Special values of shifted Dirichlet series and quasi-modular forms

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, say [Formula: see text], whose [Formula: see text]th Fourier coefficient only occurs when [Formula: see text] is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. The [Formula: see text]th Fourier coefficients [Formula: see text] of a Hilbert modular form of parallel weight [Formula: see text] lying in the generalized Kohnen plus space does not vanish only if [Formula: see text] is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These series give a generalization of the one introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over [Formula: see text].

The purpose of this course is to give an introduction to the theory of p-adic integration with values in spaces of modular forms (elliptic modular forms, Siegel modular forms, . . .). We show that very general p-adic families of modular forms can be constructed as moments of certain p-adic measures on a profinite group Y = lim ←− Yi with values in a formal q-expansion ring like Zp[[q ]] where B is an additive semi-group, and q = {q |ξ ∈ B} the corresponding formally written multiplicative semi-group (for example B = Bn = {ξ = ξ ∈ Mn(Q)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms). We discuss some applications of this theory to the construction of certain new p-adic families of modular forms (families of Klingen-Eisenstein series, families of theta-series with spherical polynomials. . .). Main sources of this theory are: • Serre’s theory of p-adic forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, LNM 350 (1973) 191-268) [Se73]. • Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi93]). • Construction of p-adic Siegel-Eisenstein series by the author, see [PaSE]. As an application, we describe a solution of a problem of Coleman-Mazur in [PaTV], using the RankinSelberg method and the p-adic integration in a Banach algebra A. An introductory cours given on November 29 in POSTECH (Pohang, Korea) 0 Introduction Let p be a prime number (we often assume p≥ 5). There are two different ways of introducing p-adic modular forms: the first approach uses formal q-expansions with coefficients in a p-adic ring [Se73], and the second approach is the p-adic interpolation of Galois representations attached to classical automorphic forms. The first approach was extensively developped by Katz [Ka78] for the group G = GL2 over a totally real number field, in order to construct p-adic L-functions for CM-fields using p-adic Hilbert-Eisenstein series. In general, in this q-expansion method a typical p-adic family φ of modular (automorphic) forms is an element of the Serre ring: φ ∈ Λ[[q]] where Λ = Zp[[T ]] is the Iwasawa algebra. In the second approach one considers Λ-adic Galois representations of type ρ : Gal(Q/Q) → GLm(Λ) (“Big Galois representations”, see [Hi86], [Til-U]). These two theories are essentially equivalent if we start from holomorphic automorphic forms on the group G = GL2 over a totally real field, but in other cases there is no direct link between φ and ρ. On the other hand there exist interesting examples of p-adic L-functions Lφ,p and Lρ,p attached to φ and to ρ. In general Lφ,p and Lρ,p should belong to the quotient field L = QuotΛ or to its finite extensions. If ρ interpolates a p-adic family of motives then there are conjectural general definitions of Lρ,p (see [Co-PeRi], [Colm98], [PaAdm]). It would be very interesting to formulate a general Langlands-type conjecture relating Λ-adic automorphic forms and Λ-adic Galois representations. As an application, we describe a solution of a problem of Coleman-Mazur, using the Rankin-Selberg method and the theory of p-adic integration with values in a p-adic algebra A. This problem was stated in "The Eigencurve" (1998), R.Coleman and B.Mazur stated the following as follows: Given a prime p and Coleman’s family {fk′} of cusp eigenforms of a fixed positive slope σ = ordp(αp(k )) > 0, to construct a two variable p-adic L-function interpolating on k the Amice-Velu p-adic L-functions Lp(fk′ ). Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. Such measures come from Eisenstein distributions with values in certain Banach A-modules M = M(N ;A) of families of overconvergent forms over A.

We extend the relation between quasi-modular forms and modular forms to a wider class of functions. We then relate both forms to vector-valued modular forms with symmetric power representations, and prove a general structure theorem for these vector-valued forms.

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest-weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2. Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight $det^\ell$ sym^m with respect to an arbitrary congruence subgroup of Sp_4(Q). We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight $det^{\ell'} sym^{m'}$ with $(\ell', m')$ varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight $det^{3}sym^{m'}$ that cannot be obtained from holomorphic forms. As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction.

Pitale, Saha and Schmidt studied the representation theoretic aspects of nearly holomorphic modular forms. By their theory, we obtain a classification of [Formula: see text]-modules which occur in the space of nearly holomorphic modular forms. In this paper, we give two constructions of nearly holomorphic Siegel modular forms of degree [Formula: see text] which generate reducible indecomposable modules. One construction is given by the Rankin–Cohen bracket of Shimura’s Eisenstein series and the other by Klingen Eisenstein series.

