerned with the sets S±={pprime :χ(p) = ±1}and A±={n∈N:χ(n) = ±1} for a multiplicative arithmetic function χ:N→ {−1,0,+1}, as explained in the introduction. We found it necessary to assume more than just that S±has a natural density in order to conclude something about the density of A±; namely, we obtain our results under the assumption that S±is (weakly) regular (see below). We also show that (weak) regularity is a consequence of a sufficiently good error bound for the convergence of the natural density of S±. 3 2.1 Notions of density Definition 2.1.1. Let P⊂Nbe the set of all prime numbers. For a set of primes S⊆Pwe make the following definitions: •For x∈R, denote πS(x) := #{p≤x:p∈S}. As usual denote πPby π. •PS(z) := Pp∈S1 pz. This defines a holomorphic function on {Re(z)>1}. •For a multiplicative function χ:N→Rwe let Dχ(z) := Pn≥1 χ(n) nzbe the corresponding Dirichlet series. If |χ|is bounded, it also defines a holomorphic function on {Re(z)>1}. In particular, D1=ζ(z)is the Riemann-zeta function. •A function χ:N→Ris said to be characteristic on Sif χis multiplicative and its restriction to Pis the characteristic function of the set S. The following lemma links the Dirichlet series Dχfor some χthat is characteristic on Sto PS. This link is the key to relating density statements on subsets of Pto subsets of N. Lemma 2.1.2. Let χ:N→ {−1,0,1}be a multiplicative function. Then on {Re(z)>1}one has log Dχ(z)=X p∈P χ(p) pz+g(z), where g(z)is a function that is holomorphic on {Re(z)>1/2}. In particular, if χis characteristic on S, the equality becomes log Dχ(z)=PS(z) + g(z). Proof. We use the Euler product Dχ(z) = Qp∈P1 + Pn≥1 χ(pn) pnz , which is absolutely convergent on {Re(z)>1}in the sense that Pp∈PPn≥1 χ(pn) pnz converges absolutely in this region. We first treat the following special case. Let S⊆Pand χ:N→ {0,1}be multiplicative such that for any prime pone has χ(pn) = 1 if and only if p∈Sand n= 1. Then the Euler factor of Dχ at pis either 1 + 1 pzor 1, depending on whether p∈Sor not. We take the logarithm of the Euler product log Dχ(z) = X p∈S log(1 + 1 pz) = X p∈S 1 pz+g(z)with g(z) := X p∈SX m≥2 (−1)m+1 m1 pzm . It is elementary to prove that g(z)defines a holomorphic function on {Re(z)>1 2}. In order to tackle the general case, let S±:= {p∈P:χ(p) = ±1}and define the multiplicative functions χ±on prime powers by χ±(pn) =    1if p∈S±and n= 1, 0otherwise. . Define Φ(z) := Dχ(z)·Dχ−(z) Dχ+(z). Then we have on {Re(z)>1} log(Φ(z)) = log(Dχ(z)) + log(Dχ−(z)) −log(Dχ+(z)) = log(Dχ(z)) + PS−(z)−PS+(z) + g(z), where g(z)is holomorphic on {Re(z)>1 2}. On {Re(z)>1}the function Φis described by an absolutely converging product Φ(z) = Qp∈PΦp(z), where Φp(z)satisfies |1−Φp(z)| ≤ 20 p2z. It easily follows that this product converges absolutely on {Re(z)>1 2}, which implies the assertion. 4 The density of a set of prime numbers (if it exists) measures its size. There are several notions of density, e.g. Dirichlet density and natural density, which in general are distinct. In a similar way, one can define analogous notions of density for subsets of N. Here we recall the definitions. By the symbol limz→1+we denote the limit defined by letting ztend to 1on the real interval (1,∞). Definition 2.1.3. Let S⊆Pbe a set of primes. The set Sis said to have Dirichlet density equal to δ(S)if the limit lim z→1+Pp∈S1 pz Pp∈P1 pz = lim z→1+Pp∈S1 pz log ζ(z)= lim z→1+Pp∈S1 pz log 1 z−1 exists and is equal to δ(S). Moreover, Sis said to have natural density equal to d(S)if the limit lim x→∞ πS(x) π(x) exists and is equal to d(S). Let now A⊆Nbe a subset. It is said to have Dedekind-Dirichlet density δ(A)if the limit lim z→1+Pn∈A1 nz Pn∈N1 nz = lim z→1+Pn∈A1 nz ζ(z)= lim z→1+(z−1) X n∈A 1 nz exists and is equal to δ(A). Similarly, Ais said to have natural density d(A)if the limit lim x→∞ #{n≤x:n∈A} x exists and is equal to d(A). The equalities in the statements all follow