# Singular modular forms and Theta relations

by E. Freitag

Publisher: Springer-Verlag in Berlin, New York

Written in English

## Subjects:

• Forms, Modular.,
• Series, Theta.

## Edition Notes

Includes bibliographical references (p. [170]-172).

Classifications The Physical Object Statement Eberhard Freitag. Series Lecture notes in mathematics ;, 1487, Lecture notes in mathematics (Springer-Verlag) ;, 1487. LC Classifications QA3 .L28 no. 1487, QA243 .L28 no. 1487 Pagination 172 p. ; Number of Pages 172 Open Library OL1552284M ISBN 10 3540547045, 0387547045 LC Control Number 91032180

However, we state every definition and every essential fact concerning classical modular forms of one variable. One of the principal new features of this book is the theory of modular forms of half-integral weight, another being the discussion of theta functions and Eisenstein series of holomorphic and nonholomorphic s: 1. A Classical Introduction to Modern Number Theory: Edition 2 - Ebook written by Kenneth Ireland, Michael Rosen. Read this book using Google Play Books app on your PC, android, iOS devices. Download for offline reading, highlight, bookmark or take notes while you read A Classical Introduction to Modern Number Theory: Edition 2. They are modular in one variable and abelian (or double periodic) in another variable. The theory of Jacobi modular forms became an independent research subject after the famous book of Martin Eichler and Don Zagier “Jacobi modular forms” (Progress in Mathematics, vol. 55, ) which was cited more than a thousand times in research papers. The purpose of this book is to present in the form of a self-contained text-book the contemporary state of the theory of Hecke operators on the spaces of hoi om orphic modular forms of integral weight (the Siegel modular forms) for congruence subgroups of integral symplectic groups. The Siegel Modular Group and the Theta-Group.-

Chapter VI gives the proof of the "theorem on arithmetic progressions" due to Dirichlet; this theorem is used at a critical point in the first part (Chapter Ill, no. ). Chapter VII deals with modular forms, and in particular, with theta functions. Some of the quadratic forms of Chapter V reappear here. It is well-known that many modular forms can be expressed as infinite products. For instance, the most famous one is probably the expansion $$\Delta(q) = q \prod_{n=1}^\infty (1-q^n)^{24}$$ for the discriminant cusp form of weight $12$ and level $1$. Another example is the cusp form . Harmonic Maass Forms and Mock Modular Forms: Theory and Applications | Kathrin Bringmann, Amanda Folsom, Ken Ono, Larry Rolen | download | B–OK. Download books for free. Find books. In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last letter to G. H. Hardy and in his lost notebook. Sander Zwegers (, ) discovered that adding certain non-holomorphic.

Mock Modular Result Theorem (Folsom, Garthwaite, Kang, S-, Treneer) The functions V mn are mock modular forms of weight 1=2 with respect to the congruence subgroups A mn. Moreover, the shadow of V mn is given by a constant multiple of the odd eta-theta function E m 2˝ c2 m. In particular, the functions V mn may be completed to form harmonic.   The theory of modular forms is a fundamental tool used in many areas of mathematics and physics. It is also a very concrete and fun'' subject in itself and abounds with an amazing number of surprising comprehensive textbook, which includes numerous exercises, aims to give a complete picture of the classical aspects of the subject, with an emphasis on explicit formulas. In mathematics, theta functions are special functions of several complex are important in many areas, including the theories of abelian varieties and moduli spaces, and of quadratic have also been applied to soliton theory. When generalized to a Grassmann algebra, they also appear in quantum field theory.. The most common form of theta function is that occurring in.

## Singular modular forms and Theta relations by E. Freitag Download PDF EPUB FB2

Singular Modular Forms and Theta Relations | Eberhard Freitag (auth.) | download | B–OK. Download books for free. Find books. This research monograph reports on recent work on the theory of singular Siegel modular forms of arbitrary level. Singular modular forms are represented as linear combinations of theta series.

The reader is assumed toknow only the basic theory of Siegel modular forms. This research Singular modular forms and Theta relations book reports on recent work on the theory of singular Siegel modular forms of arbitrary level.

Singular modular forms are represented as linear combinations of theta series. The reader is assumed toknow only the basic theory of Siegel modular : Springer-Verlag Berlin Heidelberg. Siegel modular forms.- Theta series with polynomial coefficients.- Singular weights.- Singular modular forms and theta series.- The fundamental lemma.- The results.

Series Title: Lecture notes in mathematics, Responsibility: Eberhard Freitag. More information: Inhaltsverzeichnis. Get this from a library. Singular modular forms and Theta relations. [E Freitag] -- This research monograph reports on recent work on the theory of singular Siegel modular forms of arbitrary level.

Singular modular forms are represented as linear combinations of theta series. The. Cite this chapter as: Freitag E. () Siegel modular forms. In: Singular Modular Forms and Theta Relations.

Lecture Notes in Mathematics, vol Cite this chapter as: Freitag E. () Singular modular forms and theta series. In: Singular Modular Forms and Theta Relations. Lecture Notes in Mathematics, vol We discuss some examples of mod $\frak{p}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms.

Finally, we apply our results to properties mod $\frak{p}$ of Klingen-Eisenstein series. Cite this chapter as: Freitag E. () Singular weights. In: Singular Modular Forms and Theta Relations. Lecture Notes in Mathematics, vol holds for Siegel modular forms. Moreover, we deﬁne the notion of mod p singular modular forms and discuss some relations between their weights and the corresponding prime p.

