Abstract:
This article is a research exposition based on the author's talk at the International Colloquium on Automorphic Representations and L-Functions, 2012, held at TIFR, Mumbai. We consider some special cases of the following question: when is a natural subset of the Fourier coefficients sufficient to uniquely determine a "modular form"? Two kinds of modular forms are considered in this article: a) classical modular forms of half-integral weight, and b) Siegel modular forms of genus 2 and integral weight. These two apparently different scenarios turn out to be closely related. Our results were motivated by, and have several interesting connections to, automorphic L-functions and Bessel models.

Abstract:
We consider a variant of a question of N. Koblitz. For an elliptic curve $E/\Q$ which is not $\Q$-isogenous to an elliptic curve with torsion, Koblitz has conjectured that there exists infinitely many primes $p$ such that $N_p(E)=#E(\F_p)=p+1-a_p(E)$ is also a prime. We consider a variant of this question. For a newform $f$, without CM, of weight $k\geq 4$, on $\Gamma_0(M)$ with trivial Nebentypus $\chi_0$ and with integer Fourier coefficients, let $N_p(f)=\chi_0(p)p^{k-1}+1-a_p(f)$ (here $a_p(f)$ is the $p^{th}$-Fourier coefficient of $f$). We show under GRH and Artin's Holomorphy Conjecture that there are infinitely many $p$ such that $N_p(f)$ has at most $[5k+1+\sqrt{\log(k)}]$ distinct prime factors. We give examples of about hundred forms to which our theorem applies.

Abstract:
Let $d(n)$ denote the number of divisors of $n$. In this paper, we study the average value of $d(a(p))$, where $p$ is a prime and $a(p)$ is the $p$-th Fourier coefficient of a normalized Hecke eigenform of weight $k \ge 2$ for $\Gamma_0(N)$ having rational integer Fourier coefficients.

Abstract:
Let r : G_Q -> GL_n Q_l be a motivic l-adic Galois representation. For fixed m > 1 we initiate an investigation of the density of the set of primes p such that the trace of the image of an arithmetic Frobenius at p under r is an m^th power residue modulo p. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals 1/m whenever the image of r is open. We further conjecture that for such r the set of these primes p is independent of any set defined by Cebatorev-style Galois theoretic conditions (in an appropriate sense). We then compute these densities for certain m in the complementary case of modular forms of CM-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the CM case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at p in abelian extensions of imaginary quadratic fields unramified away from p.

Abstract:
A thorough analysis is made of the Fourier coefficients for vector-valued modular forms associated to three-dimensional irreducible representations of the modular group. In particular, the following statement is verified for all but a finite number of equivalence classes: if a vector-valued modular form associated to such a representation has rational Fourier coefficients, then these coefficients have "unbounded denominators", i.e. there is a prime number p, depending on the representation, which occurs to an arbitrarily high power in the denominators of the coefficients. This provides a verification in the three-dimensional setting of a generalization of a long-standing conjecture about noncongruence modular forms.

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

Abstract:
We prove the following theorem. Suppose that $F=(f_1, f_2)$ is a 2-dimensional vector-valued modular form on $SL_2(Z)$ whose component functions $f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.

Abstract:
In this note, we show that the algebraicity of the Fourier coefficients of half-integral weight modular forms can be determined by checking the algebraicity of the first few of them. We also give a necessary and sufficient condition for a half-integral weight modular form to be in Kohnen's +-subspace by considering only finitely many terms.

Abstract:
We prove a formula of Petersson's type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficient. The method in this paper is essentially a generalization of Kitaoka's previous work which studied the full modular case, but some modification is necessary to obtain estimates which are sharp with respect to the level aspect.

Abstract:
In this note, we generalize the isomorphisms to the case when the discriminant form is not necessarily induced from real quadratic fields. In particular, this general setting includes all the subspaces with epsilon-conditions, only two spacial cases of which were treated before. With this established, we shall prove the Zagier duality for canonical bases. Finally, we raise a question on the integrality of the Fourier coefficients of these bases elements, or equivalently we concern the existence of a Miller-like basis for vector valued modular forms.