Showing papers on "Modular form published in 2011"
22 Oct 2011
TL;DR: This work presents a new unified continuous greedy algorithm which finds approximate fractional solutions for both the non-monotone and monotone cases, and improves on the approximation ratio for many applications.
Abstract: The study of combinatorial problems with a sub modular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics, algorithmic game theory and combinatorial optimization. Classical works on these problems are mostly combinatorial in nature. Recently, however, many results based on continuous algorithmic tools have emerged. The main bottleneck of such continuous techniques is how to approximately solve a non-convex relaxation for the sub modular problem at hand. Thus, the efficient computation of better fractional solutions immediately implies improved approximations for numerous applications. A simple and elegant method, called ``continuous greedy'', successfully tackles this issue for monotone sub modular objective functions, however, only much more complex tools are known to work for general non-monotone sub modular objectives. In this work we present a new unified continuous greedy algorithm which finds approximate fractional solutions for both the non-monotone and monotone cases, and improves on the approximation ratio for many applications. For general non-monotone sub modular objective functions, our algorithm achieves an improved approximation ratio of about $1/e$. For monotone sub modular objective functions, our algorithm achieves an approximation ratio that depends on the density of the poly tope defined by the problem at hand, which is always at least as good as the previously known best approximation ratio of $1 - 1/e$. Some notable immediate implications are an improved $1/e$-approximation for maximizing a non-monotone sub modular function subject to a matroid or $O(1)$-knapsack constraints, and information-theoretic tight approximations for Sub modular Max-SAT and Sub modular Welfare with $k$ players, for {\em any} number of players $k$. A framework for sub modular optimization problems, called the \emph{contention resolution framework}, was introduced recently by Chekuri et al. The improved approximation ratio of the unified continuous greedy algorithm implies improved approximation ratios for many problems through this framework. Moreover, via a parameter called \emph{stopping time}, our algorithm merges the relaxation solving and re-normalization steps of the framework, and achieves, for some applications, further improvements. We also describe new monotone balanced contention resolution schemes for various matching, scheduling and packing problems, thus, improving the approximations achieved for these problems via the framework.
380 citations
TL;DR: In this article, the Fourier coecients of these modular integrals are given in terms of cycle integrals of modular functions, which can be interpreted as a Shimura-type lift of a mock modular form of weight 1/2 and yields a real quadratic analogue of a Borcherds product.
Abstract: In this paper we construct certain mock modular forms of weight 1/2 whose Fourier coecients are given in terms of cycle integrals of the modular j-function. Their shadows are weakly holomorphic forms of weight 3/2. These new mock modular forms occur as holomorphic parts of weakly harmonic Maass forms. We also construct a generalized mock modular form of weight 1/2 having a real quadratic class number times a regulator as a Fourier coecient. As an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers. The Fourier coecients of these modular integrals are given in terms of cycle integrals of modular functions. Such a modular integral can be interpreted in terms of a Shimura-type lift of a mock modular form of weight 1/2 and yields a real quadratic analogue of a Borcherds product.
150 citations
TL;DR: In this article, the authors give an interpretation of the Omega deformed B-model that leads naturally to the generalized holomorphic anomaly equations, and show that the failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation.
Abstract: We give an interpretation of the Omega deformed B-model that leads naturally to the generalized holomorphic anomaly equations. Direct integration of the latter calculates topological amplitudes of four dimensional rigid N=2 theories explicitly in general Omega-backgrounds in terms of modular forms. These amplitudes encode the refined BPS spectrum as well as new gravitational couplings in the effective action of N=2 supersymmetric theories. The rigid N=2 field theories we focus on are the conformal rank one N=2 Seiberg-Witten theories. The failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation. Our formalism applies irrespectively of whether a Lagrangian formulation exists. In the class of rigid N=2 theories arising from compactifications on local Calabi-Yau manifolds, we consider the theory of local P2. We calculate motivic Donaldson-Thomas invariants for this geometry and make predictions for generalized Gromov-Witten invariants at the orbifold point.
126 citations
TL;DR: In this article, the Sato-Tate conjecture for Hilbert modular forms has been proved for regular algebraic cuspidal automorphic representations of GL2(AF ), F a totally real field which is not of CM type.
Abstract: We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of GL2(AF ), F a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a “topological” argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of potential automorphy theorems for non-ordinary n-dimensional Galois representations.
