Classical Modular Forms Research Articles

Local Newforms with Global Applications in the Jacobi Theory

The study of spherical representations of the Jacobi group begun in (R. Schmidt, 1998, Abh. Math. Sem. Univ. Hamburg68, 273–296) is continued. Using certain index shifting operators, the notion of age of such a representation is introduced, as well as the notion of a local newform. The precise structure of the space of spherical vectors, in particular its dimension, is determined in terms of the age. The age of the spherical principal series representations is computed, and the local newforms amongst these are determined. Restricting to the classical situation, it is shown that the local index shifting operators essentially coincide with well-known Hecke operators on classical modular forms. This leads to some global applications of the local results.

Isogeny Covariant Differential Modular Forms Modulo p

An interesting theory arises when the classical theory of modular forms is expanded to include differential analogs of modular forms. One of the main motivations for expanding the theory of modular forms is the existence of differential modular forms with a remarkable property, called isogeny covariance, that classical modular forms cannot possess. Among isogeny covariant differential modular forms there exists a particular modular form that plays a central role in the theory. The main result presented in the article will be the explicit computation modulo p of this fundamental isogeny covariant differential modular form.

Icosahedral Galois representations

Lifting projective representations.- Local primitive galois representations.- Two-dimensional representations over Q.- The L-series of an icosahedral representation.- Classical modular forms of weight one.- An icosahedral form.

Classical and overconvergent modular forms

The purpose of this article is to use rigid analysis to clarify the relation between classical modular forms and Katz’s overconvergent forms. In particular, we prove a conjecture of F. Gouvea [G, Conj. 3] which asserts that every overconvergent p-adic modular form of sufficiently small slope is classical. More precisely, let p > 3 be a prime, K a complete subfield of Cp, N be a positive integer such that (N, p) = 1 and k an integer. Katz [K-pMF] has defined the spaceMk(Γ1(N)) of overconvergent p-adic modular forms of level Γ1(N) and weight k over K (see §2) and there is a natural map from weight k modular forms of level Γ1(Np) with trivial character at p to Mk(Γ1(N)). We will call these modular forms classical modular forms. In addition, there is an operator U on these forms (see [G-ApM, Chapt. II §3]) such that if F is an overconvergent modular form with q-expansion F (q) = ∑ n≥0 anq n then UF (q) = ∑

On relations between analytic modular forms and Maass waveforms forPSL(2,Z)

In the standard Hilbert space L2(F; dμ) of even automorphic functions, where F is the fundamental domain of the modular group PSL(2, ℤ), a full system of functions (in terms of classical analytic modular forms) is constructed, up to a finite-dimensional subspace. Therefore, in particular, almost every Maass waveform can be represented by a series in analytic forms. Bibliography: 9 titles.

Classical and overconvergent modular forms

The purpose of this article is to use rigid analysis to clarify the relation between classical modular forms and Katz’s overconvergent forms. In particular, we prove a conjecture of F. Gouvea [G, Conj. 3] which asserts that every overconvergent p-adic modular form of sufficiently small slope is classical. More precisely, let p > 3 be a prime, K a complete subfield of Cp, N be a positive integer such that (N, p) = 1 and k an integer. Katz [K-pMF] has defined the spaceMk(Γ1(N)) of overconvergent p-adic modular forms of level Γ1(N) and weight k over K (see §2) and there is a natural map from weight k modular forms of level Γ1(Np) with trivial character at p to Mk(Γ1(N)). We will call these modular forms classical modular forms. In addition, there is an operator U on these forms (see [G-ApM, Chapt. II §3]) such that if F is an overconvergent modular form with q-expansion F (q) = ∑ n≥0 anq n then UF (q) = ∑

