Serre’s conjecture and its consequences

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.

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.

In 1973, Drinfeld introduced the notion of elliptic modules, which is now known as Drinfeld modules. After that the analogies between number fields and function fields have many interesting new aspects. Drinfeld modular function theory is one of these. Drinfeld modular functions were studied by several mathematicians and known to have many properties analogous to those of classical elliptic functions, such as the generators of modular function fields, Galois groups between them. ([5],[8],[9]). In the first part of this note, we establish some more properties of Drinfeld modular functions in analogy with those obtained by Shimura. In [12], Shimura proved his exact-sequence and his reciprocity law. Lang proved the exact-sequence another way in [11] using the isogeny theory. Shimura's proof of the reciprocity law is not easy. For example he used the parametrizations of the models of a modular function field over Q. In [11], Lang avoided Shimura's method using the decomposition gorups which is well-known in algebraic number theory. In this article, we will follow Shimura's method to prove the exact-sequence, and Lang's method to prove the reciprocity law in the function field case. In the second part, we go on to study two variable Drinfeld modular functions in analogy with two variable elliptic functions studied by Berndt. ([!]). In [2], he also gerneralized Shimura's exact-sequence and his reciprocity law corresponding to this extended modular function fields. We discuss the analogies of these in the Drinfeld setting. 1. Definitions and basic facts Let Λ=/yT], k=Fq(T k^ be the completion of k at 00= (£), and C the completion of the algebraic closure of k^. Then C has an absolute value extending that of k^. By an ^-lattice in C, we mean a projective Λ-submodule A of C which is discrete in the topology of C. A meromorphic function /on C is said to be even if/(μz)=/(z) for every μeF*. A meromorphic function /on C is

Let p\geq5 be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between eigenforms of weights 2 and p+1 , in terms of group cohomology. We also illustrate how our method works for inert primes p\geq5 in the contexts of quadratic imaginary fields (where there is no Hasse invariant available) and Hilbert modular forms over totally real fields, cyclic and of even degree over the rationals.

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

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.

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.

We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular form, and we give both individual and statistical results. The second problem, which has been considered by a number of authors, is to determine the size, in terms of the conductor and weight, of the first sign-change of Hecke eigenvalues. Here we improve the recent estimate of Iwaniec, Kohnen and Sengupta.

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 $\mathbb P^1_{a,b,c}$ 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.

In this paper we deal with monogenic and k-monogenic Clifford-valued automorphic forms in two and several vector variables. We focus on the construction of several families of Eisenstein and Poincare series within the setting of this function theory. The functions that we construct provide non-trivial examples for generalizations of the classical Hilbert modular forms within the framework of real and complex Clifford analysis.

The Kohnen plus space for Hilbert–Siegel modular forms