Twist-minimal trace formula for holomorphic cusp forms

We derive an explicit formula for the trace of an arbitrary Hecke operator on spaces of twist-minimal holomorphic cusp forms with arbitrary level and character, and weight at least 2. We show that this formula provides an efficient way of computing Fourier coefficients of basis elements for newform or cusp form spaces. This work was motivated by the development of a twist-minimal trace formula in the non-holomorphic case by Booker, Lee and Strömbergsson, as well as the presentation of a fully generalised trace formula for the holomorphic case by Cohen and Strömberg.

Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

Let {j(z)} be the modular j-invariant function. Let τ be an algebraic number in the complex upper half plane {\mathbb{H}}. It was proved by Schneider and Siegel that if τ is not a CM point, i.e., {[\mathbb{Q}(\tau):\mathbb{Q}]\neq 2}, then {j(\tau)} is transcendental. Let f be a harmonic weak Maass form of weight 0 on {\Gamma_{0}(N)}. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of f on Hecke orbits of τ. For a positive integer m, let {T_{m}} denote the m-th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp {i\infty} are algebraic, and that f has its poles only at cusps equivalent to {i\infty}. We prove, under a mild assumption on f, that, for any fixed τ, if N is a prime such that {N\geq 23} and {N\notin\{23,29,31,41,47,59,71\}}, then {f(T_{m}.\tau)} are transcendental for infinitely many positive integers m prime to N.

Statistics of Hecke eigenvalues for GL(𝑛)

A two-dimensional central limit theorem for the eigenvalues of {\mathrm{GL}(n)} Hecke–Maass cusp forms is newly derived. The covariance matrix is diagonal and hence verifies the statistical independence between the real and imaginary parts of the eigenvalues. We also prove a central limit theorem for the number of weighted eigenvalues in a compact region of the complex plane, and evaluate some moments of eigenvalues for the Hecke operator {T_{p}} which reveal interesting interferences.

UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ARISING FROM HILBERT MODULAR SURFACES

Let $p$ be a prime number and $F$ a totally real number field. For each prime $\mathfrak{p}$ of $F$ above $p$ we construct a Hecke operator $T_{\mathfrak{p}}$ acting on $(\text{mod}\,p^{m})$ Katz Hilbert modular classes which agrees with the classical Hecke operator at $\mathfrak{p}$ for global sections that lift to characteristic zero. Using these operators and the techniques of patching complexes of Calegari and Geraghty we prove that the Galois representations arising from torsion Hilbert modular classes of parallel weight $\mathbf{1}$ are unramified at $p$ when $[F:\mathbb{Q}]=2$. Some partial and some conjectural results are obtained when $[F:\mathbb{Q}]>2$.

A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform

Although links between values of finite field hypergeometric functions and eigenvalues of elliptic modular forms are well known, we establish in this paper that there are also connections to eigenvalues of Siegel modular forms of higher degree. Specifically, we relate the eigenvalue of the Hecke operator of index p of a Siegel eigenform of degree 2 and level 8 to a special value of a 4F3-hypergeometric function.

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

On the Kohnen plus space for Hilbert modular forms of half-integral weight I

In this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula.

HECKE-TYPE CONGRUENCES FOR TWO SMALLEST PARTS FUNCTIONS

We prove infinitely many congruences modulo 3, 5, and powers of 2 for the overpartition function [Formula: see text] and two smallest parts functions: [Formula: see text] for overpartitions and M2spt(n) for partitions without repeated odd parts. These resemble the Hecke-type congruences found by Atkin for the partition function p(n) in 1966 and Garvan for the smallest parts function spt(n) in 2010. The proofs depend on congruences between the generating functions for [Formula: see text], [Formula: see text], and M2spt(n) and eigenforms for the half-integral weight Hecke operator T(ℓ2).

QUASIMODULAR FORMS AND COHOMOLOGY

We construct linear maps from the spaces of quasimodular forms for a discrete subgroup Γ of SL(2,ℝ) to some cohomology spaces of the group Γ and prove that these maps are equivariant with respect to appropriate Hecke operator actions. The results are obtained by using the fact that there is a correspondence between quasimodular forms and certain finite sequences of modular forms.