123 citations
TL;DR: In this article, a family of L-series specialising to both Dirichlet characters over F_q[T] and integral values of Carlitz-Goss zeta function was introduced, with the use of the theory of deformations of vectorial modular forms.
Abstract: We introduce a family of L-series specialising to both L-series associated to certain Dirichlet characters over F_q[T] and to integral values of Carlitz-Goss zeta function associated to F_q[T]. We prove, with the use of the theory of deformations of vectorial modular forms, a formula for their value at 1, as well as some arithmetic properties of other values at positive integers
88 citations
Book•
12 Oct 2011
TL;DR: A collection of articles in memory of Irene Dorfman and her research in mathematical physics is presented in this article, where the topics covered are: the Hamiltonian and bi-Hamiltonian nature of continuous and discrete integrable equations; the t-function construction; the r-matrix formulation of integrably-integrable systems; pseudo-differential operators and modular forms; master symmetries and the Bocher theorem; asymptotic integrability of the equations of associativity; invariance under Laplace-darboux transformations.
Abstract: A collection of articles in memory of Irene Dorfman and her research in mathematical physics. Among the topics covered are: the Hamiltonian and bi-Hamiltonian nature of continuous and discrete integrable equations; the t-function construction; the r-matrix formulation of integrable systems; pseudo-differential operators and modular forms; master symmetries and the Bocher theorem; asymptotic integrability; the integrability of the equations of associativity; invariance under Laplace-darboux transformations; trace formulae of the Dirac and Schrodinger periodic operators; and certain canonical 1-forms.
71 citations
TL;DR: In this article, the authors presented three new analogues of Clausen's identities, which were motivated by the study of relations between modular forms of weight 2 and modular functions associated with modular groups of genus 0.
62 citations
TL;DR: In this article, the main conjecture for CM modular forms at supersingular primes has been formulated for normalised new forms of arbitrary weights and a formulation of the Iwasawa main conjecture has been given.
Abstract: We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.
55 citations
TL;DR: In this article, the authors proved many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod p Hilbert modular forms, by making use of modularity lifting theorems and computations in p-adic Hodge theory.
Abstract: We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod p Hilbert modular forms, by making use of modularity lifting theorems and computations in p-adic Hodge theory.
55 citations
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TL;DR: In this paper, an explicit Waldspurger formula for toric Hilbert modular forms is presented, which generalizes the construction of Bertolini, Darmon and Prasanna in the elliptic case.
Abstract: In this article, we prove an explicit Waldspurger formula for the toric Hilbert modular forms. As an application, we construct a class of anticyclotomic p-adic Rankin-Selberg L-functions for Hilbert modular forms, generalizing the construction of Bertolini, Darmon and Prasanna in the elliptic case. Moreover, building on works of Hida, we give a necessary and sufficient condition when the Iwasawa mu-invariant of this p-adic L-function vanishes and prove a result on the non-vanishing modulo $p$ of central Rankin-Selberg L-values with anticyclotomic twists.
53 citations
TL;DR: A general theory of overconvergent p-adic modular forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre is presented in this paper.
Abstract: A general theory of overconvergent p-adic modular forms and eigenvarieties is presented for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions due to Buzzard, Chenevier and Yamagami for forms of GLn. This leads to some new phenomena, including the appearance of intermediate spaces of ‘semi-classical’ automorphic forms; this gives a hierarchy of interpolation spaces (eigenvarieties) interpolating classical automorphic forms satisfying different finite slope conditions (corresponding to a choice of parabolic subgroup of G at p). The construction of these spaces relies on methods of locally analytic representation theory, combined with the theory of compact operators on Banach modules.
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TL;DR: In this paper, the authors give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form, based on the period relations proved by Raghuram and Shahidi for periods attached to regular algebraic cuspidal automorphic representations.
Abstract: The purpose of this semi-expository article is to give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form. Our proof is along the lines of previous work of Harder and Hida (independently). What is different is an organizational principle based on the period relations proved by Raghuram and Shahidi for periods attached to regular algebraic cuspidal automorphic representations. The point of view taken in this article is that one need only prove an algebraicity theorem for the most interesting L-value, namely, the central critical value of the L-function of a sufficiently general type of a cuspidal automorphic representation. The period relations mentioned above then gives us a result for all critical values. To transcribe such a result into a more classical context we also discuss the arithmetic properties of the dictionary between holomorphic Hilbert modular forms and automorphic representations of GL(2) over a totally real number field F.
