Showing papers on "Modular form published in 2002"
Journal Article•
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TL;DR: In this article, the authors consider several of the examples that Ramanujan gave of mock theta functions, and relate them to real-analytic modular forms of weight 1/2 and show that the transformation behaviour becomes that of a Jacobi form if we add a (relatively simple) correction term.
Abstract: The mock theta functions were invented by the Indian mathematician Srinivasa Ramanujan, who lived from 1887 until 1920. He discovered them shortly before his death. In this dissertation, I consider several of the examples that Ramanujan gave of mock theta functions, and relate them to real-analytic modular forms of weight 1/2.
In Chapter 1, I consider a sum, which was also studied by Lerch. This Lerch sum transforms almost as a Jacobi form under substitutions in (upsilon, nu, tau ). I show that the transformation behaviour becomes that of a Jacobi form if we add a (relatively simple) correction term. This correction term is real-analytic in (upsilon, nu, tau) but not holomorphic. For special values of (upsilon, nu), we could call the Lerch sum (considered as a function of tau ) a mock theta function, although these examples were not considered by Ramanujan.
In Chapter 2, I consider theta functions for indefinite quadratic forms. These indefinite theta functions are modified versions of theta functions introduced by Gottsche and Zagier. I find elliptic and modular transformation properties of these functions, similar to the properties of theta functions associated to positive definite quadratic forms. In the case of positive definite quadratic forms, the theta functions are holomorphic. The theta functions in the chapter are not holomorphic. By taking special values of certain parameters, we get most of the examples of mock theta functions given by Ramanujan.
Andrews gave most of the fifth order mock theta functions as Fourier coefficients of meromorphic Jacobi forms, namely certain quotients of ordinary Jacobi theta-series. This is the motivation for the study of the modularity of Fourier coefficients of meromorphic Jacobi forms, in Chapter 3. We find that modularity follows on adding a real-analytic correction term to the Fourier coefficients.
In Chapter 4, I use the results from Chapter 2 to get the modular transformation properties of the seventh-order mock v-functions and most of the fifth-order functions. The final result is that we can write each of these mock theta-functions as the sum of two functions ? and G, where:
- ? is a real-analytic modular form of weight 1/2 and is an eigenfunction of the appropriate Casimir operator with eigenvalue 3/16 (this is also the eigenvalue of holomorphic modular forms of this weight); and
- G is a theta series associated to a negative definite unary quadratic form. Moreover G is bounded as ? tends vertically to any rational limit.
Many of the results of Chapter 4 could be deduced using the methods from Chapter 1 or Chapter 3 instead of Chapter 2. This means that I have actually given three approaches to proving modularity properties of the mock theta -functions.
731 citations
Book•
10 Apr 2002
TL;DR: In this article, the authors introduce vector valued modular forms for the metaplectic group and the regularized theta lift for the Fourier theta lifts, as well as a lifting into cohomology.
Abstract: Introduction.- Vector valued modular forms for the metaplectic group. The Weil representation. Poincare series and Einstein series. Non-holomorphic Poincare series of negative weight.- The regularized theta lift. Siegel theta functions. The theta integral. Unfolding against F. Unfolding against theta.- The Fourier theta lift. Lorentzian lattices. Lattices of signature (2,l). Modular forms on orthogonal groups. Borcherds products.- Some Riemann geometry on O(2,l). The invariant Laplacian. Reduction theory and L^p-estimates. Modular forms with zeros and poles on Heegner divisors.- Chern classes of Heegner divisors. A lifting into cohomology. Modular forms with zeros and poles on Heegner divisors II.
472 citations
TL;DR: In this article, the slice-independent gauge-fixed superstring chiral measure in genus 2 is evaluated explicitly in terms of theta-constants and the degeneration limits for the contribution of each spin structure are determined, and the divergences, before the GSO projection, are the ones expected on physical grounds.
148 citations
TL;DR: In this article, a theory of Fourier coefficients for modular forms on the split exceptional group G2 over ℚ was developed, where the coefficients are derived from the Fourier coefficient theory of modular forms.
Abstract: We develop a theory of Fourier coefficients for modular forms on the split exceptional group G2 over ℚ.
