Modular Forms and Functions

Recently G. Shimura [1] constructed modular forms of integral weight from the forms of half integral weight. His construction is rather indirect. Indeed, he proved that the Dirichlet series, obtained from a form of half integral weight, multiplied by a certain L-function, corresponds to a modular form of an integral weight by means of the characterization of modular forms due to Weil.

We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit ‘weak harmonic lifts’ for every eigenform of integral weight k and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to n^{1-k}(a_n^{prime } + rho a_n) for nne 0, where rho is the normalised permanent of the period matrix of the corresponding motive, and a_n, a_n^{prime } are the Fourier coefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform, respectively. More generally, this framework provides a conceptual explanation for the algebraicity of the coefficients of mock modular forms in the CM case.

A modular grid is a pair of sequences ( f m ) m $ (f_m ) _m$ and ( g n ) n $ (g_n ) _n$ of weakly holomorphic modular forms such that for almost all m and n, the coefficient of q n $q^n$ in f m $f_m$ is the negative of the coefficient of q m $q^m$ in g n $g_n$ . Zagier proved this coefficient duality in weights 1/2 and 3/2 in the Kohnen plus space, and such grids have appeared for Poincaré series, for modular forms of integral weight, and in many other situations. We give a general proof of coefficient duality for canonical row-reduced bases of spaces of weakly holomorphic modular forms of integral or half-integral weight for every group Γ ⊆ SL 2 ( R ) $\Gamma \subseteq {\text{\rm SL}}_2(\mathbb {R})$ commensurable with SL 2 ( Z ) ${\text{\rm SL}}_2(\mathbb {Z})$ . We construct bivariate generating functions that encode these modular forms, and study linear operations on the resulting modular grids.

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

We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for $\Gamma_1(N)$: an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.

Modular forms play an essential role in Number Theory and its importance continues to grow in many areas of Mathematics. Also, we have seen most diverse branches of Mathematics coming together in the theory of Elliptic Curves and Modular Forms to solve one of the outstanding problems in Number Theory, viz., ‘The Fermat’s Last Theorem’. In Part I of this volume, various aspects of the theory of Elliptic Curves are given. In Part II, we discuss some aspects of the theory of Modular Forms. The main topic in this part is the Eichler-Shimura correspondence.

This is the second volume of a 2-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology du ring the last 25 years. The second volume presupposes a background in number theory com¬ parable to that provided in the first volume, together with a knowledge of the basic concepts of complex analysis. Most of the present volume is devoted to elliptic functions and modular functions with some of their number-theoretic applications. Among the major topics treated are Rademacher's convergent series for the partition function, Lehner's congruences for the Fourier coefficients of the modular functionj( r), and Hecke's theory of entire forms with multiplicative Fourier coefficients. The last chapter gives an account of Bohr's theory of equivalence of general Dirichlet series. Both volumes of this work emphasize classical aspects of a subject wh ich in recent years has undergone a great deal of modern development. It is hoped that these volumes will help the nonspecialist become acquainted with an important and fascinating part of mathematics and, at the same time, will provide some of the background that belongs to the repertory of every specialist in the field. This volume, like the first, is dedicated to the students who have taken this course and have gone on to make notable contributions to number theory and other parts of mathematics. T. M. A. January, 1976.

The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. The second edition of this text includes an updated bibliography indicating the latest, dramatic changes in the direction of proving the Birch and Swinnerton conjecture. It also discusses the current state of knowledge of elliptic curves.

It was Ramanujan who in a fundamental paper of 1916 introduced his τ-function as the Fourier coefficient of a modular form and then attached a Dirichlet series to it. He established an analytic continuation of the series and a functional equation for it. He then made his famous conjectures about the multiplicativity of these coefficients and their size. The multiplicativity conjecture would allow him to write his Dirichlet series as an Euler product thereby establishing an analogy with classical zeta and L-functions. Subsequently Mordell proved that τ(n) is a multiplicative function but it was left to Hecke to develop a more elaborate theory and establish the existence of an infinite family of such examples. Ramanujan’s conjecture on estimating the size of τ(n) however defied immediate attack. The fundamental method of Rankin and Selberg did allow one to get good estimates for them but they were not optimal. The final resolution of Ramanujan’s conjecture came from algebraic geometry when it was shown to be a consequence of Deligne’s proof of the celebrated Weil conjectures. In this chapter, we will give a brief introduction to the fundamental concepts and study the oscillations of the Fourier coefficients from the standpoint of the non-vanishing of various L-functions.KeywordsEntire FunctionAnalytic ContinuationModular FormFourier CoefficientElliptic CurfThese 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.

* 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.