TL;DR: In this article, the authors studied periodic torus orbits on spaces of lattices and showed that the ideal classes of a cubic totally real field are equidistributed in the modular 5-fold SL3(Z)/SL3(R)/SO3.
Abstract: We study periodic torus orbits on spaces of lattices. Using the action of the group of adelic points of the underlying tori, we define a natural equivalence relation on these orbits, and show that the equivalence classes become uniformly distributed. This is a cubic analogue of Duke's theorem about the distribution of closed geodesics on the modular surface: suitably interpreted, the ideal classes of a cubic totally real field are equidistributed in the modular 5-fold SL3(Z)\SL3(R)/SO3. In particular, this proves (a stronger form of) the folklore conjecture that the collection of maximal compact flats in SL3(Z)\SL3(R)/SO3 of volume infinity.
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TL;DR: In this article, it was shown that the GW invariants of the elliptic orbifold lines with weights (3, 3, 3), (4, 4, 2), (5, 5, 2) and (6, 6, 3 2) are quasi-modular.
Abstract: In this paper we prove that the GW invariants of the elliptic orbifold lines with weights (3,3,3), (4,4,2), and (6,3,2) are quasi-modular forms Our method is based on Givental's higher genus reconstruction formalism applied to the settings of Saito's Frobenius structures for simple elliptic singularities Our results are part of a larger project whose goal is to prove the Landau-Ginzburg/Calabi-Yau correspondence for simple elliptic singularities The correspondence describes a relation between Gromov-Witten theory (of a certain hypersurface) and Fan-Jarvis-Ruan-Witten theory (of a certain Landau-Ginzburg potential) Roughly, the main statement is that the Saito's Frobenius manifold for simple elliptic singularities has some special points such that locally near these points the Frobenius structure governs one of the two theories The local part of the correspondence is established in a companion article by M Krawitz and Y Shen, while here we describe the global picture
TL;DR: In this article, the transcendence of certain Eichler integrals associated to Eisenstein series and more generally to modular forms using functional identities due to Ramanujan, Grosswald, Weil et al.
Abstract: We study the transcendence of certain Eichler integrals associated to Eisenstein series and more generally to modular forms using functional identities due to Ramanujan, Grosswald, Weil et al. The special values of such integrals at algebraic points in the upper half-plane are linked to Riemann zeta values at odd positive integers.
TL;DR: In this article, a method of computing higher-genus amplitudes along the lines of the direct integration formalism was developed, making full use of the Seiberg-Witten curve expressed in terms of modular forms and E_8-invariant Jacobi forms.
Abstract: We study topological string amplitudes for the local half K3 surface. We develop a method of computing higher-genus amplitudes along the lines of the direct integration formalism, making full use of the Seiberg-Witten curve expressed in terms of modular forms and E_8-invariant Jacobi forms. The Seiberg-Witten curve was constructed previously for the low-energy effective theory of the non-critical E-string theory in R^4 x T^2. We clarify how the amplitudes are written as polynomials in a finite number of generators expressed in terms of the Seiberg-Witten curve. We determine the coefficients of the polynomials by solving the holomorphic anomaly equation and the gap condition, and construct the amplitudes explicitly up to genus three. The results encompass topological string amplitudes for all local del Pezzo surfaces.
Book•
8 Feb 2011TL;DR: Lozano-Robledo as mentioned in this paper gave an introductory survey of elliptic curves, modular forms, and $L$-functions, and provided the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory.
Abstract: Many problems in number theory have simple statements, but their solutions require a deep understanding of algebra, algebraic geometry, complex analysis, group representations, or a combination of all four. The original simply stated problem can be obscured in the depth of the theory developed to understand it. This book is an introduction to some of these problems, and an overview of the theories used nowadays to attack them, presented so that the number theory is always at the forefront of the discussion. Lozano-Robledo gives an introductory survey of elliptic curves, modular forms, and $L$-functions. His main goal is to provide the reader with the big picture of the surprising connections among these three families of mathematical objects and their meaning for number theory. As a case in point, Lozano-Robledo explains the modularity theorem and its famous consequence, Fermat's Last Theorem. He also discusses the Birch and Swinnerton-Dyer Conjecture and other modern conjectures. The book begins with some motivating problems and includes numerous concrete examples throughout the text, often involving actual numbers, such as 3, 4, 5, $\frac{3344161}{747348}$, and $\frac{2244035177043369699245575130906674863160948472041} {8912332268928859588025535178967163570016480830}$. The theories of elliptic curves, modular forms, and $L$-functions are too vast to be covered in a single volume, and their proofs are outside the scope of the undergraduate curriculum. However, the primary objects of study, the statements of the main theorems, and their corollaries are within the grasp of advanced undergraduates. This book concentrates on motivating the definitions, explaining the statements of the theorems and conjectures, making connections, and providing lots of examples, rather than dwelling on the hard proofs. The book succeeds if, after reading the text, students feel compelled to study elliptic curves and modular forms in all their glory.