from Lemma 2.1.2 and the well-known fact that the Riemann-zeta function has a simple pole of residue 1at 1. It is well known that if a set of prime numbers S(resp. a set of natural numbers A) has a natural density, then it also has a Dirichlet density (resp. a Dedekind-Dirichlet density) and they coincide. A function χ:N→ {0,1}that is characteristic on S⊆Plinks the set Sto the set of natural numbers A={n∈N:χ(n) = 1}. The following proposition, the proof of which is evident in view of Lemma 2.1.2, makes clear the nature of the relation between the Dirichlet density of Sand the Dedekind-Dirichlet density of A. Proposition 2.1.4. Let Sbe a set of primes and χ:N→ {0,1}be a multiplicative function characteristic on Sand let A={n∈N:χ(n) = 1}. Then the Dirichlet density of S, if it exists, equals δ(S) = lim z→1+ log Dχ(z) log ζ(z) and the Dedekind-Dirichlet density of A, if it exists, equals δ(A) = lim z→1+ Dχ(z) ζ(z)= exp lim z→1+(log Dχ(z)−log ζ(z)). 5 We now prove a precise relationship between the densities of Aand S. This result will be strengthened below in Proposition 2.2.2 under the extra assumption of weak regularity, which is introduced in the next section. Proposition 2.1.5. Let Sbe a set of primes and χ:N→ {0,1}be characteristic on Sand let A={n∈N:χ(n) = 1}. If δ(A)6= 0 (in particular, the limit exists), then δ(S) = 1. Proof. As δ(A)6= 0, it follows from Proposition 2.1.4 that lim z→1+(log Dχ(z)−log ζ(z)) exists. But we have by Lemma 2.1.2 log Dχ(z)−log ζ(z) = X p∈S 1 pz−X p∈P 1 pz+g(z) = −X p6∈S 1 pz+g(z),(2.1) where gis a function that is holomorphic on {Re(z)≥1}. This implies the convergence of Pp6∈S1 p, showing that P\Sis a set of Dirichlet density 0, thus Sis of Dirichlet density 1. 2.2 Regular and weakly regular sets of primes Definition 2.2.1. Let S⊆Pbe a set of primes. We call Sweakly regular if there is a∈Rand a function g(z)which is holomorphic on {Re(z)>1}and continuous (in particular, finite) on {Re(z)≥1} such that PS(z) = alog 1 z−1+g(z). As in [13] (and [7]) we say that Sis regular1if the function gis holomorphic on {Re(z)≥1}. Clearly, every regular set Sis weakly regular. If Sis weakly regular, it directly follows that it has a Dirichlet density, namely δ(S) = a. If Sis regular (weakly regular) of density 0, then PSis holomorphic (continuous) on {Re(z)≥1}. Proposition 2.2.2. Let Sbe a weakly regular set of primes and χ:N→ {0,1}be a multiplicative function characteristic on primes with respect to Sand let A={n∈N:χ(n) = 1}. Then δ(A)6= 0 ⇔δ(S) = 1. Proof. The direction ‘⇒’ was proved in Proposition 2.1.5 without the assumption of weak regularity. Hence, we now assume that Sis weakly regular such that δ(S) = 1. It follows that P\Sis weakly regular of density 0, meaning that Pp6∈S1 pzdefines a continuous function on {Re(z)≥1}. From Equation (2.1) we get that log Dχ(z)−log ζ(z)is continuous on {Re(z)≥1}, in particular the limit limz→1+exists, whence by Proposition 2.1.4 it follows that δ(A)exists and is nonzero. 1Added in proof: The notion of a regular set of primes already appeared in [4]. 6 We next show that sets of primes that have a natural density and additionally satisfy certain error bounds for the convergence of the limit defining the natural density are (weakly) regular. In [7], Proposition 2.2, we proved such a statement. We will now weaken the assumption on the error term in a way that still allows to conclude weak regularity instead of regularity. Proposition 2.2.3. Let Sbe a set of primes having natural density d(S). Let E(x) := πS(x) π(x)−d(S) be the error function. If the integral R∞ 2|E(x)| xlog(x)dx converges, then Sis a weakly regular set of primes having Dirichlet density δ(S) = d(S). Proof. The proof follows the proof of [7], Proposition 2.2, very closely and the reader is referred there for some of the calculations. We put g(x) := E(x)π(x)and f(z) = R∞ 2 g(x) xz+1 dx. Then PS(z) = d(S)P(z) + zf(z). Hence, it