Modular Form

Using maps due to Ozeki and Broué-Enguehard between graded spaces of invariants for certain finite groups and the algebra of modular forms of even weight we equip these invariants spaces with a differential operator which gives them the structure of a Rankin-Cohen algebra. A direct interpretation of the Rankin-Cohen bracket in terms of transvectant for the group SL(2, C) is given.

This paper treats in detail the life and work of Otto Blumenthal, one of the most tragic figures of the 188 emigré mathematicians from Germany and the Nazi-occupied continent. Blumenthal, the first doctoral student of David Hilbert, was crucial in the publication and communication system of German mathematics between the two World Wars. There has been an unusual revival of interest in his mathematical work in the last three decades. Thus his work on orthogonal polynomials whose zeros are dense in intervals, called the Blumenthal theorem by T.S. Chihara (1972), lead to over two dozen recent papers in the field. The Blumenthal-Nevai theorem, with applications to scattering theory in physics, is one example. In modern work on Hilbert modular forms, increasingly being called Hilbert-Blumenthal modular forms, many recent papers even contain the word Blumenthal in their titles. This paper contains 212 references.

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of Qp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2.

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over Z with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diamond Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of Jacobians of modular curves. Contents §1. Introduction §2. Cubic structures and categories §3. The main calculation and result §4. Galois structure of modular forms §5. An equivariant Birch and Swinnerton-Dyer relation 1 Corollary 1.5. Suppose θ 2 [P χ ] = 0 in Cl(Z[ζ r , 1 2 ]). Then at least one of s χ ([J ′ H (Q)]) or s χ ([X(J H )]) is non-trivial if the Birch Swinnerton-Dyer conjecture of §5 holds. Suppose in addition that C(p) = J 1 (p)(Q) tor , as conjectured in [CES]. Then either the χ-eigenspace of

The present notes are the expanded and polished version of three lectures given in Stanford, concerning the analytic and arithmetic properties of weight one modular forms. The author tried to write them in a style accessible to non-analytically oriented number theoritists: in particular, some effort is made to be precise on statements involving uniformity in the parameters. On the other hand, another purpose was to provide an introduction, together with a set of references, consciously kept small, to the realm of Galois representations, for non-algebraists -- like the author. The proofs are sketched, at best, but we tried to motivate the results, and to relate them to interesting conjectures.

The present notes are the expanded and polished version of three lectures given in Stanford, concerning the analytic and arithmetic properties of weight one modular forms. The author tried to write them in a style accessible to non-analytically oriented number theoritists: in particular, some effort is made to be precise on statements involving uniformity in the parameters. On the other hand, another purpose was to provide an introduction, together with a set of references, consciously kept small, to the realm of Galois representations, for non-algebraists -- like the author. The proofs are sketched, at best, but we tried to motivate the results, and to relate them to interesting conjectures.

A hydrological forecasting model is presented that attempts to combine the important distributed effects of channel network topology and dynamic contributing areas with the advantages of simple lumped parameter basin models. Quick response flow is predicted from a storage/contributing area relationship derived analytically from the topographic structure of a unit within a basin. Average soil water response is represented by a constant leakage infiltration store and an exponential subsurface water store. A simple non-linear routing procedure related to the link frequency distribution of the channel network completes the model and allows distinct basin sub-units, such as headwater and sideslope areas to be modelled separately. The model parameters are physically based in the sense that they may be determined directly by measurement and the model may be used at ungauged sites. Procedures for applying the model and tests with data from the Crimple Beck basin are described. Using only measured and estimated parameter values, without optimization, the model makes satisfactory predictions of basin response. The modular form of the model structure should allow application over a range of small and medium sized basins while retaining the possibility of including more complex model components when suitable data are available. Un modèle à base physique de zone d'appel variable de l'hydrologie du bassin versant

We give new examples of noncongruence subgroups Γ ⊂ SL2(Z) whose space of weight 3 cusp forms S3(Γ) admits a basis satisfying the Atkin-Swinnerton-Dyer congruence relations with respect to a weight 3 newform for a certain congruence subgroup.

The moduli space of (1, 3)-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y , respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common thirdétale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group Γ 1 (6). By Tate's conjecture, this should imply that Y , the fibred square of the universal elliptic curve S 1 (6), and Verrill's rigid Calabi-Yau Z A3 , which all have the same L-function, are in correspondence over Q. We show that this is indeed the case by giving explicit maps.