Searching for $p$-adic eigenfunctions

In doing p-adic analysis on spaces of classical modular functions and forms, it is convenient and traditional to broaden the notion of “modular form” to a class called “overconvergent p-adic modular forms.” Critical for the analysis of the p-adic Banach spaces composed of this wider class of forms is the “Atkin U -operator”, which is completely continuous and whose spectral theory (still not very well understood) seems to be the key to a good deal of arithmetic. The part of the spectrum of U corresponding to eigenvalues which are p-adic units is somewhat more understood, thanks to the work of Hida. As for the rest of the spectrum, it is surprising how fragmentary our information is (although recent work of Coleman, resolving in part some prior conjectures of ours, has improved the situation). We have begun an experimental search for nonclassical (but “overconvergent”) eigenfunctions in a fairly simple way. We take a number of classical modular functions which are p-adically overconvergent (e.g., j, 1/j,...), and try to find their p-adic “U -eigenfunction expansion.” There is a straightforward computational procedure to approximate such eigenfunction expansions, even though, on a theoretical level, we do not even know that the “expansions” that our algorithm produces converge in any sense, or even settle down numerically. Experimentally, they seem to, and they produce candidate Fourier expansions. In our computations, we specialize principally to p = 5. The same eigenfunctions (produced by our algorithm) seem to occur as terms in each of the eigenfunction expansions we have calculated. This leads us to suspect that we have in fact encountered all the 5-adic eigenfunctions

A simple example of modular forms as tau-functions for integrable equations

We show how classical modular forms and functions appear as tau-functions for a certain integrable reduction of the self-dual Yang-Mills equations obtained by S. Chakravarty, M. Ablowitz, and P. Clarkson [6]. We discuss possible consequences of this novel phenomenon in integrable systems which indicate deep connections between integrable equations, group representations, modular forms, and moduli spaces.

The Poisson kernel for Drinfeld modular curves

In an earlier work [T 1], the author described a technique for constructing rigid analytic modular forms on the p-adic upper half plane by means of an integral transform (a Poisson Kernel). In this paper, we apply these methods to the study of rigid analytic modular forms on the upper half plane over a complete local field k of p. Arising out of the theory of Drinfeld modules [D], these modular forms, first studied by Goss (see [Gol, Go2]), are in many ways analogous to the classical modular forms for SL2 (Z). For a brief, but effective, introduction to the theory of these characteristic p modular forms, see the introduction to Gekeler's paper [Gekl]. Also useful is Gekeler's book [Gek2]. We establish three main results. First of all, we show that if, k is the function field of a complete, geometrically irreducible curve C over the finite field lFq with q elements, k is a completion of k, F is an arithmetic subgroup of GL2(k), and ,u is a measure on Pi coming from a harmonic cocycle of weight n for I, (see Definition 1 below) then

Jacobiformen und Thetareihen

We give a characterisation of Jacobi forms by classical modular forms from which we obtain dimension formulas for the spaces of Jacobi forms in certain cases. Then we consider the ordinary theta series to the quaternary quadratic forms of discriminant q2 (q an odd prime) representing 2; these possess a ‘natural’ continuation to Jacobi forms for which we give a sufficient condition of linear independence. If this condition is fulfilled and if there is no cusp form of weight 4 with respect to Γo(q) which vanishes at the cusp 0 with a certain order then the classical theta series are also linear independent.

Indefinite theta functions arising in Gromov–Witten theory of elliptic orbifolds

In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modularity properties of several simpler pieces of the last, and most mysterious, function by developing several identities between functions with properties generalizing those of the mock modular forms in Zwegers' thesis. Here, we complete the analysis of all pieces of Cho, Hong, Kim, and Lau's functions, inspired by recent work of Alexandrov, Banerjee, Manschot, and Pioline on similar functions. Combined with the work of Lau and Zhou, as well as the aforementioned work of Zwegers and two of the authors, this affords a complete understanding of the modularity transformation properties of the open Gromov-Witten potentials of elliptic orbifolds of the form P-a, b, c(1) computed by Cho, Hong, Kim, and Lau. It is hoped that this will provide a fuller picture of the mirror-symmetric properties of these orbifolds in subsequent works.