Hecke theory over arbitrary number fields

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We consider logarithmic vector- and matrix-valued modular forms of integral weight $k$ associated with a $p$-dimensional representation $\rho: SL_2(\mathbb{Z}) \to GL_p(\mathbb{C})$ of the modular group, subject only to the condition that $\rho(T)$ has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincar\'e series associated to $\rho$ for all large enough weights. The component functions are logarithmic $q$-series, i.e., finite sums of products of $q$-series and powers of $\log q$. We derive several consequences, in particular we show that the space $\mathcal{H}(\rho)=\oplus_k \mathcal{H}(k, \rho)$ of all holomorphic logarithmic vector-valued modular forms associated to $\rho$ is a free module of rank $p$ over the ring of classical holomorphic modular forms on $SL_2(\mathbb{Z})$.

TSUYUMINE 0. Let $k,$ $N\in N,$ $4|N$ .Let $M_{k+1/2}(N,\chi_{0})$ denote the space of modular forms for $\Gamma_{0}(N)$ of weight $k+1/2$ with a character $\chi_{0}(mod N)$ .Lifting maps of cusp forms in $M_{k+1/2}(N,\chi_{0})$ to modular forms of integral weight was first studied by Shimura [7] and later by Niwa [4].The domain of the map is extended to $M_{k+1/2}(N,\chi_{0})$ by van Asch [1] in case that $\mathcal{X}0$ is real and $N=4p$ for $p$ prime, and by Pei [5] in case that $\mathcal{X}0$ is real and $N/4$ is square-free.In the present paper we consider the lifting map without any condition on $N$ and $\mathcal{X}0$ ' and extend the domain of the map to $M_{k+1/2}(N,\chi_{0})$ for $k\geq 2$ .To show the assertion, we take some specific modular forms in $M_{k+1/2}(N,\chi_{0})$ which together with cusp forms, span $M_{k+1/2}(N,\chi_{0})$ .Further we construct their liftings explicitly.It proves our main result.It may be expected to have further application to study of special values of L-series of Hecke eigen cusp forms, as in Zagier [9], Kohnen-Zagier [3] where the lifting of some particular modular forms plays an important role.1. We denote by $N,$ $Z,$ $C$ , the set of natural numbers, the ring of integers and the complex number field respectively.For a prime $p\in N,$ $v_{p}$ denotes the p-adic valuation.For $N\in N,$ $(Z/N)^{*}$ denotes the group of Dirichlet characters $(mod N)$ .When $N=1$ , the group is consisting of a constant 1.The identity element of $(Z/N)^{*}$ is denoted by $1_{N}$ .A group consisting of invertible elements in $Z/N$ is denoted by $(Z/N)^{\times}$ If $\chi\in(Z/N)^{*}$ and $e\in N$ , the $\chi^{(e)}$ denotes a character $(mod eN)$ obtained by $\chi^{(e)}(d)=\chi(d)((d, e)=1),$ $0((d, e)\neq 1)$ .In case that all prime factors of $e$ appear as factors of $N$ , then $\chi^{(e)}$ is equal to $\chi$ .For $a\in Z$ and for an odd $b\in N,$ $(a/b)$ denotes the Jacobi-Legendre symbol where it is $0$ if $(a, b)\neq 1$ .If $D$ is a discriminant of a quadratic field, then $\chi_{D}$ denote the Kronecker-Jacobi-Legendre symbol.We put $\chi_{D}=1$ for $D=1$ .Let $\mathfrak{H}$ denote the upper-half plane $\{z\in C|{\rm Im} z>0\}$ .The group $SL_{2}(Z)$ acts on $\mathfrak{H}$ by the usual modular transformation sending $z\in \mathfrak{H}$ to $Mz=(az+b)/$

In previous papers, it has been shown that the coefficients of terms in the large-N expansion of a certain integrated four-point correlator of superconformal primary operators in $$\mathcal {N}=4$$ N = 4 supersymmetric Yang–Mills theory are rational sums of real-analytic Eisenstein series and “generalised Eisenstein series”. The latter are novel modular functions first encountered in the context of graviton amplitudes in type IIB superstring theory. Similar modular functions, known as two-loop modular graph functions, are also encountered in the low-energy expansion of the integrand of genus-one closed superstring amplitudes. In this paper, we further develop the mathematical structure of such generalised Eisenstein series emphasising, in particular, the occurrence of L-values of holomorphic cusp forms in their Fourier mode decomposition. We show that both the coefficients in the large-N expansion of this integrated correlator and two-loop modular graph functions admit a unifying description in terms of four-dimensional lattice sums generated by theta lifts of certain local Maass functions, which generalise the structure of real-analytic Eisenstein series. Through the theta lift representation, we demonstrate that elements belonging to these two families of non-holomorphic modular functions can be expressed as rational linear combinations of generalised Eisenstein series for which all the L-values of holomorphic cusp forms precisely cancel.