We discuss some examples of mod p singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Singular Modular Forms and Theta relations, Lecture Notes in Mathematics,Springer-Verlag, Berlin/Heidelberg ().

Abstract. In this chapter we introduce vector valued modular forms and show how to derive their theta transforms. As we concentrate only on the properties needed for our classification purpose given in sections andfor an in depth study of these objects, the reader can consult the following books: [Borc11], [EicZ], [Frei], [Mi], [Shi], [Serr2].

In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory.

Throughout the s [24–26], the second author in joint work with Millson employed the Weil representation and the theta correspondence to systematically construct holomorphic (Siegel) modular forms associated to indefinite quadratic forms.

More precisely, they obtain a lift from the (co)homology of the underlying locally symmetric space to. In the case of Siegel modular forms, the authors found interesting examples of such modular forms. For example, Igusa’s odd weight cusp form is an element of mod 23 kernel of the theta operator.

Hilbert Modular Forms. Springer-Verlag, Grundlehren der mathematischen Wissenschaften,ISBN pbk reprint; Singular Modular Forms and Theta Relations. In: Lecture Notes in Mathematics.

vol. Springer-Verlag,ISBN ; pbk reprint. Siegel modular forms.- Theta series with polynomial coefficients.- Singular weights.- Singular modular forms and theta series.- The fundamental lemma.- The results.

Series Title: Lecture notes in mathematics (En ligne), Responsibility: Eberhard Freitag. This is a master course given in Moscow at the Laboratory of Algebraic Geometry of the National Research University Higher School of Economics by Valery Gritsenko, a professor of University Lille 1, France.

Jacobi forms are holomorphic functions in two complex variables. They are modular in one variable and abelian (or double periodic) in another variable. We discuss some examples of mod $\frak{p}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms.

Finally, we apply our results to properties mod $\frak{p}$ of Klingen-Eisenstein series. MODULAR FORMS OF ORTHOGONAL TYPE AND JACOBI THETA-SERIES 3 D(L2).Then O+(L2) is the intersection of the integral orthogonal group O(L2) with O+(L 2 ⊗R).

We use the similar notation SO+(L2) for the special orthogonal group. In this paper we assume that L2 is an even lattice of signature (2,n0 + 2) con- taining two hyperbolic planes.

The construction uses the singular theta lifting by considering a suitable twist of a Siegel theta function. Though the main emphasis of this book is on the geometry of Hilbert modular. It is generated by 10 modular forms (5 of weight 1 and 5 of weight 3) and there are 20 relations (5 in weight 5 and 15 in weight 6).

The proof consists of two steps. In a first step we prove that. We write M (Γ g) for the ring of the Siegel modular forms. The Siegel modular forms are considered to be Γ g-invariant functions (see, for details about Siegel modular forms). Next, we introduce some typical examples of Siegel modular forms.

For g ∈ N, let Δ g, 0: = {(⁎ ⁎ 0 n ⁎) ∈ Γ g}, where 0 g is the zero matrix of degree g. Singular Modular Forms and Theta Relations. Home. WorldCat Home About WorldCat Help. Search. Search for Library Items Search for Lists Search for Contacts Search for a Library. Create lists, bibliographies and reviews: or Search WorldCat.

Find items in libraries near you. Mock theta functions and quantum modular forms Article (PDF Available) in Forum of Mathematics Pi 1 January with Reads How we measure 'reads'. It is proved that the ring of Siegel modular forms in any genus is determined by doubly even self-dual codes and the theta relations.

The (higher) weight polynomials of such codes are proved to be the generators of the ring of invariants of a polynomial ring in 2 g variables under a certain specified finite group. Moreover codes are uniquely determined by their weight polynomials. We discuss some examples of mod $\frak{p}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms.

Finally, we apply our results to properties mod $\frak{p}$ of Klingen-Eisenstein t: 21 page. Reflective modular forms are of great importance. Such modular forms play a vital role in classifying interesting Lorentzian Kac-Moody algebras, as their denominator identities are usually. 4 D. Zagier The modular group takes its name from the fact that the points of the quotient space Γ1\H are moduli (= parameters) for the isomorphism classes of elliptic curves over C.

To each point z∈ H one can associate the lattice Λ z = Z.z+ Z.1 ⊂C and the quotient space E z = C/Λ z, which is an elliptic curve, i.e., it is at the same time a complex curve and an abelian group. Hilbert Modular Forms by Eberhard Freitag Singular Modular Forms and Theta Relations (Lecture Notes in Mathematics) Dec 2, by Eberhard Freitag Paperback.

 39 More Information Audible Listen to Books & Original Audio Performances. Eberhard Freitag: free download. Ebooks library. On-line books store on Z-Library | B–OK.

Download books for free. Find books.SINGULAR KUMMER SURFACES AND HILBERT MODULAR FORMS* by H. L. ResnilcofS For my brother Kumrner surface is a quartic surface in P, which has the maximum ibIe number of point singularities, namely, sixteen. The first exampIe surface of this type was published in by the physicist Fresne1 [IStanford Libraries' official online search tool for books, media, journals, databases, government documents and more.