suffices to show that fis continuous on {Re(z)≥1}. We use π(x)< x log(x)−4for x > 55 (by Theorem 29 of [17]) in order to obtain the estimate |g(x)|=|E(x)π(x)| ≤ x|E(x)| log(x)−4. We now use this to estimate f(z)for Re(z)≥1: |Z∞ 56 g(x)dx xz+1 | ≤ Z∞ 56 |g(x)| xRe(z)+1 dx ≤Z∞ 56 |E(x)| x(log(x)−4)dx ≤2Z∞ 56 |E(x)| xlog(x)dx The assumption ensures that the last integral is convergent. Let now ǫ > 0. There is hence some N such that |R∞ N g(x)dx xz+1 |< ǫ/4for any zwith Re(z)≥1. Moreover, fN(z) := RN 2 g(x)dx xz+1 is continuous in a neighbourhood of any such z. In particular, for any z1with Re(z1)≥1close enough to zwe have |fN(z1)−fN(z)|< ǫ/2. This implies |f(z1)−f(z)|< ǫ, as required. The following corollary for an explicit error function will be applied in the situation of CM modular forms in Section 3 (see also Proposition 2.2.7 below). Corollary 2.2.4. Let Sbe a set of primes having natural density d(S). Let E(x) := πS(x) π(x)−d(S)be the error function. If there are α > 0,C > 0and B > 0such that |E(x)| ≤ C log(x)αfor all x > B, then Sis a weakly regular set of primes having Dirichlet density δ(S) = d(S). Proof. Note that the derivative of −1 αlog(x)αis 1 xlog(x)1+α. Thus the former is a primitive function for the upper bound of the error term. As it clearly tends to 0for x→ ∞, it follows that the assumptions of Proposition 2.2.3 are satisfied. The Chebotarev Density Theorem, which plays an essential role in Section 3, provides us with examples of (weakly) regular sets of primes (see Proposition 2.2.7 below), which areused inSection 4. Definition 2.2.5. Let K/Qbe a finite Galois extension with Galois group G. We will say that a set Sof finite rational primes is a Chebotarev set for K/Qif for all p∈S,pis unramified in K/Qand moreover there exists a subset C⊆G, invariant under conjugation, such that S= {prational prime: Frobp∈C}, where Frobpdenotes a lift to Gof the Frobenius element of the residual extension of K/Qat a prime p|p. 7 We quote the effective version of the Chebotarev Density Theorem from [20]. Theorem 2.2.6 (Chebotarev Density Theorem).Let K/Qbe a finite Galois extension, and let Sbe a Chebotarev set, which corresponds to C⊂Gal(K/Q). Then the following hold: (a) For all sufficiently large x,πS(x) = |C| |G|π(x) + O(xexp(−cplog(x))) for some constant c > 0. (b) If we assume the Riemann Hypothesis for the Dedekind zeta function of K, then for all sufficiently large x,πS(x) = |C| |G|π(x) + O(x1/2log(x)). Proposition 2.2.7. Let K/Qbe a finite Galois extension and Sa Chebotarev set. Then Sis weakly regular. If the Riemann Hypothesis for the Dedekind zeta function of Kholds, then Sis regular.2 Proof. Let C⊂Gal(K/Q)be the set corresponding to S. Then by part (a) of Theorem 2.2.6, and taking into account that x log(x)+2 < π(x)for x≥55 (see Theorem 29 of [17]), it follows that, for all sufficiently large x,  πS(x) π(x)−|C| |G|≤c1 xexp(−c2plog(x)) π(x)≤c1(log x+ 2) exp(−c2plog(x)) for some positive constants c1and c2. It is clear that this quantity is less than or equal to c3 log(x)α for sufficiently large x, where αand c3are any positive constants. Thus by Corollary 2.2.4 we can conclude that Sis weakly regular. If we assume that the Dedekind zeta function of Ksatisfies the Riemann Hypothesis, then part (b) of Theorem 2.2.6 yields  πS(x) π(x)−|C| |G|≤c1 x1/2log(x) π(x)≤c1(log x+ 2) log(x)x−1/2 for all big enough values of x, where c1is some positive constant. Proposition 2.2 of [7] implies that Sis regular. 2.3 An application: weak regularity yields Dedekind-Dirichlet density In this section we derive an equidistribution result, which will allow us to establish our results towards the Bruinier-Kohnen conjecture in Section 4. Proposition 2.3.1. Let P=P=0 ∪P>0∪P<0be a partition of the set of all primes into three weakly regular sets such that P=0 is of Dirichlet density 0and the Dirichlet density of P<0is not zero. Let ψ:N→ {0,1,−1}be a multiplicative arithmetic function such that, for every prime p,ψ(p) = 0 (resp. ψ(p) = 1,ψ(p) = −1) if and only if p∈P=0 (resp. p∈P>0,p∈P<0). Then {n:ψ(n)>0}and {n:ψ(n)<0}have a Dedekind-Dirichlet density, which for both is 1/2of the Dedekind-Dirichlet density of {n:ψ(n)6= 0}. 