On the equality case of the Ramanujan Conjecture for Hilbert modular forms Liubomir Chiriac Abstract The generalized Ramanujan Conjecture for unitary cuspidal automorphic representations π on GL 2 posits that a v (π) 2. We prove that this inequality is strict if π is generated by a Hilbert modular form with complex multiplication and v is a finite place of degree one. Equivalently, the Satake parameters at v are necessarily distinct. We also give examples where the equality case does occur for primes of degree two. 1 Introduction Let π = v π v be a unitary cuspidal automorphic representation of GL 2 (A F ), where A F is the adèles ring of a number field F. For almost all finite places v the representation π v of the local group GL 2 (F v ) is unramified. For such v, the associated conjugacy class in GL 2 (C) is given by a diagonal matrix [α v, β v ], where α v and β v are complex numbers called the Satake parameters at v. The trace of this class is denoted by a v (π). The generalized Ramanujan Conjecture (GRC) predicts that all unramified π v are tempered, i.e., α v = β v = 1. The GRC is known to be true in several special cases, such as when π is attached to a classical holomorphic modular form or to a Hilbert modular form (see Livné [9] for a detailed account on the latter). In general, one knows that α v, β v (Nv) 7/64 by the work of Kim and Sarnak [7] (over Q), and Blomer and Brumley (over arbitrary number fields). In another direction, Ramakrishnan [10] proved that at least 90% of the local components satisfy the GRC. This bound was further improved by Kim and Shahidi [8] who showed that there exists a set of primes v for which GRC holds that has a lower Dirichlet density of 34/35 0.971. An equivalent way 1 of stating the GRC is that a v (π) 2. In the context of classical or Hilbert modular forms this is also referred to as the Ramanujan-Petersson conjecture. The primary goal of the present paper is to address the sharpness of this inequality. In other words, we are interested whether the Satake parameters at some v can be equal to each other. Our main result is the following. 1 If n 3, the temperedness condition for π on GL n (A F ) clearly implies, but is not equivalent to, a v (π) n. 1
Liubomir Chiriac Theorem 1. Let F be a totally real number field with discriminant disc(f ). Let π = v π v be a cuspidal automorphic representation of GL 2 (A F ) associated to a CM Hilbert modular form f of parallel weight two and trivial character. Let v be an unramified prime of F of degree one whose residual characteristic is prime to the level of f and disc(f ). Then the Satake parameters at v are distinct. If π is attached to a classical modular form of weight k 2, Coleman and Edixhoven [3] have shown that the Satake paramaters are always distinct, provided that the action of the crystalline Frobenius is semisimple. This is known for k = 2 unconditionally, whereas for higher weights it would follow from the Tate s conjecture. On the other hand, using some congruences of Hatada [6], it is not too difficult to see (cf. Gouvêa [5]) that the conclusion of Theorem 1 holds for classical modular forms of level one. To prove Theorem 1 we distinguish two cases. If the degree [F : Q] is odd or if [F : Q] is even and π has some square-integrable local compotent one can transfer to a quaternionic Shimura curve via the Jacquet-Langlands correspondence and then proceed as in [3] (cf. Proposition 2). Thus, the novelty here is when [F : Q] is even and there are no square-integrable local components. In this case, we make use of a construction based on the Grunwald-Wang theorem for which we have to assume that f has complex multiplication (CM), i.e., it arises from a Hecke character of a totally imaginary quadratic extension of F. This way, we can reduce the problem to the previous case. If f does not have CM, the proved Sato-Tate conjecture for Hilbert modular forms [1] immediately gives that the set of primes v where a v (π) = 2 is of density zero. We expect much more to be true, namely that a v (π) < 2 for every v of odd degree. It is worth pointing out that the situation is quite different if we allow v to have even degree. In Section 4 we give a recipe of producing examples of Hilbert newforms that have equal Satake paramaters at primes v of degree two. This is achieved by lifting classical modular forms of weight two and integer Fourier coefficients to Hilbert modular forms over a real quadratic field, and specializing at certain inert primes. 2 The Jacquet-Langlands condition In this section we remove the requirement that f have CM, but instead assume that π has a square-integrable local component when [F : Q] is even. This ensures the existence of an abelian variety that arises as a quotient of the Jacobian of a suitably defined Shimura curve, via the Jacquet-Langlands correspondence. Then we can take advantage of the semisimplicty of the Frobenius endomorphism. Proposition 2. Let F be a totally real number field with discriminant disc(f ). Let π = v π v be a cuspidal automorphic representation of GL 2 (A F ) associated to a Hilbert modular form of parallel weight two. If [F : Q] is even assume that there exists a finite place of F where the local component of π is square-integrable. Let v be an unramified prime of F of degree one whose residual characteristic is prime to the level of f and disc(f ). Then the Satake parameters at v are distinct. 2
