On modular signs

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

The theme of this paper is the correspondence between classical modular forms and pseudodifferential operators (ΨDO’s) which have some kind of automorphic behaviour. In the simplest case, this correspondence is as follows. Let Γ be a discrete subgroup of PSL 2(ℝ) acting on the complex upper half-plane H in the usual way, and f(z) a modular form of even weight k on Γ. Then there is a unique lifting from f to a Γ-invariant ΨDO with leading term f(z)∂-k/2, where ∂ is the differential operator \( \frac{d}{{dz}} \). This lifting and the fact that the product of two invariant ΨDO’s is again an invariant ΨDO imply a non-commutative multiplicative structure on the space of all modular forms whose components are scalar multiples of the so-called Rankin-Cohen brackets (canonical bilinear maps on the space of modular forms on Γ defined by certain bilinear combinations of derivatives; the definition will be recalled later). This was already discussed briefly in the earlier paper [Z], where it was given as one of several “raisons d’etre” for the Rankin-Cohen brackets.

Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm {GL}_2$ . Here, we use ‘hyperendoscopy’ techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$ , eventually getting an analog of the Eichler–Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard and Taïbi to compute dimensions of spaces of level- $1$ quaternionic representations. On the way, we prove a Jacquet–Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form $G_2^c$ .The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being ‘regular’. A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series.We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.

It is known that, given a vector-valued modular form of negative weight, its Fourier coefficients can be calculated based on the principal part of the form. In this paper we start with an arbitrary principal part and complete the Fourier expansion using the calculation. We show that the so-obtained function is a vector-valued modular integral of negative weight on the full modular group. Next, we construct the supplementary function associated to a vector-valued modular cusp form of positive weight. The constructions are inspired by the construction of Eichler integrals by Knopp. We conclude with a comparison of these forms and their integrals to vector-valued weak harmonic Maass forms.

Drinfeld modular forms were introduced by D. Goss in 1980 for congruence subgroups of ${\\rm GL}_2(\\mathbb{F}_q[T])$. They are a counterpart of classical modular forms in the function field world. In this thesis I study Drinfeld modular forms for inner forms of ${\\rm GL}_2$ that correspond to unit groups $\\Lambda^\\star$ of quaternion division algebras over $\\mathbb{F}_q(T)$ split at the place $\\infty = 1/T$. I show, following work of Teitelbaum for ${\\rm GL}_2(\\mathbb{F}_q[T])$, that these forms have a combinatorial interpretation as certain maps from the edges of the Bruhat-Tits tree $\\mathcal{T}$ associated to ${\\rm PGL}_2(K_\\infty)$. Here $K_\\infty$ denotes the completion of $K$ at $\\infty$. A major focus of this thesis is on computational aspects: I present an algorithm for computing a fundamental domain for the action of $\\Lambda^\\star$ on $\\mathcal{T}$ with an edge pairing, and describe how to obtain a basis of the space of these forms out of this fundamental domain. On this basis one can compute the Hecke action.
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We develop a theory connecting the following three areas: (a) the mean field equation (MFE) 4u + eu = ρ δ0, ρ ∈ R>0 on flat tori Eτ = C/(Z + Zτ), (b) the classical Lame equations and (c) modular forms. A major theme in part I is a classification of developing maps f attached to solutions u of the mean field equation according to the type of transformation laws (or monodromy) with respect to Λ satisfied by f . We are especially interested in the case when the parameter ρ in the mean field equation is an integer multiple of 4π. In the case when ρ = 4π(2n + 1) for a non-negative integer n, we prove that the number of solutions is n + 1 except for a finite number of conformal isomorphism classes of flat tori, and we give a family of polynomials which characterizes the developing maps for solutions of mean field equations through the configuration of their zeros and poles. Modular forms appear naturally already in the simplest situation when ρ = 4π In the case when ρ = 8πn for a positive integer n, the solvability of the MFE depends on the moduli of the flat tori Eτ and leads naturally to a hyperelliptic curve Xn = Xn(τ) arising from the Hermite-Halphen ansatz solutions of Lame’s differential equation d2w dz2 − (n(n + 1)℘(z; Λτ) + B)w = 0. We analyse the curve Xn from both the analytic and the algebraic perspective, including its local coordinate near the point at infinity, which turns out to be a smooth point of Xn. We also specify the role of the branch points of the hyperelliptic projection Xn → P1 when the parameter ρ varies in a neighborhood of ρ = 8πn. In part II, we study a “premodular form” Zn(σ; τ), a real-analytic function in two variables associated to Xn(τ), which has many symmetries and also the property that the τ-coordinates of zeros of Zn(σ; τ) correspond exactly to those flat tori where the MFE with parameter ρ = 8πn has a solution.

We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we investigate endoscopy using theta series and a theorem of Rallis. Along the way, we exhibit many examples and pose several conjectures. As a first application, we express counts of Kneser neighbours in terms of coefficients of classical or Siegel modular forms, complementing work of Chenevier–Lannes. As a second application, we prove new instances of Eisenstein congruences of Ramanujan and Kurokawa–Mizumoto type.