146 citations
Posted Content•
TL;DR: In this paper, a topological refinement of modular forms, called topological modular forms (Topological Modular Form (TFL), has been proposed, motivated by technical issues in homotopy theory.
Abstract: Modular forms appear in many facets of mathematics, and have played important roles in geometry, mathematical physics, number theory, representation theory, topology, and other areas. Around 1994, motivated by technical issues in homotopy theory, Mark Mahowald, Haynes Miller and I constructed a topological refinement of modular forms, which we call {\em topological modular forms}. At the Zurich ICM I sketched a program designed to relate topological modular forms to invariants of manifolds, homotopy groups of spheres, and ordinary modular forms. This program has recently been completed and new directions have emerged. In this talk I will describe this recent work and how it informs our understanding of both algebraic topology and modular forms.
143 citations
Book•
30 Jun 2002TL;DR: Andrews as discussed by the authors derived infinite families of non-trivial exact exact formulas for sums of squares by combining a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups.
Abstract: The problem of representing an integer as a sum of squares of integers is one of the oldest and most significant in mathematics. It goes back at least 2000 years to Diophantus, and continues more recently with the works of Fermat, Euler, Lagrange, Jacobi, Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's elliptic function approach dates from his epic Fundamenta Nova of 1829. Here, the author employs his combinatorial/elliptic function methods to derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to 4n2 or 4n(n+1) squares, respectively, without using cusp forms such as those of Glaisher or Ramanujan for 16 and 24 squares. These results depend upon new expansions for powers of various products of classical theta functions. This is the first time that infinite families of non-trivial exact explicit formulas for sums of squares have been found. The author derives his formulas by utilizing combinatorics to combine a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups. His results (in Theorem 5.19) generalize to separate infinite families each of the 21 of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions in sections 40-42 of the Fundamental Nova. The author also uses a special case of his methods to give a derivation proof of the two Kac and Wakimoto (1994) conjectured identities concerning representations of a positive integer by sums of 4n2 or 4n(n+1) triangular numbers, respectively. These conjectures arose in the study of Lie algebras and have also recently been proved by Zagier using modular forms. George Andrews says in a preface of this book, 'This impressive work will undoubtedly spur others both in elliptic functions and in modular forms to build on these wonderful discoveries.' Audience: This research monograph on sums of squares is distinguished by its diversity of methods and extensive bibliography. It contains both detailed proofs and numerous explicit examples of the theory. This readable work will appeal to both students and researchers in number theory, combinatorics, special functions, classical analysis, approximation theory, and mathematical physics.
123 citations
TL;DR: In this article, an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associated to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve was considered.
Abstract: We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.
85 citations
TL;DR: In this article, it was shown that the q-expansion of a weight two modular form has constant term equal to one and all other coefficients equal to even integers, except perhaps for its constant term, which we require merely to be an integer.
Abstract: This paper deals with two subjects and their interaction. The first is the problem of spanning spaces of modular forms by theta series. The second is the commutative algebraic properties of Hecke modules arising in the arithmetic theory of modular forms. Let p be a prime, and let B denote the quaternion algebra over Q that is ramified at p and ∞ and at no other places. If L is a left ideal in a maximal order of B then L is a rank four Z-module equipped in a natural way with a positive definite quadratic form [6, §1]. (We shall say that L is a rank four quadratic space, and remark that the isomorphism class of L as a quadratic space depends only on the left ideal class of L in its maximal order.) Eichler [5] proved that the theta series of L is a weight two modular form on Γ0(p), and that as L ranges over a collection of left ideal class representatives of all left ideals in all maximal orders of B these theta series span the vector space of weight two modular forms on Γ0(p) over Q. In this paper we strengthen this result as follows: if L is as above, then the q-expansion of its theta series Θ(L) has constant term equal to one and all other coefficients equal to even integers. Suppose that f is a modular form whose qexpansion coefficients are even integers, except perhaps for its constant term, which we require merely to be an integer. It follows from Eichler’s theorem that f may be written as a linear combination of Θ(L) (with L ranging over a collection of left ideals of maximal orders of B) with rational coefficients. We show that in fact these coefficients can be taken to be integers. Let T denote the Z-algebra of Hecke operators acting on the space of weight two modular forms on Γ0(p). The proof that we give of our result hinges on analyzing the structure of a certain T-module X . We can say what X is: it is the free Zmodule of divisors supported on the set of singular points of the (reducible, nodal) curve X0(p) in characteristic p. The key properties of X , which imply the above result on theta series, are that the natural map T −→ EndT(X ) is an isomorphism, and that furthermore X is locally free of rank one in a Zariski neighbourhood of the Eisenstein ideal of T. We remark that it is comparatively easy to prove the analogous statements after tensoring with Q, for they then follow from the fact that X is a faithful T-module. Indeed, combining this with the semi-simplicity of the Q-algebra T⊗Z Q, one deduces that X ⊗Z Q is a free T⊗Z Q-module of rank one, and in particular that the map T⊗Z Q −→ EndT⊗ZQ(X ⊗Z Q) is an isomorphism.