TL;DR: In this paper, the authors extend Howard's results on the variation of Heegner points in the Hida family of f to a general quaternionic setting, and show that the size of Nekovař's extended Selmer groups with suitable big Galois representations can be computed in a uniform way.
Abstract: Given a newform f, we extend Howard’s results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekovař’s extended Selmer groups attached to suitable big Galois representations and we formulate two-variable Iwasawa main conjectures both in the definite case and in the indefinite case. Moreover, in the definite case we propose refined conjectures a la Greenberg on the vanishing at the critical points of (twists of) the L-functions of the modular forms in the Hida family of f living on the same branch as f.
TL;DR: This paper showed that affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of affine extended versions of the root systems E_8, D_6 and A_4.
Abstract: In this paper, we show that affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of affine extended versions of the root systems E_8, D_6 and A_4. We show that the induced affine extensions of the non-crystallographic groups H_4, H_3 and H_2 correspond to a distinguished subset of the Kac-Moody-type extensions considered in Dechant et al. This class of extensions was motivated by physical applications in icosahedral systems in biology (viruses), physics (quasicrystals) and chemistry (fullerenes). By connecting these here to extensions of E_8, D_6 and A_4, we place them into the broader context of crystallographic lattices such as E_8, suggesting their potential for applications in high energy physics, integrable systems and modular form theory. By inverting the projection, we make the case for admitting different number fields in the Cartan matrix, which could open up enticing possibilities in hyperbolic geometry and rational conformal field theory.
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TL;DR: In this article, the authors consider families of $q$-hypergeometric series which converge in two disjoint domains and show that these series are often equal to one another, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases.
Abstract: Ramanujan studied the analytic properties of many $q$-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious $q$-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have $q$-expansions resembling modular theta functions, is not well understood. Here we consider families of $q$-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.
TL;DR: In this paper, a generalized Witten genus for spin$^c$ manifolds was constructed, which takes values in level 1 modular forms with integral Fourier expansion on a class of spin manifolds called string$c$ manifold, and the Landweber-Stong type vanishing theorems were proven for the generalizedWitten genus.
Abstract: We construct a generalized Witten genus for spin$^c$ manifolds, which takes values in level 1 modular forms with integral Fourier expansion on a class of spin manifolds called string$^c$ manifolds. We also construct a mod 2 analogue of the Witten genus for $8k+2$ dimensional spin manifolds. The Landweber-Stong type vanishing theorems are proven for the generalizedWitten genus and the mod 2 Witten genus on string$^c$ and string (generalized) complete intersections in (product of) complex projective spaces respectively.
TL;DR: In this paper, the authors consider logarithmic vector-valued modular forms of integral weight $k$ associated with a matrix-valued Poincar\'e series associated to $k, and derive a free module of rank $p$ over the ring of classical holomorphic modular forms on the modular group.
Abstract: 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 $SL_2(\mathbb{Z})$.
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TL;DR: In this paper, the Betti numbers of moduli spaces of semi-stable sheaves were derived for N = 4 U(r) gauge theory on a Hirzebruch surface with r = 2 and 3.
Abstract: Generating functions of BPS invariants for N=4 U(r) gauge theory on a Hirzebruch surface with r=2 and 3 are computed. The BPS invariants provide the Betti numbers of moduli spaces of semi-stable sheaves. The generating functions for r=2 are expressed in terms of higher level Appell functions for a certain polarization of the surface. The level corresponds to the self-intersection of the base curve of the Hirzebruch surface. The non-holomorphic functions are determined, which added to the holomorphic generating functions provide functions which transform as a modular form.
1 Jan 2011
TL;DR: In this paper, the Torelli type theorem was used to determine the Kodaira dimension of the moduli space of polarised K3 surfaces of degree 2d, for almost all values of d.
