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It is shown that there are infinitely many primitive cusp forms f of weight 2 with the property that for all X large enough, every interval (X, X + cX(1/4)), where c > 0 depends only on the form, contains an integer n such that the n-th Fourier coefficient of f is nonzero.

The endomorphism algebra X f X_f attached to a non-CM primitive cusp form f f of weight at least two is a 2-torsion element in the Brauer group of a number field F F . We give formulas for the ramification of X f X_f locally at primes lying above the odd supercuspidal primes of f f . We show that the local Brauer class is determined by the underlying local Galois representation together with an auxiliary Fourier coefficient.

Recently, Bruinier, Kohnen and Ono obtained an explicit description of the action of the theta-operator on meromorphic modular forms f f on S L 2 ( Z ) SL_2(\mathbb {Z}) in terms of the values of modular functions at points in the divisor of f f . Using this result, they studied the exponents in the infinite product expansion of a modular form and recurrence relations for Fourier coefficients of a modular form. In this paper, we extend these results to meromorphic modular forms on Γ 0 ( N ) \Gamma _0(N) for an arbitrary positive integer N > 1 N>1 .

We give upper bounds for the maximal order of vanishing at ∞ \infty of a modular form or cusp form of weight k k on Γ 0 ( N p ) \Gamma _0(Np) when p ∤ N p\nmid N is prime. The results improve the upper bound given by the classical valence formula and the bound (in characteristic p p ) given by a theorem of Sturm. In many cases the bounds are sharp. As a corollary, we obtain a necessary condition for the existence of a non-zero form f ∈ S 2 ( Γ 0 ( N p ) ) f\in S_2(\Gamma _0(Np)) with ord ∞ ⁡ ( f ) \operatorname {ord} _\infty (f) larger than the genus of X 0 ( N p ) X_0(Np) . In particular, this gives a (non-geometric) proof of a theorem of Ogg, which asserts that ∞ \infty is not a Weierstrass point on X 0 ( N p ) X_0(Np) if p ∤ N p\nmid N and X 0 ( N ) X_0(N) has genus zero.

Bruinier and Ono classified cusp forms of half-integral weight F(z):=∞Σn=0a(n)q n ∈ S λ+1 2 (Γ 0 (N), Χ )∩Z[[q]] whose Fourier coefficients are not well distributed for modulo odd primes l. Ahlgren and Boylan established bounds for the weight of such a cusp form and used these bounds to prove Newman's conjecture for the partition function for prime-power moduli. In this note, we give a simple proof of Ahlgren and Boylan's result on bounds of cusp forms of half-integral weight.

We determine an explicit formula for a Rankin-Cohen bracket for Siegel modular forms of degree n on a certain subgroup of the symplectic group. Moreover, we lift that bracket via a Poincare series to a Siegel cusp form on the full symplectic group.

Let f = ∑ n = 1 ∞ a f ( n ) q n f=\sum _{n=1}^\infty a_f(n)q^n be a cusp form with integer weight k ≥ 2 k\ge 2 that is not a linear combination of forms with complex multiplication. For n ≥ 1 n\ge 1 , let \[ i f ( n ) := max { i : a f ( n + j ) = 0 for all 0 ≤ j ≤ i } . i_f(n):=\max \{i:a_f(n+j)=0\quad \text {for all $0\le j\le i$}\}. \] Improving on work of Balog, Ono, and Serre we show that i f ( n ) ≪ f , ϕ ϕ ( n ) i_f(n)\ll _{f,\phi }\phi (n) for almost all n n , where ϕ ( x ) \phi (x) is any good function (e.g. such as log ⁡ log ⁡ ( x ) \log \log (x) ) monotonically tending to infinity with x x . Using a result of Fouvry and Iwaniec, if f f is a weight 2 cusp form for an elliptic curve without complex multiplication, then we show for all n n that i f ( n ) ≪ f , ε n 69 169 + ε i_f(n)\ll _{f,\varepsilon } n^{\frac {69}{169}+\varepsilon } . We also obtain conditional results depending on the Generalized Riemann Hypothesis and the Lang-Trotter Conjecture.

Given a complex number s with 0 < Res < 1, we study the existence of a cusp form of large even weight for the full modular group such that its associated symmetric square L-function L(sym 2 f, s) does not vanish. This problem is also considered in other articles.

We estimate the number of Fourier coefficients that determine a Hilbert modular cusp form of arbitrary weight and level. The method is spectral (Rayleigh quotient) and avoids the use of the maximum principle.

We introduce a simple construction of a Siegel cusp form obtained by taking the difference between the Siegel Eisenstein series and the restricted Hermitian Eisenstein series. In addition, we present applications of the Siegel cusp form.