On the equality case of the Ramanujan Conjecture for Hilbert modular forms Proof. Recall that, by global class field theory, the set of places at which a quaternion algebra B ramifies is finite and of even cardinality. Conversely, given an even number of places there exists a unique quaternion algebra that ramifies at precisely those places. Now, let B be the quaternion algebra over F with the following properties: B splits at only one infinite place; if [F : Q] is odd, B is unramified at every finite place; if [F : Q] is even, the finite place where π is square-integrable (by assumption) is the unique finite place where B is ramified. Denote by ram(b) the set of places at which B is ramified, and by disc(b) the product of all the finite places in ram(b) (which, by definition, is the discriminant of B). Let n be an ideal such that n = n disc(b), where n is the level of f. By the Jacquet-Langlands correspondence, there exists a cuspidal automorphic representation π of B(A F ) such that π v π v for every v / ram(b). To get the prescribed level structure, we take an Eichler order O(n ) in B of level n and consider the group O 1 (n ) = {θ O(n ) nrd(θ) = 1}. Since our B is split at exactly one real place, there is a map B B Q R M 2 (R) H n 1. Letting ι : B M 2 (R) be the projection onto the first factor, we obtain an arithmetic Fuchsian group (i.e., a discrete subgroup of PSL 2 (R)) Γ B (n ) = ι(o 1 (n )) GL + 2 (R). This group acts by linear fractional transformations on the upper half-plane H and the compact quotient Γ B (n )\H can be viewed as the complex points of the connected component of a Shimura curve C B, which has a canonical model defined over F. At this point, one can argue exactly as in [3] to get the conclusion. For the convenience of the reader, we sketch the argument below. Let K f be the number field generated by all the Fourier coefficients of f, and let O Kf be its ring of integers. There exists an abelian variety A f, defined over F and of dimension [K f : Q], which arises as a quotient of the Jacobian of C B. To get an action of all of O Kf take A = A f O Kf. If v be a prime of F of degree one whose residual characteristic p is prime to the level of f and disc(f ), the abelian variety A has good reduction at v. Denote by A v the reduction of A modulo v. The endomorphism ring End(A v ) of A v is semisimple, so the Frobenius endomorphism of Frob v acts (on the Tate module) as a semisimple linear operator. Assume by contradiction that the Satake parameters of π v are equal, i.e., α v = β v = λ. Then a v (π) = 2λ and q v = λ 2 = p. The Eichler-Shimura congruence relation implies that the characteristic polynomial of Frob v is T 2 a v (π)t +q v, and thus (Frob v λ) 2 = 0. By the semisimplicity of Frob v, we infer that it must act as the multiplication by the scalar λ. In other words, Frob v = λ in End(A v ). 3
Liubomir Chiriac Consider the first de Rham cohomology group M of the associated Néron model A of A over O v. It admits a Hodge filtration Fil 1 M := H 0 (A, Ω 1 ) M. Since Fil 1 (M) is locally isomorphic to O Kf O v and λ 2 = p we get that p does not divide λ, so Fil 1 (M) F v is not annihilated by λ. However, Frob v has differential zero, so by the previous paragraph we get that λ annihilates H 0 (A v, Ω 1 ) = Fil 1 (M) F v, which is a contradiction. 3 Proof of Theorem 1 Now we assume that the Hilbert modular form f has complex multiplication by an imaginary quadratic extension K/F with the associated Hecke character χ. Under this assumption we will settle the case not covered by Proposition 2 namely, when [F : Q] is even and the automorphic representation π generated by f has no discrete series component at a finite place; so one cannot move to a Shimura curve directly. Our strategy is to construct a certain character µ defined by some local relations such that the automorphic representation π corresponding to χµ will have the following two properties: (P 1 ) π and π have the same pair of Satake parameters at v. (P 2 ) π has a square-integrable local component at some auxiliary finite place v. This way, by switching from π to π we reduce the problem to the situation treated in the previous section. Proof of Theorem 1. Note that v cannot be inert in K, for otherwise a v (π) = 0. Therefore, we can assume that v splits in K. Denote by q and q the primes above p in K. If π has a discrete series component π u, the conclusion follows from Proposition 2. So assume otherwise, and pick an auxiliary finite place v of F that remains inert in K. Consider the set S = {q, q, v } of places of K. To each element of S we shall associate a local character as follows: For q and q, simply take the trivial characters of K q and K q. For v, choose λ to be a quadratic character of K v such that λ λ c, where c is the non-trivial element of Gal(K v /F v ). More concretely, we will take λ to be the quadratic character of K v attached to its unique unramified quadratic extension. By the Grunwald-Wang theorem there exists a finite order global character µ of K such that µ q = µ q = 1, µ v = λ and µ = 1. Take π to be the induced representation Ind F K(χµ). 4