2Added in proof: The assumption of the Riemann Hypothesis is not necessary, see [19], Proposition 1.5. 8 Proof. Let us record first that the set {n:ψ(n)6= 0}indeed has a positive Dedekind-Dirichlet density by Proposition 2.2.2, that is, the limit lim z→1+(z−1) X n∈N |ψ(n)| nz=d(2.2) exists with 0< d ≤1. Lemma 2.1.2 yields log(Dψ(z)) = X p∈P ψ(p) pz+g(z) = X p∈P>0 1 pz−X p∈P<0 1 pz+g(z), where g(z)is a function that is holomorphic on {Re(z)>1/2}. Using the definition of weak regularity for the sets P>0and P<0, we obtain log(Dψ(z)) = alog( 1 z−1) + h(z), or, equivalently, Dψ(z) = 1 (z−1)aexp(h(z)), where ais δ(P>0)−δ(P<0), which is strictly less than 1by assumption, and h(z)is a function that is continuous on {Re(z)≥1}. Taking the exponential yields Dψ(z) = X n∈N,ψ(n)=1 1 nz−X n∈N,ψ(n)=−1 1 nz=1 (z−1)aφ(z),(2.3) where φ(z) = exp(h(z)) is also continuous on {Re(z)≥1}. Adding Equations (2.2) and (2.3) yields lim z→1+(z−1)2X n∈N,ψ(n)=1 1 nz=d, which is the claimed formula. 2.4 Towards natural density In this section we show that regularity of density 1for a set S⊆Psuffices to conclude that the set of natural numbers corresponding to a function that is characteristic on Shas a positive natural density, and not only a Dedekind-Dirichlet density, whose existence was shown in Proposition 2.2.2. In fact, one sees that a slightly weaker assumption than regularity works, however, we are unable to prove that weak regularity is enough. Proposition 2.4.1. Let S⊆Pbe a set of primes of density 1and let χ:N→ {0,1}be a multiplicative function characteristic on S. We assume that Ssatisfies the following condition (which is implied by regularity but not weak regularity): 9 Theorem 3.2.5. Let K,m,ξ,uas above. Assume u6= 0. Then the expression f(z) := X a ξ(a)NK/Q(a)u/2qNK/Q(a)(3.9) defines a modular form f∈Su+1(N, χ), where aruns through all integral ideals of Kwith (a,m) = 1,N=|d|NormK(m)and where χis the Dirichlet character defined as χ(m) = d mξ((m))sgn(m)ufor all m∈Z.(3.10) Conversely, any modular form with CM arises in this way from some Hecke character of an imaginary quadratic field (cf. [16], Thm. 4.5). 3.3 Equidistribution of Fourier coefficients of CM modular forms Assume now that we have a normalised eigenform f∈Sk(Γ0(N)) such that fhas CM by the imaginary quadratic field K. Let ξbe the Hecke character that gives rise to fas in Theorem 3.2.5. Then the Fourier expansion of flooks like Equation (3.9). In particular, for all primes p∤N, we have ap=   ξ(p1)NK(p1)k−1 2+ξ(p2)NK(p2)k−1 2if (p) = p1p2with p16=p2; 0if (p)is inert in K. Since fhas trivial nebentypus, Equation (3.10) implies that ξ((p)) = 1 whenever psplits in K. Thus if (p) = p1p2, then ξ(p1)and ξ(p2)are complex conjugates. Therefore ap 2p(k−1)/2= Re(ξ(p1)).(3.11) We introduce the notation πK/Q,split(x) := #{prational prime :p≤xand (p)splits in K/Q} and similarly πK/Q,inert(x)and πK/Q,ram(x). Lemma 3.3.1. We have that #{pprime ideal of OK:NormK/Q≤xand p/(p∩Z)is not split }=O(√x) and πK(x) = 2πK/Q,split(x) + O(√x). Proof. The number of elements in the set of the first claim is clearly at most #{pprime :p≤√x}= O(√x). The second claim follows from the equality πK(x) = 2πK/Q,split(x) + πK/Q,inert(√x) + πK/Q,ram(x) and the fact that only finitely many primes ramify in K/Q. 16 Proof of Theorem 3.1.1. We only prove part (a), since the arguments in part (b) are entirely analogous. Let I⊆[−1,1] be a subinterval. We want to count how many primes psatisfy that ap 2p(k−1)/2∈I. We count the split and the inert primes separately and start with the inert ones: #{pprime inert in K:p≤x, p ∤N, ap 2p(k−1)/2∈I} =   #{pprime inert in K:p≤x, p ∤N}if 0∈I; 