Elementary Dirichlet series and modular forms

On the number of isomorphism classes of CM elliptic curves defined over a number field

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.

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.

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.

Multiplication by a given modular form can be viewed as a linear map on the space of modular forms. By computing its adjoint operator, one can obtain certain cusp forms whose Fourier coefficients are special values of Dirichlet series of Rankin-Selberg type associated to modular forms. We generalize this idea to the space of almost holomorphic modular forms with some cuspidal conditions. We prove that the generating function of special values of the Dirichlet series at certain points is a quasi-modular form.

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

Modular cocycles and cup product

Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let $$ f(z)\, = \,\sum\nolimits_{n = 0}^\infty {a(n)q^n } $$ be a holomorphic half integer weight modular form with integer coefficients. If l is prime, then we shall be interested in congruences of the form $$ a(lN)\, \equiv \,0\,\bmod \,l $$ where N is any quadratic residue (resp. non-residue) modulo l. For every prime l > 3 we exhibit a natural holomorphic weight $$ \frac{l} {2}\, + \,1 $$ modular form whose coefficients satisfy the congruence a(lN) ≡ 0 mod l for every N satisfying $$ \left( {\frac{{ - N}} {l}} \right)\, = \,1 $$ . This is proved by using the fact that the Fourier coefficients of these forms are essentially the special values of real Dirichlet L—series evaluated at $$ s\, = \,\left( {\frac{{1 - l}} {2}} \right) $$ which are expressed as generalized Bernoulli numbers whose numerators we show are multiples of l. From the works of Carlitz and Leopoldt, one can deduce that the Fourier coefficients of these forms are almost always a multiple of the denominator of suitable Bernoulli numbers. Using these examples as a template, we establish sufficient conditions for which the Fourier coefficients of a half integer weight modular form are almost always divisible by a given positive integer M.

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)

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

This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. The second edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten.

Different types of number theories such as elementary number theory, algebraic number theory and computational number theory; algebra; cryptology; security and also other scientific fields like artificial intelligence use applications of quadratic fields. Quadratic fields can be separated into two parts such as imaginary quadratic fields and real quadratic fields. To work or determine the structure of real quadratic fields is more difficult than the imaginary one. The Dirichlet class number formula is defined as a special case of a more general class number formula satisfying any types of number field. It includes regulator, ℒ -function, Dedekind zeta function and discriminant for the field. The Dirichlet’s class number h ( d ) formula in real quadratic fields claims that we have h ( d ) . log ε d = Δ . ℒ ( 1 , χ d ) h\left(d \right).log {\varepsilon _d} = \sqrt {\Delta} {\scr L} \left({1,\;{\chi _d}}\right) for positive d &gt; 0 and the fundamental unit ɛ d of ℚ ( d ) {\rm{\mathbb Q}}\left({\sqrt d} \right) . It is seen that discriminant, ℒ -function and fundamental unit ɛ d are significant and necessary tools for determining the structure of real quadratic fields. The focus of this paper is to determine structure of some special real quadratic fields for d &gt; 0 and d ≡ 2,3 ( mod 4). In this paper, we provide a handy technique so as to calculate particular continued fraction expansion of integral basis element w d , fundamental unit ɛ d , and so on for such real quadratic number fields. In this paper, we get fascinating results in the development of real quadratic fields.

Background and Aims: With the continuous advancement of new education curriculum reform, especially in China, the classroom roles of teachers and students are evolving. Logical reasoning is an important part of the core literacy of mathematics. The purposes of this research were: (1) Compare Mathematics Logical Reasoning Ability before and after the implementation of a Small Private Online Course (SPOC) on Elementary Number Theory course based on Constructivism theory. (2) Assess the student’s satisfaction with the Small Private Online Course (SPOC) on the Elementary Number Theory course. Methodology: The sample of this study was 40 first-year students in Xi'an University, Xi’an City, Shaanxi Province. They were selected by cluster random sampling. The research instruments were: 1) Seven lesson plans of a Small Private Online Course (SPOC) on Elementary Number Theory course based on Constructivism theory. 2) Mathematics Logical Reasoning Ability test paper. 3) Questionnaire for students’ satisfaction. The researcher used the Mathematics Logical Reasoning Ability test paper to conduct pre-tests and post-tests on the sample. Then the pre-test data and post-test data are analyzed. The content of data analysis includes: score mean, standard deviation, the correlation between pre-test data and post test data, t-test of paired dependent sample. Results: From the study, it was found that: (1) The mathematics Logical Reasoning Ability test paper data showed the mean score of post-test data were higher than pre-test scores at the.05 level of statistical significance (The mean of pretest was 19.48, the mean of post-test was 22.55, p &lt;.05). (2) The mean scores of students’ satisfaction with the Small Private Online Course (SPOC) on the Elementary Number Theory course based on Constructivism theory were very high level. Conclusion: The students who learn through the Elementary Number Theory course will have a post-test score higher than the pretest score. The students’ satisfaction with a Small Private Online Course on Elementary Number Theory course based on Constructivism theory will be at a very high level.

The volume is devoted to the interaction of modern scientific computation and classical number theory. The contributions, ranging from effective finiteness results to efficient algorithms in elementary, analytical and algebraic number theory, provide a broad view of the methods and results encountered in the new and rapidly developing area of computational number theory. Topics covered include finite fields, quadratic forms, number fields, modular forms, elliptic curves and diophantine equations. In addition, two new number theoretical software packages, KANT and SIMATH, are described in detail with emphasis on algorithms in algebraic number theory.

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.