Abstract: Following [4], we determine the Kodaira dimension of the moduli space of polarised K3 surfaces of degree 2d, for almost all values of d. Using a Torelli type theorem, the Kodaira dimension is related to the existence of certain modular forms. The construction of such forms in turn reduces to a combinatorial problem: the existence of points in the lattice E8 that are orthogonal to few roots. This problem is resolved (partially by exhaustive computer search): we conclude that the moduli space is of general type for almost all values of d, and we obtain partial information on the Kodaira dimension for some other values of d.
TL;DR: In this article, a class of eight derivative interactions in the effective action of type IIB string theory compactified on T^2 were considered and the constraints of supersymmetry were imposed to show that each of these couplings satisfy a first order differential equation on moduli space which relate it to other couplings in the same supermuliplet.
Abstract: We consider a class of eight derivative interactions in the effective action of type IIB string theory compactified on T^2. These 1/2 BPS interactions have moduli dependent couplings. We impose the constraints of supersymmetry to show that each of these couplings satisfy a first order differential equation on moduli space which relate it to other couplings in the same supermuliplet. These equations can be iterated to give second order differential equations for the various couplings. The couplings which only depend on the SO(2)\SL(2,R) moduli satisfy Laplace equation on moduli space, and are given by modular forms of SL(2,Z). On the other hand, the ones that only depend on the SO(3)\SL(3,R) moduli satisfy Poisson equation on moduli space, where the source terms are given by other couplings in the same supermultiplet. The couplings of the interactions which are charged under SU(2) are not automorphic forms of SL(3,Z). Among the interactions we consider, the R^4 coupling depends on all the moduli.
TL;DR: In this article, a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p is defined.
Abstract: We define a power series expansion of an holomorphic modular form f in the p-adic neighborhood of a CM point x of type K for a split good prime p. The modularity group can be either a classical conguence group or a group of norm 1 elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give p-adic information on the ring of definition of f. By letting the CM point x vary in its Galois orbit, the rth coefficients define a p-adic K×-modular form in the sense of Hida. By coupling this form with the p-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin–Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a p-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K.
1 Jan 2011
TL;DR: Ghate and Kumar as mentioned in this paper studied the p-adic families of modular forms and proved the Iwasawa main conjecture on the P-adic family of modular form, which they called Λ-adic forms.
Abstract: [1] M. Emerton: p-adic families of modular forms [after Hida, Coleman, and Mazur]. Seminaire Bourbaki, 2009/2010, expose 1013, Asterisque 339 (2011), 31-61. [2] H. Hida: Iwasawa modules attached to congruences of cusp forms. Ann. Scient. Ec. Norm. Sup. 4th series 19 (1986), 231-273. [3] H. Hida: Elementary Theory of L-functions and Eisenstein series. Cambridge University Press, (1993), Book. [4] D. Banerjee, E. Ghate, V.G.N Kumar: Λ-adic forms and the Iwasawa main conjecture. http: //www.math.tifr.res.in/~eghate/lectures.pdf
TL;DR: In this paper, the non-vanishing of the spanning set for the space of cuspidal modular forms of weight m ≥ 3 constructed in SL2(ℝ) was studied.
Abstract: Let Γ ⊂ SL2(ℝ) be a Fuchsian group of the first kind. In this paper, we study the non-vanishing of the spanning set for the space of cuspidal modular forms of weight m ≥ 3 constructed in [5, Corollary 2.6.11]. Our approach is based on the generalization of the non-vanishing criterion for L1-Poincare series defined for locally compact groups and proved in [6, Theorem 4.1]. We obtain very sharp bounds for the non-vanishing of the spaces of cusp forms for general Γ having at least one cusp. We obtain explicit results for congruence subgroups Γ(N), Γ0(N), and Γ1(N) (N ≥ 1).
TL;DR: In this paper, it was shown that k = 81632 is the largest weight for which all the coefficients of Fk;0(z) are non-negative, i.e.
Abstract: A cusp form f(z) of weight k for SL2(Z) is determined uniquely by its first ` := dimSk Fourier coefficients. We derive an explicit bound on thenth coefficient off in terms of its firstcoefficients. We use this result to study the non- negativity of the coefficients of the unique modular form of weightk with Fourier expansion Fk;0(z) = 1 + O(q `+1 ): In particular, we show that k = 81632 is the largest weight for which all the coef- ficients of Fk;0(z) are non-negative. This result has applications to the theory of extremal lattices.