0if 06∈ I. This implies #{pprime inert in K:p≤x, p ∤N, ap 2p(k−1)/2∈I}=1 2δ0(I)π(x) + O(xexp(−cplog x)), (3.12) where we have used that #{pprime inert in K:p≤x, p ∤N}=1 2π(x) + O(xexp(−c√log x)) for some constant c > 0, which follows from part (a) of Theorem 2.2.6. The split primes are counted using Remark 3.2.4 and Lemma 3.3.1 as follows: #{pprime split in K:p≤x, p ∤N, ap 2p(k−1)/2∈I} =1 2#{pprime of OK:NormK/Q(p)≤x, p/(p∩Z)is split ,Re(ξ(p)) ∈I} =1 2#{pprime of OK:NormK/Q(p)≤x, Re(ξ(p)) ∈I}+O(√x) =1 21 πZI 1 √1−t2dtπK(x) + O(xexp(−cplog x)) =1 πZI 1 √1−t2dtπK/Q,split(x) + O(xexp(−cplog x)) =1 21 πZI 1 √1−t2dtπ(x) + O(xexp(−cplog x)) (3.13) for some constant c > 0. The theorem follows by adding Equations (3.12) and (3.13). 4 Application to the Bruinier-Kohnen Conjecture 4.1 Equidistribution of signs of half-integral weight modular forms - the prime case In this section, we state an analog of the Bruinier-Kohnen sign equidistribution conjecture for the family {a(tp2)}where tis a squarefree number such that a(t)6= 0 and pruns through the primes for a half-integral weight modular form whose Shimura lift is without CM or with CM. The proof will be carried out in Section 4.2. Furthermore we will give some properties of these coefficient sets. Note that the following theorem is an improvement of Theorems 4.1 and 4.2 of [7]. We start by summarising some known facts about half-integral weight modular forms and the Shimura lift. Let k≥2. According to Shimura [21] and Niwa [14], if fis a Hecke eigenform of 17 weight k+ 1/2with Fourier expansion f=P∞ n=1 a(n)qn∈Sk+1/2(N, χ)then there is a corresponding modular form Ft∈S2k(N/2, χ2)for fixed t≥1squarefree such that a(t)6= 0, named the Shimura lift of fwith respect to t, such that the Tn2-Hecke eigenvalue on fagrees with the Tn-Hecke eigenvalue on Ft. For k= 1 suppose that fis contained in the orthogonal complement with respect to the Petersson scalar product of the subspace Sk+1/2(N, χ)generated by unary theta functions as in [3]. The Fourier expansion of Ftis given by Ft(z) = Pn≥1At(n)qnwhere At(n) := X d|n χt,N (d)dk−1a(tn2 d2),(4.14) where χt,N denotes the character χt,N (d) := χ(d)(−1)kN2t d. Moreover, the Fourier coefficients are multiplicative in the sense a(tm2)a(tn2) = a(t)a(tm2n2)(4.15) for (n, m) = 1. If Fthas CM, then let µdenote µCM, otherwise put µ=µST. We assume throughout that χis trivial or quadratic and that fhas real coefficients. This implies that Ftalso has real coefficients. The following is our main theorem about the distribution of signs of the coefficients a(tp2), when p runs through the primes. In the statement we understand by an equality of two Dirichlet characters the equality of the underlying primitive characters (i.e. we allow them to differ at finitely many primes). Theorem 4.1.1. Assume the set-up above and define the set of primes P>0:= {p∈P:a(tp2)>0} and similarly P<0and P=0 (depending on fand t). (a) If Fthas no complex multiplication then the sets P>0and P<0have natural density 1/2and the set P=0 has natural density 0. (b) (i) If Fthas complex multiplication and χt,N = 1 then the set P=0 has natural density equal to zero, and the sets P>0and P<0have natural densities, respectively 1/4and 3/4if a(t)>0 and, respectively 3/4and 1/4if a(t)<0. (ii) If Fthas complex multiplication and χt,N =δ, where δis the quadratic Dirichlet character corresponding to the imaginary quadratic field by which fhas CM, then the set P=0 has natural density equal to zero, and the sets P>0and P<0have natural densities, respectively 3/4and 1/4if a(t)>0and, respectively 1/4and 3/4if a(t)<0. (iii) If Fthas complex multiplication and χt,N 6∈ {1, δ}then the set P=0 has natural density equal to zero, and the sets P>0and P<0have the same natural density which is equal to 1/2. 