Let N ∈ ℕ and let χ be a Dirichlet character modulo N. Let f be a modular form with respect to the group Γ0(N), multiplier χ and weight k. Let F be the L -function associated with f and normalized in such a way that F (s) satisfies a functional equation where s reflects in 1 – s. The modular forms f for which F belongs to the extended Selberg class S# are characterized. For these forms the factorization of F in primitive elements of S# is enquired. In particular, it is proved that if f is a cusp form and F ∈ S# then F is almost primitive (i.e., that if F = PG is a factorization with P, G ∈ S# and the degree of P is < 2 then P is a Dirichlet polynomial). It is also proved that the conductor of the polynomial factor P is bounded by N. If f belongs to the space generated by newforms and N ≤ 4 then F is actually primitive (i.e., P is a constant) (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

* Author writes in a clear and engaging style * Contains never before published elementary proofs * Author provides new results and detailed exposition * Self-contained, and suitable for use in a classroom setting or for self-study * A highly creative contribution to the theory of modular forms and dirichlet series The main topics of the book are the critical values of Dirichlet L-functions and Hecke L-functions of an imaginary quadratic field, and various problems on elliptic modular forms. As to the values of Dirichlet L-functions, all previous papers and books reiterate a single old result with a single old method. After a review of elementary Fourier analysis, the author presents completely new results with new methods, though old results will also be proved. No advanced knowledge of number theory is required up to this point. As applications, new formulas for the second factor of the class number of a cyclotomic field will be given. The second half of the book assumes familiarity with basic knowledge of modular forms. However, all definitions and facts are clearly stated, and precise references are given. The notion of nearly holomorphic modular forms is introduced and applied to the determination of the critical values of Hecke L-functions of an imaginary quadratic field. Other notable features of the book are: (1) some new results on classical Eisenstein series; (2) the discussion of isomorphism classes of elliptic curves with complex multiplication in connection with their zeta function and periods; (3) a new class of holomorphic differential operators that send modular forms to those of a different weight. The book will be of interest to graduate students and researchers who are interested in special values of L-functions, class number formulae, arithmetic properties of modular forms (especially their values), and the arithmetic properties of Dirichlet series. It treats in detail, from an elementary viewpoint, the simplest cases of a fundamental area of ongoing research, the only prerequisite being a basic course in algebraic number theory.

This book provides an introduction to the theory of elliptic modular functions and forms, a subject of increasing interest because of its connexions with the theory of elliptic curves. Modular forms are generalisations of functions like theta functions. They can be expressed as Fourier series, and the Fourier coefficients frequently possess multiplicative properties which lead to a correspondence between modular forms and Dirichlet series having Euler products. The Fourier coefficients also arise in certain representational problems in the theory of numbers, for example in the study of the number of ways in which a positive integer may be expressed as a sum of a given number of squares. The treatment of the theory presented here is fuller than is customary in a textbook on automorphic or modular forms, since it is not confined solely to modular forms of integral weight (dimension). It will be of interest to professional mathematicians as well as senior undergraduate and graduate students in pure mathematics.

Modular form identities lying in the framework of Shimura’s theory of nearly holomorphic modular forms are obtained by Lie theoretic means as consequences of identities relating the Maass–Shimura operator and the Rankin–Cohen brackets, which in turn follow from change-of-basis formulae in the theory of Verma modules. The Lie theoretic origin of known van der Pol and Lahiri-type arithmetic identities is thus unveiled, and similar new ones are derived in a systematic way. These identities relate divisor functions, Ramanujan’s τ-function and functions defined by the Fourier coefficients of other cusp forms and involve hybrid coefficients, drawn from Lie theory and number theory, given explicitly by formulae combining the arithmetic Clebsch–Gordan coefficients and the Bernoulli numbers. A few side results, interesting in their own right, such as Leibniz-type rules satisfied by the Rankin–Cohen brackets, are also obtained.

Modular cocycles and cup product

Quasimodular forms generalize modular forms and have been studied actively in recent years in connection with various topics in number theory and geometry. One of their interesting properties is that they correspond to finite sequences of modular forms of certain types. We extend such a correspondence to the case of Hilbert quasimodular forms. As an application we construct Poincare series for Hilbert quasimodular forms.

Introduction. Several authors have developed the theory of lifting from the space of modular forms of one variable to that of modular forms on the orthogonal groups attached to quadratic forms over Q (cf. [1, 4–6, 8]). Shimura [9], [10] dealt with the problem of construction of arithmetic modular forms on orthogonal groups over totally real algebraic number fields. However, he did not take up the explicit calculation of the Fourier coefficients of lifted modular forms. On the other hand, in [3], [4] we have established a correspondence Ψ k between the space S(2k−1)/2(M,χ) of modular cusp forms of half integral weight (2k − 1)/2 of level M to the space M (2) k (M,χ) of Maass forms of Siegel modular cusp forms of degree two of weight k of level M in such a way that it commutes with the actions of Hecke operators. We evaluated explicitly the Fourier coefficients of Ψ k (f) with a form f in S(2k−1)/2(M,χ), and made clear a coincidence with Shimura’s zeta functions attached to f and Andrianov’s zeta functions attached to Ψ k (f). We note that these results are closely related to Saito–Kurokawa’s conjecture concerning Siegel modular forms of degree two. Using the technique in the theory of group representation of Jacquet and Langlands, PiatetskiShapiro [7] discussed Saito–Kurokawa’s conjecture in the case of Siegel modular forms on GpSp(2, AF ) where AF is the adele ring of an arbitrary number field F . Unfortunately, it seems that his approach is difficult to use for an explicit calculation of the Fourier coefficients of the lifted forms. The first purpose of the present note is to show the existence of a correspondence ΨN ′ between Hilbert modular forms f of half integral weight with respect to the principal congruence group and Hilbert–Siegel modular forms ΨN ′(f) of degree two attached to totally real number fields. The second one