67 citations
TL;DR: In this article, a matrix model is used to describe all the massive vacua of the N=1*, or mass deformed N=4, theory including the Higgs vacuum.
Abstract: In this note we show how Dijkgraaf and Vafa's hypothesis relating the exact superpotential of an N=1 theory to a matrix model can be used to describe all the massive vacua of the N=1*, or mass deformed N=4, theory including the Higgs vacuum. The matrix model computation of the superpotential for each massive vacuum independently yields a modular function of the associated effective coupling in that vacuum which agrees with previously derived results up to a vacuum-independent additive constant. The results in the different massive vacua can be related by the action of SL(2,Z) on the N=4 coupling, thus providing evidence for modular invariance of the underlying N=4 theory.
64 citations
TL;DR: In this paper, the authors examined crossing probabilities and free energies for conformally invariant critical 2D systems in rectangular geometries, derived via conformal field theory and Stochastic Lowner Evolution methods.
Abstract: We examine crossing probabilities and free energies for conformally invariant critical 2-D systems in rectangular geometries, derived via conformal field theory and Stochastic Lowner Evolution methods. These quantities are shown to exhibit interesting modular behavior, although the physical meaning of modular transformations in this context is not clear. We show that in many cases these functions are completely characterized by very simple transformation properties. In particular, Cardy's function for the percolation crossing probability (including the conformal dimension 1/3), follows from a simple modular argument. A new type of "higher-order modular form" arises and its properties are discussed briefly.
59 citations
1 Jan 2002
TL;DR: In this paper, a canonical projection operator is used onto the characteristic subspace of an eigenvalue α of the Atkin-Lehner operator Up to construct p-adic measures associated with modular cusp eigenforms.
Abstract: We give a new method of constructing admissible p-adic measures associated with modular cusp eigenforms, starting from distributions with values in spaces of modular forms. A canonical projection operator is used onto the characteristic subspace of an eigenvalue α of the Atkin–Lehner operator Up. An algebraic version of nearly holomorphic modular forms is given and used in constructing p-adic measures. 2000 Math. Subj. Class. 11F33, 11F67, 11F30.
TL;DR: In this paper, an analogue of Teitelbaum's conjecture was formulated in which the cyclotomic Zp extension of Q is replaced by the anticyclotomic zp-extension of an imaginary quadratic field and proved by using the Cerednik-Drinfeld theory of /7-adic uniformisation of Shimura curves.
Abstract: Teitelbaum formulated a conjecture relating first derivatives of the Mazur-Swinnerton Dyer /?-adic L-functions attached to modular forms of even weight k > 2 to certain /^-invariants arising from Shimura curve parametrizations. This article formulates an analogue of Teitelbaum's conjecture in which the cyclotomic Zp extension of Q is replaced by the anticyclotomic Zp-extension of an imaginary quadratic field. This analogue is then proved by using the Cerednik-Drinfeld theory of /7-adic uniformisation of Shimura curves. Introduction. Let 0 = Y, an 2 on To(N). The classical L-function L(cf),s) admits an analytic continuation to the entire complex plane, and a functional equation which relates its values at s and k ? s. Of special arithmetic interest for the present work is the central value L((j>,k/2). For example, when k = 2, the Birch and Swinnerton-Dyer conjecture relates the behavior of L((j),s) at s = 1 to the arithmetic of the abelian variety A associated to (j) by the Eichler-Shimura construction. In (MTT), a/?-adic variant of the Birch and Swinnerton-Dyer conjecture is formulated with L((\>, s) replaced by a /7-adic analogue Lp((f), s) attached to the cyclotomic Zp-extension of Q. When p divides N exactly and ap = 1 (which implies that A^ has split multiplicative reduction at p), the function Lp((f),s) vanishes at s = 1. In this case the conjectures of (MTT) imply the following relationship between the first
Book•
1 Jan 2002
TL;DR: In a recent paper as discussed by the authors, Yui presents a survey of logarithmic approximations of Diophantine equations in the context of modular curves and their application to algebraic independence.