18 (c) If Fthas no complex multiplication then we additionally assume that there are C > 0and α > 0 such that for all subintervals [a, b]⊆[−1,1] one has  #{p≤xprime |At(p) a(t)2pk−1/2∈[a, b]} π(x)−µ([a, b])≤C log(x)α. Then the sets P>0,P<0, and P=0 are weakly regular sets of primes. (d) Assume here that there are C > 0and α > 0such that for all subintervals [a, b]⊆[−1,1] one has  #{p≤xprime |At(p) a(t)2pk−1/2∈[a, b]} π(x)−µ([a, b])≤C xα (note that this condition is satisfied if Ft/a(t)fulfills the assumptions of Theorem 3.1.2, see also Remark 3.1.3). Then the sets P>0,P<0, and P=0 are regular sets of primes. Example 4.1.2. Consider the elliptic curve defined by the equation y2=x3−x. This elliptic curve has conductor 32 and has CM by Z[i]. Let F=P∞ n=1 A(n)qn∈S2(32) be the associated cuspidal eigenform. We have that, for all p≡ −1 (mod 4),A(p) = 0, that is, Fhas CM by Q(i). In [22], Tunnell has shown that there exist modular forms f1∈S3/2(128) (trivial character) and f2∈S3/2(128, χ2), where χ2=2 ·, such that their Shimura lifts with t= 1 coincide with F. •For f1, we have χ1,128(p) = −1·4 p, which coincides with the character by which Fhas CM. Thus, P>0has natural density 3/4and P<0has natural density 1/4. •For f2, we have χ1,128(p) = −2 p, which is different from the trivial character and the character by which Fhas CM. In this case the densities of P>0and P<0coincide and they are equal to 1/2. Remark 4.1.3. (a) For fixed squarefree tsuch that a(t)6= 0 we use the notation: A(p) := a(tp2) a(t)2pk−1/2and B(p) := At(p) a(t)2pk−1/2. Note that Equation (4.14) implies A(p) = B(p)−χt,N (p) 2√p.(4.16) The main point in our approach is that we view the sequence A(p)as a ‘perturbed’ version of the sequence B(p). 19 (b) We remark that ‘small’ perturbations preserve the property of a sequence to be equidistributed. More precisely, let µbe a nonnegative regular normed Borel measure on [−1,1] and (xn)n∈N⊆ [−1,1] be a µ-equidistributed sequence. Let (yn)n∈N⊆[−1,1] be a sequence such that lim n→∞|xn−yn|= 0. Then also (yn)n∈Nis µ-equidistributed. This follows from a straight forward calculation using the definition of µ-equidistribution and the compactness of [−1,1]. (c) Returning to our set-up of modular forms, we first remark that the set Sof primes psuch that a(tp2) 2a(t)pk−1/26∈ [−1,1] has natural density 0(this is an easy consequence of Theorem 4.2.1 below). Part (b) above together with Equation (4.16) thus implies that the elements a(tp2) 2a(t)pk−1/2p∈P\S are µ-equidistributed. We stress that equidistribution of a(tp2) 2a(t)pk−1/2is not enough to imply equidistribution of signs if the measure has points of positive mass (like µCM). See for instance Example 4.1.2. This is the reason why we are not only interested in equidistribution in the sense of the definition, but, are studying the limits limN→∞ #{n≤N:xn∈I} Nfor all intervals I, even those having a boundary of positive measure. 4.2 Densities of perturbed sequences In this section we provide a treatment of an abstract setting modeled on the relation between coefficients of half-integral and integral weight modular forms under the Shimura lift (see, in particular, Remark 4.1.3), and we will use it to prove Theorem 4.1.1. Theorem 4.2.1. Let χbe a Dirichlet character of order dividing 2. Let B:P→Rbe a map and define A:P→Rby the formula A(p) := B(p)−χ(p) y√pfor some 06=y∈R. Let D={x1,...,xn} ⊂ [−1,1]. For any I⊆[−1,1] define SI:= {p∈P:B(p)∈I}and T′ I:= {p∈P:A(p)∈I, B(p)6∈ D}. Let f: (−1,1) →R≥0be an integrable function and w1,...,wn≥0. Define a measure on [−1,1] by µ(I) = RIf(t)dt +Pn i=1 wiδxi(I), where δxiis the Dirac measure at the point xi, for any Borel measurable subset I⊆[−1,1]. Assume that µ([−1,1]) = 1 and that for all intervals I⊆[−1,1] (open, closed or half-open) the set SIhas natural density µ(I). (a) Then for any interval I⊆[−1,1] (open, closed or half-open), the set T′ Ihas natural density RIf(t)dt. 20 (b) Assume that there are m∈N≥1and M > 0such that for all ǫ > 0small enough the integrals |R1 1−ǫf(t)dt|and |R−1+ǫ −1f(t)dt|are bounded above by Mǫ1/m. Assume moreover that there is a function E(x)tending to 0as x→ ∞such that for all intervals I⊆[−1,1]  πSI(x) π(x)−µ(I)≤E(x). Then for any interval I⊆[−1,1] (open, closed or half-open), there is C > 0such that for all big enough x  πT′ I(x) π(x)−ZI f(t)dt≤C·E(x) + 1 x1/(2m+2) . Proof. For any interval I⊆[−1,1] define S′ I(x) := {p∈P:B(p)∈I, B(p)6∈ D}=SI\D. By assumption the set S′ Ihas natural density RIf(t)dt. Let abe the start point and bthe end point of I. Let ǫ > 0be small enough. For all p > 1 y2ǫ2one has  χ(p) y√p< ǫ. One observes the inequalities πS′ [a+ǫ,b−ǫ](x)−π(1/(yǫ)2) π(x)≤ πT′ [a,b](x) π(x)≤ πS′ [max{−1,a−ǫ},min{1,b+ǫ}](x) + π(1/(yǫ)2) π(x).(4.17) (a) From Equation (4.17) we obtain the inequalities Zb−ǫ a+ǫ f(t)dt ≤lim inf x→∞ πT′ [a,b](x) π(x)and lim sup x→∞ πT′ [a,b](x) π(x)≤Zmin{1,b+ǫ} max{−1,a−ǫ} f(t)dt. Letting ǫtend to 0we obtain lim sup x→∞ πT′ [a,b](x) π(x)≤Zb a f(t)dt ≤lim inf x→∞ πT′ [a,b](x) π(x), implying the result. (b) Equation (4.17) yields −Za+ǫ a f(t)dt −Zb b−ǫ f(t)dt −(n+ 1)E(x)−π(1/(yǫ)2) π(x)≤ πT′ [a,b](x) π(x)−Zb a f(t)dt ≤Za max{−1,a−ǫ} f(t)dt +Zmin{1,b+ǫ} b f(t)dt + (n+ 1)E(x) + π(1/(yǫ)2) π(x), which is valid for all (small enough) ǫ > 0and all (big enough) x. Using the assumptions we obtain  πT′ [a,b](x) π(x)−Zb a f(t)dt≤2Mǫ1/m +π(1/(yǫ)2) π(x)+ (n+ 1)E(x). We may (and do) assume that E(x)≥1 x1/(2m+2) for large enough x. Let ǫ:= E(x)m. One finds π(1/(yǫ)2) π(x)=π(1/(y2E(x)2m)) π(x)∼log(x) y2·E(x)2m·log(1/(y2E(x)2m)) ·x≤C·E(x) for xbig enough and suitable C > 0. Thus we obtain the claimed inequality. 21 Remark 4.2.2. For the applications below we remark that for I⊆[−1,1] we have {p∈P:A(p)∈I}=T′ I⊔ n G i=1{p∈P:B(p) = xi, A(p)∈I} =T′ I⊔ n G i=1 {p∈P:B(p) = xi}∩{p∈P:xi−χ(p) y√p∈I}. Note that we have {p∈P:xi−χ(p) y√p∈I}=   finite set if xi6∈ I, P\finite set if xi∈◦ I, where Idenotes the closure and ◦ Ithe interior of I. If I= [xi, b]with b > xi, then moreover {p∈P:xi−χ(p) y√p∈I}={p∈P:χ(p) = −y |y|}\finite set, and analogously for I= [a, xi]with a < xi, {p∈P:xi−χ(p) y√p∈I}={p∈P:χ(p) = y |y|}\finite set. The same formulas hold if the intervals are open or half-open. In particular, for any interval I, the set {p∈P:xi−χ(p) y√p∈I}has a density, which is one of 0,1 2,1. Proof of Theorem 4.1.1. We use the notation introduced in Remark 4.1.3 (a). (a) See [7], Theorem 4.1. (b) Assume that Fthas complex multiplication. Put D={0},f=1 2π 1 √1−t2,I= (0,1] and J= [−1,0). Take SI(x) := {p∈P:B(p)∈I}, T′ I:= {p∈P:A(p)∈I, B(p)6= 0} and similarly SJ(x) := {p∈P:B(p)∈J}, T′ J:= {p∈P:A(p)∈J, B(p)6= 0}. The sets SIand SJhave natural densities, respectively µCM (I)and µCM (J)by Theorem 3.1.1, so that we can apply Theorem 4.2.1. For simplicity we assume a(t)>0. The arguments in the other case a(t)<0are exactly the same. We have {p∈P:A(p)>0}=P>0. By Remark 4.2.2, we conclude that P>0=T′ I⊔{p∈P:B(p) = 0}∩{p∈P:−χt,N (p) 2√p∈I}(4.18) P<0=T′ J⊔{p∈P:B(p) = 0}∩{p∈P:−χt,N (p) 2√p∈J}(4.19) 22 In order to compute d(P>0), we compute the sum of d(T′ I)and the density of the intersection, and similarly for d(P<0). We have d(T′ I) = µ(I) = 1 4and d(T′ J) = µ(J) = 1 4by Theorem 4.2.1. (bi) Assume that χt,N = 1 (recall that by an equality of Dirichlet characters we understand that the underlying primitive characters agree). In this case, since the intersection in Equation (4.18) is finite and therefore has density 0, we conclude that the set P>0has density 1/4. Similarly, the intersection in Equation (4.19) has density 1/2, therefore P<0has density 3/4. It is clear that the set P=0 has natural density equal to zero. (bii) We will do the same computation as above. Note that in this case we have {p∈P:B(p) = 0}={p∈P:δ(p) = −1} up to finitely many primes. These sets have natural density 1/2. Suppose that χt,N =δ. Then the density of the intersection in Equation (4.18) is 1/2by Remark 4.2.2. So we conclude that P>0has natural