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On the Fourier coefficients of the modular invariant j([tau]). by Oddmund Kolberg

Written in English

Subjects:

• Invariants.,
• Congruences and residues.

Edition Notes

Bibliography: p. 8.

Book details

Classifications The Physical Object Statement [By] O[ddmund] Kolberg. Series Årbok for Universitetet i Bergen., 1969, no. 3 LC Classifications Q1 .B455 1969, no. 3 Pagination 8 p. Open Library OL5769170M LC Control Number 71455464

In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on the upper half-plane of complex is the unique such function which is holomorphic away from a simple pole at the cusp such that (/) =.

Rational functions of j are modular, and in fact give all modular functions. Lehner, J. "Divisibility Properties of the Fourier Coefficients of the Modular Invariant." Amer. Math. 71,a. Lehner, J.

"Further Congruence Properties of the Fourier Coefficients of the Modular Invariant." Amer. Math. 71,b. Morain, F. "Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm. Subsequent chapters explore the automorphisms of a compact Riemann surface, develop congruences and other arithmetic properties for the Fourier coefficients of Klein's absolute modular invariant, and discuss analogies with the Hecke theory as well as with the Ramanujan congruences for the partition function.5/5(3).

In particular, they construct modular forms which are analogs to the modular discriminant and the Klein j-invariant of the full modular group PSL (2, Z). In this article, we produce effective and practical bounds for the Fourier coefficients in the q-expansion of such generators, thus allowing for rigorous numerical inspection of the : Daniel Garbin.

The Fourier coefficients of the modular invariant J(τ) The book also proves Dirichlet’s theorem on primes in arithmetic progressions, covers Dirichlet’s class number formula, and shows. vergent series for the Fourier coefficients of/(21/2; t) and/(3I/2; t).

In §2, two relations between j(21/2; r) and j(r) are established directly for a convenient calculation in closed form of the first few coefficients in terms of those ol j(r).

This supplies a partial check on our main theorem. Recursive formulae satisfied by the Fourier coefficients of meromorphic modular forms on groups of genus zero have been investigated by several authors. Bruinier et al. [17] Lehmer, D.

H., ‘ Properties of the coefficients of the modular invariant J. A cusp form is a modular form with a zero constant coefficient in its Fourier series. It is called a cusp form because the form vanishes at all cusps. Generalizations. There are a number of other usages of the term "modular function", apart from this classical one; for example, in the theory of Haar measures, it is a function Δ(g) determined by the conjugation action.

We give congruences modulo powers of p ∈ {3, 5, 7} for the Fourier coefficients of certain modular functions in level p with poles only at 0, answering a question posed by Andersen and the first author and continuing work done by the authors and Moss.

The congruences involve a modulus that depends on the base p expansion of the modular form's order of vanishing at ∞. FOURIER COEFFICIENTS OF VVMFS OF DIMENSION 2 5 Remark The lowest terms in the q-expansions of f 1 and f 2 have exponents m 2 and m 1 respectively.

Note that we have rescaled the hypergeometric series by j a and j b, rather than J aand J b, simply to avoid introducing a. We obtain new lower bounds for the number of Fourier coefficients of a weakly holomorphic modular form of half-integral weight not divisible by some prime $$\ell$$.Among the applications of this we show that there are $$\gg On the Fourier coefficients of the modular invariant j book \log X$$ integers $$n \le X$$ for which the partition function p(n) is not divisible by $$\ell$$, and that there are $$\gg \sqrt{X}/\log \log X$$ values of.

CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. Using an elementary approach, we give precise effective lower and upper bounds for the Fourier coefficients of powers of the mod-ular invariant j. Moreover, a straightforward adaptation of an old result of Rademacher yields a convergent series expansion of these Fourier coef-ficients and we show that.

Lectures on Modular Forms - (Dover Books on Mathematics) by Joseph J Lehner (Paperback) $develop congruences and other arithmetic properties for the Fourier coefficients of Klein's absolute modular invariant, and discuss analogies with the Hecke theory as well as with the Ramanujan congruences for the partition function. InLehner showed that certain coefficients of the modular invariant j(τ) are divisible by high powers of small primes. Kolberg refined Lehner's results and proved congruences for these coefficients modulo high powers of these primes. inﬁnitely many non-zero fundamental Fourier coeﬃcients. Introduction The purpose of this article is to shed some light on Fourier coeﬃcients of cuspidal paramodular forms. Paramodular forms are Siegel modular forms of degree 2 that are invariant under the action of the paramodular group Γpara(N):= Sp 4(Q) ∩ Z NZ Z Z Z Z Z Z/N Z NZ Z Z. Using an elementary approach based on careful handlings of Cauchy integrals, we give precise effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. Moreover, we adapt an old result of Rademacher to get a convergent series expansion of these Fourier coecients and we show that this expansion allows to. I'm reading about Fourier expansion of modular functions, but I have trouble understanding the following equation: Is it some inherent property of the denominator, as it is. Stack Exchange Network Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn. While this is an infinite sum, in practice you only need to add the first few terms because you already know the coefficients are integers. More generally, if you want coefficients of any (weakly holomorphic) modular form of non-positive weight, you can apply the circle method, and you need only input the principal parts of the expansions at cusps. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms, preprint (). Google Scholar M. Dewar and M. Murty, A derivation of the Hardy–Ramanujan formula from an arithmetic formula, to appear in Proc. Amer. Math. Soc. THE FOURIER COEFFICIENTS OF THE MODULAR INVARIANT J(7) * By HANS RADEMACHER. Recently Dr. Zuckerman and I have developed general formulae for the Fourier coefficients of modular forms of positive dimensions.' We remarked at the end of our paper that the series obtained would be convergent also for. Effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j Article January with 29 Reads How we measure 'reads'. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Using an elementary approach based on careful handlings of Cauchy integrals, we give precise effective lower and upper bounds for the Fourier coefficients of powers of the modular invariant j. Moreover, we adapt an old result of Rademacher to get a convergent series expansion of these Fourier coecients and we show. There is a surprising connection of these Fourier coefficients with the representations of the largest of the sporadic finite simple groups (the Monster; cf. also Simple finite group).All of the early Fourier coefficients of are simple linear combinations of degrees of characters of the Monster group. This led J.H. Conway to refer to the "moonshine" properties of the Monster. [5) Rademacher, Hans, The Fourier coefficients of the modular invariant J (T), American Journal of Mathemat (). [6) Rademacher, Hans, The Fourier series and the functional equation of the absolute modular invariant J (T), American Journal of. Lectures on Modular Forms (Dover Books on Mathematics) - Kindle edition by Lehner, Joseph J. Download it once and read it on your Kindle device, PC, phones or tablets. Use features like bookmarks, note taking and highlighting while reading Lectures on Modular Forms (Dover Books Reviews: 3. Ramanujan Math. Soc. 20, No.4 () – Effective lower and upper bounds for the Fourier coefﬁcients of powers of the modular invariant j Nicolas Brisebarre∗ and Georges Philibert† ∗LArAl, Universite J. Monnet, 23, rue du Dr P. Michelon, F Saint-´ Etienne´ Cedex, France and Arenaire, LIP,´ Ecole Normale Sup´ erieure de Lyon, 46, All´ ee´. Project Euclid - mathematics and statistics online. Featured partner The Tbilisi Centre for Mathematical Sciences. The Tbilisi Centre for Mathematical Sciences is a non-governmental and nonprofit independent academic institution founded in November in Tbilisi, general aim of the TCMS is to facilitate new impetus for development in various areas of mathematical sciences in Georgia. On the Fourier coefﬁcients of modular forms of half-integral weight Kartik Prasanna (Communicated byPeter Sarnak) Abstract. We prove a formula relating the Fourier coefﬁcients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the. On the estimation of Fourier coefficients of modular forms. In: Theory of numbers, Proceedings of symposia in pure mathematics, vol. 8, Providence: Am. Math. Soc.pp. 1–15 Google Scholar Subsequent chapters explore the automorphisms of a compact Riemann surface, develop congruences and other arithmetic properties for the Fourier coefficients of Klein's absolute modular invariant, and discuss analogies with the Hecke theory as well as with the. Citation: Huichi Huang. Fourier coefficients of$\times p$-invariant measures. Journal of Modern Dynamics,doi: /jmd Title: Fourier series of modular graph functions Authors: Eric D'Hoker, William Duke (Submitted on 26 Aug (v1), last revised 14 Aug (this version, v2)). Congruence properties of the coefficients of the complete modular invariant 1 j(r) = /(t) = £ c(«)*" =-h + x + • • •, n=—1 X x = exp2Trir, imr>0, have been given by D. Lehmer [l], J. rived from some general congruences for the coefficients of certain modular forms and an explicit formula for the coefficients c(n. Among the major topics treated are Rademacher's convergent series for the partition function, Lehner's congruences for the Fourier coefficients of the modular functionj(r), and Hecke's theory of entire forms with multiplicative Fourier coefficients. The last chapter gives an account of Bohr's theory of equivalence of general Dirichlet series. Lehner, J., Divisibility properties of the Fourier coefficients of the modular invariant j(r) Amer. Math. 71 (), 2. Theorem The elliptic modular function$j(\tau)$is invariant under the action the of modular group$SL_{2}(\mathbb{Z})$; in particular, it has a Fourier series. Kaneko, The Fourier coefficients and the singular moduli of the elliptic modular function j(tau), Memoirs Faculty Engin. Sci., On the coefficients of the modular invariant J(tau), Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 56 (), It is a standard result that$\Delta$has integer Fourier coefficients. I wish to prove that that$1/\Delta$has integer Fourier coefficients. why does a certain formula in Lang's book on modular forms hold. On the estimation of fourier coefficients of modular forms. Eisenstein Series, discriminant and cusp forms. Fourier. ON THE FOURIER COEFFICIENTS OF CERTAIN MODULAR FORMS OF POSITIVE DIMENSION BY HANS RADEMACHER AND HERBERT S. ZUCKERMANI (Received Octo ) 1. Introduction One of us has recently obtained an exact formula for the number p(n) of unrestricted partitions of n.2 The proof, which makes use of the Hardy. Expressing the Fourier coefficients in terms of integrals with limits of integration symmetric about the origin allows us to exploit symmetries of the function. If is even (), then from equation () for all m, and the resulting Fourier series will contain only the terms in this is known as a Fourier cosineseries. ON THE FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF-INTEGRAL WEIGHT KARTIK PRASANNA Abstract: We prove a formula relating the Fourier coe–cients of a modular form of half-integral weight to the special values of L-functions. The form in question is an explicit theta lift from the multiplicative group of an indeﬂnite quaternion algebra over Q.$\begingroup$Regarding 2) there is no general formula for the Fourier coeffs of the trace of a general modular form. This is because the elements of$\Gamma_0 (N) \$ used in the definition of the trace will move the infinity cusp to various other cusps.An illustration of an open book.

Books. An illustration of two cells of a film strip. Video. An illustration of an audio speaker. Audio An illustration of a " floppy disk. On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups Item Preview remove-circle.

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