Abstract: Introduction 1. One century of logarithmic forms G. Wustholz 2. Report on p-adic logarithmic forms Kunrun Yui 3. Recent progress on linear forms in elliptic logarithms Sinnou David and Noriko Hirata-Kohno 4. Solving Diophantine equations by Baker's theory Kalman Gyoery 5. Baker's method and modular curves Yuri F. Bilu 6. Application of the Andre-Oort conjecture Paula B. Cohen and Gisbert Wustholz 7. Regular dessins Jurgen Wolfart 8. Maass cusp forms with integer coefficients Peter Sarnak 9. Modular forms, elliptic curves and the ABC-conjecture Dorian Goldfeld 10. On the algebraic independence of numbers Yu. V. Nesterenko 11. Ideal lattices Eva Bayer-Fluckiger 12. Integral points and Mordell-Weil lattices Tetsuji Shioda 13. Forty years of effective results in Diophantine theory Enrico Bombieri 14. Points on subvarieties of tori Jan-Hendrik Evertse 15. A new application of Diophantine approximations G. Faltings 16. Search bounds for Diophantine equations D. W. Masser 17. Regular systems and ubiquity V. V. Beresnevich, V. I. Bernik and M. M. Dodson 18. Diophantine approximation, lattices and flows Gregory Margulis 19. Baker's constant and Vinogradov's bound Ming-Chit Liu and Tianze Wang 20. Powers in arithmetic progression T. N. Shorey 21. Greatest common divisor A. Schinzel 22. Heilbronn's exponential sum and transcendence theory D. R. Heath-Brown.
1 Jan 2002
TL;DR: The purpose of this note is to introduce the reader to some of the basic concepts in the theory of congruences between modular forms.
Abstract: The purpose of this note is to introduce the reader to some of the basic concepts in the theory of congruences between modular forms. Our exposition here has been distilled from various sources. We have especially benefited from reading the papers of Hida and Ribet some of which are listed in the references.
1 Jan 2002
TL;DR: In this paper, the authors describe some examples of modular forms whose Fourier coefficients involve quantities from algebraic geometry, and suggest that they should be viewed as a kind of arithmetic analogue of theta series and that there should be an arithmetic Siegel-Weil formula relating suitable averages of them to special values of derivatives of Eisenstein series.
Abstract: The aim of these notes is to describe some examples of modular forms whose Fourier coefficients involve quantities from arithmeticla algebraic geometry. Ath the moment, no general theory of such forms exists, but the examples suggest that they should be viewed as a kind of arithmetic analogue of theta series and that there should be an arithmetic Siegel-Weil formula relating suitable averages of them to special values of derivatives of Eisenstein series. We will concentrate on the case for which the most complete picture is available, the case of generating series for cycles on the arithmetic surfaces associated to Shimura curves over ?, expanding on the treatment in [40]. A more speculative overview can be found in [41].
TL;DR: In this article, it was shown that any asymptotically nonexpansive self-map defined on a convex subset of L 1 (Ω,μ) which is compact for the topology of local convergence in measure has a fixed point.
1 Jan 2002
TL;DR: In this article, a solution to the extension of Lie's problem on classificatio n of "local continuous transformation groups of a finite-dimensi onal manifold" to the case of supermanifold s is presented.