Plus/minus p-adic L-functions for Hilbert modular forms

In this paper we consider the hyperbolic Kac–Moody algebra \(\mathcal {F}\) associated with the generalized Cartan matrix Open image in new window. Its connection to Siegel modular forms of genus 2 was first studied by A. Feingold and I. Frenkel. The denominator function of \(\mathcal{F}\) is not an automorphic form. However, Gritsenko and Nikulin extended \(\mathcal{F}\) to a generalized Kac–Moody algebra whose denominator function is a Siegel modular form. Using the Borcherds denominator identity, the denominator function can be written as an infinite product. The exponents that appear in the product are given by Fourier coefficients of a weak Jacobi form. P. Niemann also constructed a generalized Kac–Moody algebra which contains \(\mathcal {F}\) and whose denominator function is related to a product of Dedekind η-functions. In particular, root multiplicities of the generalized Kac–Moody algebra are determined by Fourier coefficients of a modular form. As the main results of this paper, we compute asymptotic formulas for these Fourier coefficients using the method of Hardy–Ramanujan–Rademacher, and obtain an asymptotic bound for root multiplicities of the algebra \(\mathcal{F}\). Our method can be applied to other hyperbolic Kac–Moody algebras and to other modular forms as demonstrated in the later part of the paper.

The Kohnen plus space for Hilbert–Siegel modular forms

The purpose of this paper is to construct modular forms, both for SL27Z (and certain of its congruence subgroups) and for the Hilbert modular group of a real quadratic field. In w 1 we fix a real quadratic field K and even integer k > 2 and construct a series of functions co,,(Zl, z2) (m=0, 1, 2, . . .) which are modular forms of weight k for the Hilbert modular group SL2(9 ((9=ring of integers in K). The form co o is a multiple of the Hecke-Eisenstein series for K, while all of the other co,. are cusp forms. The Fourier expansion of co,. (z 1, z2) is calculated in w 2; each Fourier coefficient is expressed as an infinite sum whose typical term is the product of a finite exponential sum (analogous to a Kloosterman sum) and a Bessel function of order k 1 . The main result is that, for any points z 1 and z 2 in the upper half-plane .~, the numbers m k-1 co,,(z 1, z2) (m= 1, 2, ...) are the Fourier coefficients of a modular form (in one variable) of weight k. More precisely, let D be the discriminant of K, e.=(D/ ) the character of K, and S(D, k, ~) the space of cusp forms of weight k for Fo(D ) with character e; then for fixed z 1, zzc .~, the function

A classical Hubert modular form / e J^^jV, S), Xjr) cannot be an eigenform for the full Hecke algebra.We develop a means of lifting a classical form to a modular form F e -^(Tq^, J^), Xjr) which is an eigenform for the full Hecke algebra.Using this lift, we develop the newform theory for a space of cusp forms ^(T^jV , J), Xjr) ; we also use theta series to construct eigenforms for the full Hecke algebra.

Recursive formulae satisfied by the Fourier coefficients of meromorphic modular forms on groups of genus zero have been investigated by several authors. Bruinieret al. [‘The arithmetic of the values of modular functions and the divisors of modular forms’,Compositio Math.140(3) (2004), 552–566] found recurrences for SL(2,ℤ); Ahlgren [‘The theta-operator and the divisors of modular forms on genus zero subgroups’,Math. Res. Lett.10(5–6) (2003), 787–798] investigated the groups Γ0(p); Atkinson [‘Divisors of modular forms on Γ0(4)’,J. Number Theory112(1) (2005), 189–204] considered Γ0(4), and S. Y. Choi [‘The values of modular functions and modular forms’,Canad. Math. Bull.49(4) (2006), 526–535] found the corresponding formulae for the groups Γ+0(p). In this paper we generalize these results and find recursive formulae for the Fourier coefficients of any meromorphic modular formfon any genus-zero group Γ commensurable with SL(2,ℤ) , including noncongruence groups and expansions at irregular cusps. The form of the recurrence relations is well suited for the computation of the Fourier coefficients of the functions and forms on the groups which occur in monstrous and generalized moonshine. The required initial data has, in many cases, been computed by Norton (private communication).

We present and discuss an algorithm and its implementation that is capable of directly determining Fourier expansions of any vector-valued modular form of weight at least 2 2 associated with representations whose kernel is a congruence subgroup. It complements two available algorithms that are limited to inductions of Dirichlet characters and to Weil representations, thus covering further applications like Moonshine or Jacobi forms for congruence subgroups. We examine the calculation of invariants in specific representations via techniques from permutation groups, which greatly aids runtime performance. We explain how a generalization of cusp expansions of classical modular forms enters our implementation. After a heuristic consideration of time complexity, we relate the formulation of our algorithm to the two available ones, to highlight the compromises between level of generality and performance that each them makes.

After some generalities about the absolute Galois group of $$\mathbb Q$$ , we present the historical context in which Serre made his modularity conjecture. This was recently proved by Wintenberger and the author ([22], [23]), with an input of Kisin ([24]). The focus of these notes is on the applications of the conjecture. Some of the applications are based on the methods used in the proof. We stress some applications to base change and descent for classical modular forms. For base change we have no new results, aside from a fragment for odd A 5-extensions of $$\mathbb Q$$ . Descent for Hilbert modular forms to $$\mathbb Q$$ on the other hand is a simple corollary of Serre’s conjecture.