4.1 Congruences between modular forms and p-adic integration 4.1.1 Integration in nearly holomorphic Siegel modular forms 4.1.2 Arithmetical nearly holomorphic Siegel modular forms 4.1.3 The group 4.1.4 Canonical projection 4.1.5 The standard zeta function of a Siegel cusp eigenform 4.2 Algebraic differential operators and Siegel-Eisenstein distributions 4.2.1 Operatots of Maass and Shimura 4.2.2 Formulas for Fourier expansions 4.2.3 Siegel-Eisenstein series. 4.2.4 Normaized Siegel-Eisenstein series 4.2.5 Distributions with values in nearly holomorphic Siegel modular forms. 4.2.6 Convolutions of distributions with values in nearly holomorphic Siegel modular forms. 4.3 A general result on admissible measures 4.3.1 Profinite group 4.3.2 Measures and sequences of distributions 4.4 The standard L-function 4.4.1 The standard L function 4.4.2 Theta series 4.4.3 The Rankin zeta function 4.4.4 The standard zeta function D(s,f,x) as the Rankin convolution 4.4.5 Algebraic properties of the special values of normalized distributions. 4.4.6 Integral representation for the functions D±(s,f,x) 4.4.7 Action of the group Autℂ on scalar products of modular forms. 4.4.8 Algebraicity properties and Fourier coefficients 4.5 Algebraic linear forms on modular forms 4.5.1 Convolutions of theta distributions and Eisenstein distributions with values in nearly holomorphic Siegel modular forms. 4.5.2 Evaluation of algebraic linear forms 4.6 Congruences and proof of the Main theorem 4.6.1 Regularized distributions in Siegel modular forms. 4.6.2 Sufficient conditions for admissibility of measures with values in nearly holomorphic Siegel modular forms. 4.6.3 Fourier expansions of distributions with values in nearly holomorphic Siegel modular forms. 4.6.4 Fourier expansions of regularized distributions. 4.6.5 Main congruences for the Fourier expansions of regularized distributions. 4.6.6 Kummer congruences and Mazur’s measure. 4.6.7 Reduction of the Main congruence to congruences for partial sums. 4.6.8 Proof of the Main congruence. 4.6.9 Proof of Theorem 4.23

Doi and Naganuma (see [6]) constructed a lifting map from elliptic modular forms to Hilbert modular forms in the case of a real quadratic field with narrow class number one. A Converse Theorem for Hilbert modular forms was one of their basic tools. This gives rise to the question of constructing a lifting map in the case of Jacobi forms. Here we do the first step in this direction and prove a Converse Theorem for Hilbert-Jacobi forms. Studying the connection between functions that satisfy certain transformation laws and the functional equation of their associated L-functions has value on its own and a long history. In a celebrated paper (see [9]), Hecke showed that the automorphy of a cusp form with respect to SL2(Z) is equivalent to the functional equation of its associated L-functions. That only one functional equation is needed is in a way atypical and highly depends on the fact that SL2(Z) is generated by the matrices ( 1 1 0 1 ) and ( 0 −1 1 0 ). This situation already changes if one considers cusp forms with respect to a subgroup of SL2(Z) which have a character. In this case the functional equation of twists is required (see [18]). Hecke’s work has inspired an astonishing number of people and a lot of generalizations of his “Converse Theorem” have been made, e.g. generalizations to Hilbert modular forms as mentioned above (see [6]), Siegel modular forms (see [1], [10]) or Jacobi forms (see [14],[15]). Maass showed an analogue of Hecke’s result for nonholomorphic modular forms (see [13]). He proved that these correspond to certain L-functions in quadratic fields. An outstanding generalization of a Converse Theorem for GL(n) was done by Jacquet and Langlands for n = 2 (see [11]), Jacquet, Piatetski-Shapiro, and Shalika for n = 3 (see [12]) and Cogdell and Piatetski-Shapiro for general n (see [5]). In this paper, we prove a Converse Theorem for Hilbert-Jacobi cusp forms over a totally real number field K of degree g := [K : Q] with discriminant DK and narrow class number 1. The case g = 1, i.e., Jacobi forms over Q as considered by Eichler and Zagier (see [7]), is treated in two interesting papers by Martin (see [14] and [15]). To describe our result, we consider functions φ(τ, z) from H × C into C that have a Fourier expansion with certain conditions on the Fourier coefficients (see (3.4),(3.5), and (3.6)). We show that φ is a Hilbert-Jacobi cusp form (for the definition see Section 2) if

There is a sophisticated theory of nearly holomorphic Siegel modular forms by Shimura. Using previous results by Nagaoka and myself on Rankin-Cohen operators and theta-operators we will present a proof that quasimodular forms (defined as constant terms or as holomorphic part of a nearly holomorphic Siegel modular form) are always p-adic. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.