Let G be a semisimple algebraic group defined over a number field k. In the paper are studied unramified irreducible components of irreducible automorphic cuspidal representations in the space of cusp forms A_cusp(G(k) \ G(A)) using the action of unramified Hecke algebra on compactly supported cuspidal Poincare series.

To each rank 2 parabolic subgroup of a Kleinian group T, we as- sociate a cusp form and a linear functional on the Eichler cohomology space PH (r, n2._2) • We explore the relation between these two objects and as a consequence we evaluate certain Poincare series for rank 2 parabolic groups. This paper is a continuation of the study of automorphic forms associated to elements of Kleinian groups. We treated the case of cusp forms associated to loxodromic elements in (K3) and the holomorphic forms associated to parabolic elements in (Kl). We combine ideas from (K3) and (Kl) to investigate the cusp forms associated to rank 2 parabolic subgroups of Kleinian groups. Let A and F be generators of a rank 2 parabolic subgroup G of a Kleinian group r. We associate (in §2) to the pair (A, B) a linear functional

The norms of modular cusp forms, viewed as belonging to the Bers' spaces of integrable and bounded forms, are estimated in terms of the Fourier coefficients of the cusp form. In [5] R. A. Smith found upper and lower bounds for the L2-norms of Maass wave functions which were also simultaneous eigenfunctions of the Hecke operators. His bounds depended on the size of the first Fourier coefficient Cf (1) of f and elucidated the connection between Icf (1)1 and the L2-norm of f under two different normalizations of f. Analogous results are examined here for cusp forms on subgroups of the classical modular group which can be viewed as belonging to the Bers' spaces Br (G), Ar (G), and A' (G). Estimates are given which have bearing on the numerical calculation of the various norms and which emphasize the connection between the first nonzero Fourier coefficient of the cusp form and the value of the norm. 1. Definitions and notation. Let G denote a subgroup of finite index in the classical modular group G(1). A function f analytic in the upper half-plane Z is a cusp form on G of positive weight r with multiplier system iv of modulus one if for all z E X (1) f(Mz) = v(M)(cz + d)rf(z) for each M= (a b) EG and f(z) -, 0 as z approaches any cusp of G from within a fundamental region R. Then f has a Fourier expansion of the form f(z) = E c(n)e((n + k)z/A) where e(z) = e2,iZ n+k>O and k is defined by v(SA) = e(k), O < k < 1, with SAz = z + A being the minimal positive translation of G. The Bers spaces ATP (G) of p-integrable and bounded forms are defined as follows. A function f belongs to AP (G) if f is analytic in ', satisfies (1), and p 1= G 1 ypr/2-2If(z)IPdxdy < 00, p < ?? or lIf 110 = sup yr/2If(Z)| < 00. zE,r Received by the editors Maxch 30, 1986 and, in revised form, March 28, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary llFll; Secondary 11F30. ?)1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page

The spectral theory of Roelke and Selberg provides a decomposition of the space of square integrable automorphic forms for the group S L ( 2 ) SL(2) in terms of eigenfunctions of the non-Euclidean Laplacian and of the Hecke operators. The main result of the paper uses the Roelke-Selberg theory to give an interpretation of the L L -functions of Rankin type as "multiplicity factors" in the decomposition of the product of a nonholomorphic Eisenstein series and a cusp form.

x = exp 2ri-T, im r > 0, have been given by D. H. Lehmer [1], J. Lehner [2; 3], and A. van Wijngaarden [4]. The moduli for which congruence properties have been determined are products of powers of 2, 3, 5, 7, 11. Thus Lehner has shown that if n>I is divisible by 2a3b5c7dl le, where a, b, c, d> 1 and e= 1, 2, 3 then c(n) is divisible by 23a+832b+35c+1 7dl 1e. In this note we give several congruence properties modulo 13, derived from some general congruences for the coefficients of certain modular forms and an explicit formula for the coefficients c(n). These general congruences are of interest in themselves and will be proved here as well. If n is a non-negative integer, define p,(n) as the coefficient of xn in fl(i -xn)r; otherwise define Pr(n) as zero.2 (Here and in what follows all products are extended from 1 to oo and all sums from 0 to 0o, unless otherwise stated.) Special cases of identities proved by the author in [5] and [6] follow: Let p be a prime >3. Set 8=(p-1)/12, A=(p2-1)/12. Then

Let G be a finitely generated Fuchsian group of the first kind. Let ^be a cusp form, and/a solution to 8j/ = ^, where Bi is the Schwarzian derivative. Then for every A G that is