Abstract: In the first part of my talk I will explain a solution to the extension of Lie's problem on classificatio n of "local continuous transformation groups of a finite-dimensi onal manifold" to the case of supermanifold s (More precisely, the problem is to classify simple linearly compact Lie superalgebras, ie toplogical Lie superalgebras whose underlying space is a topological product of finite-dimensional vector spaces) In the second part I will explain how this result is used in a classification of superconforma i algebras The list consists of affine superalgebras and certain super extensions of the Virasoro algebra In the third part I will discuss representation theory of affine superalgebras and its relation to "almost" modular forms Furthermore, I will explain how the quantum reduction of these representations leads to a unified represen tation theory of super extensions of the Virasoro algebra In the forth part I will discuss representation theory of exceptional simple infinite-dimensional linearly compact Lie superalgebras and will speculate on its relation to the Standard Model
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TL;DR: In this paper, the authors show how to produce congruences between forms of weights 2 and p+1, in terms of group cohomology, and also show how their method works in the contexts of quadratic imaginary fields and Hilbert modular forms over totally real fields of even degree.
Abstract: Let p be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between forms of weights 2 and p+1, in terms of group cohomology. We also show how our method works in the contexts of quadratic imaginary fields (where there is no Hasse invariant available) and Hilbert modular forms over totally real fields of even degree.
1 Jan 2002
TL;DR: In this paper, the Picard modular image surface is constructed from lifted quotients of elliptic Jacobi theta functions in higher dimensions, which yield explicit homogeneous equations for the Picard image surface.
Abstract: Starting from a fixed elliptic curve with complex multiplication we compose lifted quotients of elliptic Jacobi theta functions to abelian functions in higher dimension. In some cases, where complete Picard–Einstein metrics have been discovered on the underlying abelian surface (outside of cusp points), we are able to transform them to Picard modular forms. Basic algebraic relations of basic forms come from different multiplicative decompositions of these abelian functions in simple ones of the same lifted type. In the case of Gaus numbers the constructed basic modular forms define a Baily–Borel embedding in P. The relations yield explicit homogeneous equations for the Picard modular image surface.
TL;DR: In this paper, a modular function ηψ(z) similar to the Dedekind eta function is given by an analogue of Borcherds type liftings.
Abstract: In this paper, we study certain modular functions ηψ(z) similar to the Dedekind eta function η(z). The functions are given by an analogue of Borcherds type liftings. It turns out that the functions ηψ(z) have some good properties, similarly as in the case of the Dedekind eta function.
TL;DR: In this article, it was shown that the Hasse-Weil modular form determined by the arithmetic structure of the Fermat type elliptic curve is related in a natural way to a modular form arising from the character of a conformal field theory derived from an affine Kac-Moody algebra.
Abstract: The Shimura-Taniyama conjecture states that the Mellin transform of the Hasse-Weil L-function of any elliptic curve defined over the rational numbers is a modular form. Recent work of Wiles, Taylor-Wiles and Breuil-Conrad-Diamond-Taylor has provided a proof of this longstanding conjecture. Elliptic curves provide the simplest framework for a class of Calabi-Yau manifolds which have been conjectured to be exactly solvable. It is shown that the Hasse-Weil modular form determined by the arithmetic structure of the Fermat type elliptic curve is related in a natural way to a modular form arising from the character of a conformal field theory derived from an affine Kac-Moody algebra.
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TL;DR: In this paper, the moduli space of marked cubic surfaces has been studied, starting from a toric variety and proceeding with blow ups and contractions, and the Chow groups and the Chern classes of the modulus space have been computed.
Abstract: Naruki gave an explicit construction of the moduli space of marked cubic surfaces, starting from a toric variety and proceeding with blow ups and contractions. Using his result, we compute the Chow groups and the Chern classes of this moduli space.As an application we relate a recent result of Freitag on the Hilbert polynomial of a certain ring of modular forms to the Riemann-Roch theorem for the moduli space.
Book•
1 Jan 2002
TL;DR: In this paper, Li et al. present a survey of algebraic number theory and analytic number theory with a focus on the relationship between algebraic numbers and analytic numbers, and their applications.