On the equality case of the Ramanujan Conjecture for Hilbert modular forms Lemma 3. π satisfies the properties (P 1 ) and (P 2 ). Proof. Since v = qq, the coefficient a v (π) can be written as a v (π) = χ q (ω) + χ q (ω), where ω is an uniformizer of F v = Kq = Kq. The condition µ q = µ q χ q µ q = χ q and χ q µ q = χ q. Therefore = 1 implies that a v (π) = a v (π ). Consider the quadratic character ε of F corresponding to K/F such that f = f ε. By hypothesis, π has trivial central character, i.e., χ F = ε. To establish property (P 1 ) it suffices to show that π has trivial character at v. This is equivalent to showing that χ v µ v = ε v, which is true because v splits. Moreover, by our construction it follows that π v = IndF v K v (χ v µ v ) = IndF v K v (χ v λ). Since π has no supercuspidal component, we get that χ v was chosen such that λ λ c, we obtain = χ v c. Therefore, since λ χ v λ χ v λ c. This means that the local Galois representation at v is irreducible. By local-global compatibility, known for CM forms, one gets the local component π v to be supercuspidal and so (P 2 ) also holds. By Lemma 3 one can replace π with a related π, for which Proposition 2 can be applied. As a result, the Satake paramaters of π at v are distinct. Consequently, the same is true for the Satake parameters of π at v. 4 Examples for primes of degree 2 In this section we give a general recipe of constructing cuspidal automorphic representations on GL(2) where the equality case of the GRC occurs for certain places v of degree 2. The underlying idea is that inert supersingular primes can be lifted to produce extremal Hecke coefficients (i.e., those of absolute value equal to 2). The example below was kindly explained to us by Emerton and is originally due to Kottwitz. Example 4. Start with an elliptic curve E over Q and a prime p of good supersingular reduction, i.e., the trace of Frobenius a p (E) is 0. By a classical result of Elkies [4], it is known that there exist infinitely many such p. Pick any real quadratic field F in which p is inert. Again, for a given p, there are infinitely many quadratic fields with this property. 5
Liubomir Chiriac Let f be a newform associated to E by the modularity theorem, so that the p-th Fourier coefficient a p (f) of f is a p (E). Under the base-change from Q to F, one can lift f to a parallel weight 2 Hilbert modular form f for F. Denote by p the prime of F lying above p; it has degree 2 over p. Since p is inert in F, it follows that the ideal p has norm N(p) = p 2 and a p ( f) = a p (f) 2 2p = 2p. Therefore, if π is the cuspidal automorphic representation generated by f then so the Satake parameters at p are equal. a p (π) = a p( f) = 2, N(p) 1/2 The above construction can be summarized in a slightly more general setting. Lemma 5. Let K/F be a quadratic extension of totally real fields. Let f be a Hilbert newform for F of weight 2 and trivial character, and let f be its base change to K. Denote by π and π the cuspidal automorphic representations generated by f and f, respectively. Assume P is an unramified prime of K, which is of degree 2 over a prime p of F. Then a P (π ) = 2 if and only if either a p (π) = 2 or a p (π) = 0. References [1] T. Barnet-Lamb, Thomas, T. Gee, D. Geraghty The Sato-Tate conjecture for Hilbert modular forms, J. Amer. Math. Soc. 24 (2011), no. 2, 411-469. [2] V. Blomer, F. Brumley, On the Ramanujan conjecture over number fields, Ann. of Math. (2) 174 (2011), no. 1, 581-605. [3] R.F. Coleman, B. Edixhoven, On the semi-simplicity of the U p -operator on modular forms, Math. Ann. 310 (1998), no. 1, 119-127. [4] N. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over Q, Invent. Math. 89 (1987), no. 3, 561-567. [5] F.Q. Gouvêa, Fernando, Where the slopes are, J. Ramanujan Math. Soc. 16 (2001), no. 1, 75-99. [6] K. Hatada, Eigenvalues of Hecke operators on SL(2, Z), Math. Ann. 239 (1979), no. 1, 75-96. 6
On the equality case of the Ramanujan Conjecture for Hilbert modular forms [7] H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selberg Conjectures, J. Amer. Math. Soc. 16 (2003), 139-183, Appendix to H. Kim, Functoriality for the exterior square of GL(4) and symmetric fourth of GL(2). [8] H. Kim, F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), no. 1, 177-197. [9] R. Livné, Communication networks and Hilbert modular forms, in Applications of algebraic geometry to coding theory, physics and computation, NATO Sci. Ser. II Math. Phys. Chem. 36, Kluwer Acad. Publ., Dordrecht, (2001), 255-270. [10] D. Ramakrishnan, On the coefficients of cusp forms, Math. Res. Lett., 4 (1997), pp. 295 307. Liubomir Chiriac Department of Mathematics University of Massachusetts Amherst 710 N Pleasant St, Amherst, MA 01003 USA E-mail address: chiriac@math.umass.edu 7