Automorphic Pseudodifferential Operators

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a “determinant” map from this moduli surface to (Z/N Z)*; its fibers are the components of the surface. We define spaces of modular forms on these components and Hecke correspondences between them, and study how those spaces of modular forms behave as modules for the Hecke algebra. We discover that the component with determinant −1 is somehow the “dominant” one; we characterize the difference between its spaces of modular forms and the spaces of modular forms on the other components using forms with complex multiplication. In addition, we prove Atkin–Lehner-style results about these spaces of modular forms. Finally, we show some simplifications that arise when N is prime, including a complete determination of such CM-forms, and give numerical examples.

Let p be a prime not dividing the integer n. By an Ihara result, we mean existence of a cokernel torsion-free injection from a full lattice in the space of p-old modular forms into a full lattice in the space of all modular forms of level pn. In this paper, we will prove an Ihara result in the number field case, for Siegel modular forms. The case of elliptic modular forms is discussed in Ihara (Discrete subgroups of Lie groups and applications to moduli, Oxford University Press, Bombay, 1975). We will use a geometric formulation for the notion of p-old Siegel modular forms (Rastegar in BIMS 43(7):1–23, 2017). Then, we apply an argument by Pappas, and prove the Ihara result using density of Hecke orbits (Chai in Invent Math 121(3):439–479, 1995). This result is meant to pave the way for modularity results in higher genera.

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

Identities Arising From Hecke Transformations of Modular Forms Over Q( [formula omitted]) and Q( [formula omitted

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.

The conjecture is made that the rational structures on spaces of modular forms coming from the rationality of Fourier coefficients and the rationality of periods are not compatible. A consequence would be that ζ(2k-1)/π 2k-1 (ζ(s)=Riemann zeta function;k∈ℕ,k≥2) is irrational or even transcendental.

Let $T^n$ denote the n-dimensional torus. The class of the bounded operators on $L^2(T^n)$ with analytic orbit under the action of $T^n$ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on $T^n$ whose discrete symbol $(a_j)_{j\in Z^n}$ is uniformly analytic, in the sense that there exists $C>1$ such that the derivatives of $a_j$ satisfy $|\partial^\alpha a_j(x)|\leq C^{1+|\alpha|}\alpha!$ for all $x\in T^n$, all $j\in Z^n$ and all $\alpha\in N^n$. This implies that this class of pseudodifferential operators is a spectrally invariant *-subalgebra of the algebra of all bounded operators on $L^2(T^n)$.

The calculus of classical pseudo-differential operators has been used in a fundamental way in the study of boudary value problems associated to (systems) of elliptic differential and pseudo-differential operators. This calculus was used by Hörmander in [7] to construct parametrices for the elliptic operators, using which the boundary value problem is reduced to a system of pseudo-differential operators on the boundary. In order to treat the boundary value problems for parabolic operators in bounded cylindrical domains Piriou introduced in [15] and [16] a class of operators which called pseudo-differential operators of Volterra type. The basic idea consists in developping a calculus of an appropriate class of anisotropic pseudo-differential operators as in an earlier note of Hunt and Piriou [8].KeywordsVector BundleParabolic SystemPrincipal SymbolCylindrical DomainVolterra TypeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

Eichler–Selberg type identities for mixed mock modular forms

We describe connections between pseudodifferential operators and modular forms in terms of vector bundles over a Riemann surface whose fibers are the spaces of certain pseudodifferential operators.

Structural theorems for spaces of modular forms with respect to congruence subgroups are proved. The Dedekind η-function plays an important role in our study.

Construction of supermodular forms using differential operators from a given supermodular form

We give a condition under which a pseudodifferential operator with symbol in \(S^{m}\left (\mathbb {R}^{n}\times \mathbb {R}^{n}\right )\) cannot be a Fredholm operator when acting on suitable Besov and Triebel-Lizorkin spaces. As a corollary, we show that, if a classical pseudodifferential operator on \(\mathbb {R}^{n}\) is Fredholm in one of these spaces, then this operator must be elliptic.

This chapter is meant as another motivation (cf. introduction) for the construction of the ascending pseudodifferential analysis, rather than as an introduction to a new point of view in modular form theory: we hope to come back to possible developments in this direction at some later occasion.We here wish to show that the parallel treatments of anaplectic analysis and associated alternative pseudodifferential analysis on one hand, of usual analysis and pseudodifferential analysis on the other hand, extend, up to some point, as parallel sources of holomorphic modular forms on one hand, nonholomorphic modular forms on the other hand. In particular, let us direct the interested reader to Remark 5.1.1 at the end of this section, which puts on an absolutely equal footing the Rankin–Cohen brackets of holomorphic modular forms, well known to modular form theorists, and their analogues in nonholomorphic modular form theory, which do not seem to have kept their attention. As explained in the introduction, making this possible was one of the initial aims of the present work.