Abstract: Preface. A limiting form of the q Dixon 4pi3 summation and related partition identities K. Alladi, A. Berkovich. Arithmetical properties of solutions of linear second order q-difference equations M. Amou, T. Matala-aho. Ramanujan's contributions to Eisenstein series, especially in his lost notebook B.C. Berndt, Ae Ja Yee. New applications as a result of Galochkin on linear independence P. Bundschuh. Partitions, modulo prime powers and binomial coefficients Tianxin Cai. Infinite sums, diophantine equations and Fermat's last theorem H. Darmon, C. Levesque. On the nature of the 'explicit formulas' in analytic number theory a simple example C. Deninger. Product representations by rationals P.D.T.A. Elliott. On the distribution of alphaP modulo 1 Chaohua Jia. Ramanujan's formula and modular forms S. Kanemitsu, Y. Tanigawa, M. Yoshimoto. Waldspurger's formula and central critical values of L-functions of newforms in weight aspect W. Kohnen, J. Sengupta. Primitive roots: a survey Shuguang Li, C. Pomerance. Zeta-functions defined by two polynomials K. Matsumoto, Lin Weng. Some aspects on interactions between algebraic number theory and analytic number theory K. Miyake. On G-functions and Pade approximations M. Nagata. A penultimate step toward cubic theta-Weyl sums Y. Nakai. Some results in view of Nevanlinna theory J. Noguchi. A historical comment about the GVT in short interval Pan Chingbiao. Convexity and intersection of random spaces M. Shcherbina, B. Tirozzi. Generalized hypergeometric series and the symmetries of 3-J and 6-J coefficients Stability and new non-Abelian zeta functions Lin Weng. A hybrid mean value of L-functions and general quadratic Gauss sums ZhangWenpeng. Index.
TL;DR: In this article, the authors studied homomorphisms that arise from the restriction of Siegel modular forms to modular curves, which give rise to linear relations among the Fourier coefficients of a siegel modular form.
Abstract: We study homomorphisms form the ring of Siegel modular forms of a given degree to the ring of elliptic modular forms for a congruence subgroup. These homomorphisms essentially arise from the restriction of Siegel modular forms to modular curves. These homomorphisms give rise to linear relations among the Fourier coefficients of a Siegel modular form. We use this technique to prove that dim .
1 Jan 2002
TL;DR: In this paper, the authors consider the slopes of the U2 operator acting on spaces of 2-adic overconvergent modular forms with nontrivial weight-character of tame level 1.
Abstract: In the first part of this thesis, we consider the slopes of the U2 operator acting on spaces of 2-adic overconvergent modular forms with nontrivial weight-character of tame level 1. We establish a sufficient criterion for these slopes to be given by a simple formula, and prove this criterion in several cases. This allows us to write down all of the slopes for certain weight-characters of small level. This criterion can be written quite simply in terms of modular functions. These results on overconvergent modular forms also imply similar results for certain spaces of classical modular forms, which also allows us to prove results about the field over which the Fourier expansions of the normalised classical cuspidal modular eigenforms are defined. These calculations provide evidence for an analogue of the Gouvea-Mazur conjecture that the slopes of classical cuspidal modular eigenforms vary smoothly as the weight varies. We also present new conjectures of the same form for p = 3 and p = 5, and present some numerical evidence for them. The second part of this thesis considers the Hecke algebras attached to certain spaces of classical cuspidal modular forms of prime level. It proves that some of these Hecke algebras are not Gorenstein when localised at a prime ideal above 2. This shows that the methods developed by Mazur, Gross and Edixhoven for proving that localisations of a Hecke algebra are Gorenstein fail in some cases, because we have exhibited explicit localisations which are not Gorenstein. The computations in both parts of the thesis show the power of computational methods when applied to number theory, in solving problems which can be described concretely. It also shows that computational methods can be useful in identifying patterns and generating data. These can then be investigated by more theoretical methods.
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TL;DR: The theory of theta series with harmonic coefficients allows to classify and to construct extremal lattices as well as to prove that some of them are strongly perfect and hence local maxima of the density function as discussed by the authors.
Abstract: A main goal in lattice theory is the construction of dense lattices. Most of the remarkable dense lattices in small dimensions have an additional symmetry, they are modular, i.e. similar to their dual lattice. Extremal lattices are densest modular lattices, whoses density is as high as the theory of modular forms allows it to be. The theory of theta series with harmonic coefficients allows to classify and to construct extremal lattices as well as to prove that some of them are strongly perfect and hence local maxima of the density function.