Density results for automorphic forms on Hilbert modular groups II (original) (raw)

On the oscillatory behavior of certain arithmetic functions associated with automorphic forms

Journal of Number Theory, 2011

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both "naive" and "modified"), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.

On the Fourier coefficients of 2-dimensional vector-valued modular forms

Proceedings of the American Mathematical Society, 2012

Let ρ : SL(2, Z) → GL(2, C) be an irreducible representation of the modular group such that ρ(T ) has finite order N . We study holomorphic vector-valued modular forms F (τ ) of integral weight associated to ρ which have rational Fourier coefficients. (These span the complex space of all integral weight vectorvalued modular forms associated to ρ.) As a special case of the main Theorem, we prove that if N does not divide 120 then every nonzero F (τ ) has Fourier coefficients with unbounded denominators.

Annals of Mathematics 174 (2011), 439–484 doi: 10.4007/annals.2011.174.1.12 Hilbert modular forms and

2016

Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit in the extension of F cut out by χ. In this paper we prove Gross's conjecture when F is a real quadratic field and χ is a narrow ring class character. The main result also applies to general totally real fields for which Leopoldt's conjecture holds, assuming that either there are at least two primes above p in F , or that a certain condition relating the L-invariants of χ and χ −1 holds. This condition on L-invariants is always satisfied when χ is quadratic. Contents S. DASGUPTA, H. DARMON, and R. POLLACK 4. Galois representations 477 4.1. Representations attached to ordinary eigenforms 477 4.2. Construction of a cocycle 480 References 482

Modular Forms and Automorphic Representations

2010

Let H = {z ∈ C;=(z) > 0} be the Poincaré upper-half plane. Let k and N be two integers, and, as usual, Γ0(N) be the subgroup of SL(2,Z) of matrices whose lower left entries are divisible by N . It acts on H by fractional linear transformations: [ a b c d ] · z = az+b cz+d . Let χ be a Dirichlet character modulo q: it defines a character on Γ0(N), by evaluating χ at the upper left entry. It will be convenient to define χ(n) = 0 if the integer n is not coprime with N .

Eigenvalues of Hecke operators on Hilbert modular groups

Asian Journal of Mathematics, 2013

Let F be a totally real field, let I be a nonzero ideal of the ring of integers O F of F, let Γ 0 (I) be the congruence subgroup of Hecke type of G = d j=1 SL 2 (R) embedded diagonally in G, and let χ be a character of Γ 0 (I) of the form χ a c b d = χ(d), where d → χ(d) is a character of O F modulo I. For a finite subset P of prime ideals p not dividing I, we consider the ring H I , generated by the Hecke operators T (p 2), p ∈ P (see §3.2) acting on (Γ, χ)automorphic forms on G. Given the cuspidal space L 2,cusp ξ Γ 0 (I)\G, χ , we let V run through an orthogonal system of irreducible G-invariant subspaces so that each V is invariant under H I. For each 1 ≤ j ≤ d, let λ = (λ , j) be the vector formed by the eigenvalues of the Casimir operators of the d factors of G on V , and for each p ∈ P, we take λ ,p ≥ 0 so that λ 2 ,p − N(p) is the eigenvalue on V of the Hecke operator T (p 2) For each family of expanding boxes t → Ω t , as in (3) in R d , and fixed an interval J p in [0, ∞), for each p ∈ P, we consider the counting function N(Ω t ; (J p) p∈P) := , λ ∈ Ωt : λ ,p ∈Jp ,∀p∈P |c r ()| 2. Here c r () denotes the normalized Fourier coefficient of order r at ∞ for the elements of V , with r ∈ O F p O F for every p ∈ P. In the main result in this paper, Theorem 1.1, we give, under some mild conditions on the Ω t , the asymptotic distribution of the function N(Ω t ; (J p) p∈P), as t → ∞. We show that at the finite places outside I the Hecke eigenvalues are equidistributed with respect to the Sato-Tate measure, whereas at the archimedean places the eigenvalues λ are equidistributed with respect to the Plancherel measure. As a consequence, if we fix an infinite place l and we prescribe λ , j ∈ Ω j for all infinite places j l andλ ,p ∈ J p for all finite places p in P (for fixed intervals Ω j and J p) and then allow |λ ,l | to grow to ∞, then there are infinitely many such , and their positive density is as described in Theorem 1.1. Contents 1991 Mathematics Subject Classification. 11F41 11F60 11F72 22E30.

2-ADIC Properties of Certain Modular Forms and Their Applications to Arithmetic Functions

It is a classical observation of Serre that the Hecke algebra acts locally nilpo- tently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular L-functions.