density 3/4. Similarly, from Equation (4.19) we obtain that P<0has natural density 1/4. (biii) Suppose that χt,N 6= 1, δ. By Chebotarev’s theorem, the intersections in Equations (4.18) and (4.19) have natural density 1/4. So we conclude that the sets P>0and P<0have the same natural density, which is equal to 1/2. (c) By assumption in the non-CM case and by Theorem 3.1.1 in the CM case, we have for all intervals I⊆[−1,1]  πSI(x) π(x)−µ(I)≤C log(x)α. For the CM-case we need R1 1−ǫf(t)dt =R1 1−ǫ 1 2π√1−t2dt ≤√ǫ, as a simple calculation shows. The corresponding check in the non-CM case is trivial since the density function of the measure is continuous on [−1,1]. Thus, in both cases Theorem 4.2.1 (b) yields  πT′ I(x) π(x)−ZI f(t)dt≤˜ C log(x)α for some ˜ C > 0, where fis the density function in the CM or non-CM case. Corollary 2.2.4 shows that T′ Iis weakly regular. Since P=0 =T′ Ifor I= [0,0] = {0},P≥0=T′ [0,1] and P>0=T′ (0,1] in the non-CM case, it follows that the sets P=0,P≥0and P>0are weakly regular set of primes. By a similar argument, it is easily seen that the the sets P≤0and P<0are weakly regular sets of primes. Let us consider the CM case. Then P=0 is a weakly regular set of primes, since T′ [0,0] is. We have to show that the intersections in Equations (4.18) and (4.19) are weakly regular sets, since finite disjoint unions of weakly regular sets are weakly regular. 23 So, assume that χt,N = 1. In this case the intersection in Equation (4.18) is finite and therefore weakly regular of density 0. Since the set {p∈P:B(p) = 0}is weakly regular of density 1/2by Proposition 2.2.7 and {p∈P:−χt,N (p) 2√p∈[−1,0)}is P(except for a finite set), we conclude that the intersection in Equation (4.19) is weakly regular of density 1/2. For the case χt,N =δ, the intersection in Equation (4.18) is {p∈P:B(p) = 0}up to finitely many primes, hence weakly regular of density 1/2by Proposition 2.2.7. The intersection in Equation (4.19) is finite and hence also weakly regular. In the last case χt,N 6= 1, δ, the intersections in Equations (4.18) and (4.19) are weakly regular of density 1/4by Proposition 2.2.7. (d) Similar arguments as in part (c) prove the assertions, using Proposition 2.2 of [7] instead of Corollary 2.2.4 and replacing weak regularity by regularity throughout. 4.3 Equidistribution of signs of half-integral weight modular forms - the general case We now apply the results from Section 2 and Theorem 4.1.1 to obtain an equidistribution statement for the signs of a(tn2)for n∈N, as well as many subsets of N. In order to give a uniform description of the results, let χ:N→ {0,1}be a multiplicative arithmetic function such that χ(p) = 1 for all primes p∈P. Then define Nχ={n∈N:χ(n) = 1}. For example, for k∈N∪{∞} one can take χksuch that χk(pn) =    1if n≤k, 0otherwise. Then Nk:= Nχkis the set of (k+ 1)-free integers if k∈Nand N∞=N. Corollary 4.3.1. Let χas above. Assume the setting of part (c) of Theorem 4.1.1. Then the sets {n∈N|n∈Nχand a(tn2)>0}and {n∈N|n∈Nχand a(tn2)<0} have equal positive Dedekind-Dirichlet densities, that is, both are precisely half of the density of the set {n∈N|n∈Nχand a(tn2)6= 0}. Proof. Note that without loss of generality we can assume a(t)>0. Define the arithmetic function ψ:N→ {−1,0,1}as follows: ψ(n) := χ(n)·         1if a(tn2)>0, −1if a(tn2)<0, 0if a(tn2) = 0. 24 Equation (4.15) implies that ψis a multiplicative function. Note that P>0={p∈P:ψ(p) = 1}, P<0={p∈P:ψ(p) = −1}, and P=0 ={p∈P:ψ(p) = 0}. Theorem 4.1.1 shows that these sets are weakly regular and allows us to conclude due to Proposition 2.3.1. Corollary 4.3.2. Let χas above. Assume the setting of part (d) of Theorem 4.1.1. Then the sets {n∈N|n∈Nχand a(tn2)>0}and {n∈N|n∈Nχand a(tn2)<0} have equal positive natural densities, that is, both are precisely half of the density of the set {n∈N|n∈Nχand a(tn2)6= 0}. Proof. 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