We describe the graded ring of symmetric Hermitian modular forms of even weights and degree 2 over Q ð ffiffiffiffiffiffi ffi À2 p Þ in terms of generators and relations. All the 8 generators of weight up to 12 are MaaX lifts and some of them can also be obtained from Borcherds products. Moreover, we construct generators for the module over this ring consisting of all Hermitian modular forms with respect to the commutator subgroup. As an application the field of Hermitian modular functions over Q ð ffiffiffiffiffiffi ffi À2 p Þ is determined. Finally, we construct 5 algebraically independent symmetric Hermitian modular forms in terms of theta series. r 2004 Elsevier Inc. All rights reserved.

Let p(n) denote the number of overpartitions of n. Recently, Fortin-Jacob-Mathieu and Hirschhorn-Sellers independently obtained 2-, 3-and 4-dissections of the generating function for p(n) and derived a number of congruences for p(n) modulo 4, 8 and 64 including p(5n + 2) ≡ 0 (mod 4), p(4n + 3) ≡ 0 (mod 8) and p(8n + 7) ≡ 0 (mod 64). By employing dissection techniques, Yao and Xia obtained congruences for p(n) modulo 8, 16 and 32, such as p(48n + 26) ≡ 0 (mod 8), p(24n + 17) ≡ 0 (mod 16) and p(72n+69) ≡ 0 (mod 32). In this paper, we give a 16-dissection of the generating function for p(n) modulo 16 and we show that p(16n + 14) ≡ 0 (mod 16) for n ≥ 0. Moreover, by using the 2-adic expansion of the generating function of p(n) due to Mahlburg, we obtain that p(ℓ 2 n + rℓ) ≡ 0 (mod 16), where n ≥ 0, ℓ ≡ −1 (mod 8) is an odd prime and r is a positive integer with ℓ ∤ r. In particular, for ℓ = 7, we get p(49n + 7) ≡ 0 (mod 16) and p(49n+14) ≡ 0 (mod 16) for n ≥ 0. We also find four congruence relations: p(4n) ≡ (−1) n p(n) (mod 16) for n ≥ 0, p(4n) ≡ (−1) n p(n) (mod 32) for n being not a square of an odd positive integer, p(4n) ≡ (−1) n p(n) (mod 64) for n ≡ 1, 2, 5 (mod 8) and p(4n) ≡ (−1) n p(n) (mod 128) for n ≡ 0 (mod 4).

This is a survey article about Siegel modular varieties over the complex numbers. It is written mostly from the point of view of moduli of abelian varieties, especially surfaces. We cover compactification of Siegel modular varieties; classification of the compactified varieties by Kodaira dimension, etc.; moduli of abelian surfaces and especially applications of the lifting of Jacobi forms to modular

Introduction Reminders on abstract algebraic geometry The setting Linear and commutative algebra in a symmetric monoidal model category Geometric stacks Infinitesimal theory Higher Artin stacks (after C. Simpson) Derived algebraic geometry: D −-stacks Complicial algebraic geometry: D-stacks Brave new algebraic geometry: S-stacks Relations with other works Acknowledgments Notations and conventions Part 1. General theory of geometric stacks Chapter 2.4. Brave new algebraic geometry 2.4.1. Two HAG contexts 2.4.2. Elliptic cohomology as a Deligne-Mumford S-stack Appendix A. Classifying spaces of model categories Appendix B. Strictification

We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.

We determine the space of 1-point correlation functions associated with the Moonshine module: they are precisely those modular forms of non-negative integral weight which are holomorphic in the upper half plane, have a pole of order at most 1 at infinity, and whose Fourier expansion has constant 0. There are Monster-equivariant analogues in which one naturally associates to each element in the Monster a modular form of fixed weight k, the case k=0 corresponding to the original ``Moonshine'' of Conway and Norton.

The existence of l-adic Galois representations attached to Hecke eigenforms entail congruence properties satisfied by their Fourier coefficients [SwD]. These in turn imply congruences for the Fourier coefficients of arbitrary integral weight modular forms, which may be extended to ...

Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These non-holomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outline the general facets of the theory, and we give several applications to number theory: partitions and q-series, modular forms, singular moduli, Borcherds products, extensions of theorems of Kohnen-Zagier and Waldspurger on modular L-functions, and the work of Bruinier and Yang on Gross-Zagier formulae. What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery.

A systolic array implementation of block-based Hopfield neural network architecture using completely digital circuits is presented in this paper. The design is based on modelling the energy equation of Hopfield neural network to a systolic (or modular) form. It is shown mathematically that the modified energy equation converges in all circumstances. In addition, it is shown that the architecture provides massive parallelism and can be extended to a larger network by cascading identical chips. The performance of the proposed architecture is evaluated by applying various binary inputs and it is observed that it exhibits the same characteristics of block-based Hopfield neural network in terms of convergence and pattern association. q

We define a family of Coleman maps for positive crystalline p-adic representations of the absolute Galois group of Qp using the theory of Wach modules. Let f = anq n be a normalized new eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f , we define Coleman maps Col i for i = 1, 2 with values in Q p ⊗ Zp Λ, where Λ is the Iwasawa algebra of Z × p . Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case ap = 0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a Λ-cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.

In this note, we comment on Calabi-Yau spaces with Hodge numbers h1,1 = 3 and h2,1 = 243. We focus on the Calabi-Yau space WP1,1,2,8,12 (24) and show how some of its instanton numbers are related to coefficients of certain modular forms. We also comment on the relation of four-dimensional exchange symmetries in certain N = 2 dual models to six-dimensional heterotic/heterotic string duality.

For integers k≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2-k. The operator ξ_2-k (resp. D^k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are "dual" under ξ_2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D^k-1.

For each prime ℓ, let |·|_ℓ be an extension to of the usual ℓ-adic absolute value on . Suppose g(z) = ∑_n=0^∞ c(n)q^n ∈ M_k+(N) is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but finitely many primes ℓ there are infinitely many square-free integers m for which |c(m)|_ℓ = 1. Consequently we obtain indivisibility results for "algebraic parts" of central critical values of modular L-functions and class numbers of imaginary quadratic fields. These results partially answer a conjecture of Kolyvagin regarding Tate-Shafarevich groups of modular elliptic curves. Similar results were obtained earlier by Jochnowitz by a completely different method. Our method uses standard facts about Galois representations attached to modular forms, and pleasantly uncovers surprising Kronecker-style congruences for L-function values. For example if Δ(z) is Ramanujan's cusp form and g(z)=∑_n=1^∞c(n)q^n is the cusp form for which L(Δ_D,6)=()πD^6...

Let f be a modular eigenform of even weight k ≥ 2 and new at a prime p dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D F M f and an L-invariant L F M f . The first goal of this paper is building a suitable p-adic integration theory that allows us to construct a new monodromy module D f and L-invariant L f , in the spirit of Darmon. We conjecture both monodromy modules are isomorphic, and in particular the two L-invariants are equal.

This paper is devoted to the construction of nonconforming finite elements for the discretization of fourth order elliptic partial differential operator in three spatial dimensions. The newly constructed elements include two tetrahedron nonconforming finite elements and one quasi-conforming tetrahedron element. These elements are all proved to be convergent for a model biharmonic equation in three dimensions. In particular, the quasi-conforming tetrahedron element is a modified Zienkiewicz element while the non-modified Zienkiewicz element (a tetrahedral element of Hermite type) is proved to be divergent on a special but regular grid.

We compare the spaces of Hermitian Jacobi forms (HJF) of weight $k$ and indices $1,2$ with classical Jacobi forms (JF) of weight $k$ and indices $1,2,4$. Using the embedding into JF, upper bounds for the order of vanishing of HJF at the origin is obtained. We compute the rank of HJF as a module over elliptic modular forms and prove the algebraic independence of the generators in case of index 1. Some related questions are discussed.

The paper attempts to model the strength of TRIP-assisted steels using fuzzy inference system (FIS). The system is proficient to cope up with the changing environment deterministically due to its inherent modular structure and distributed approach creating a flexible and robust system. The problem domain has been divided into several sub-problems and autonomous agents are employed to execute the tasks and decision making by designing respective FIS. The mechanistic constituents of TRIP process are expressed in modular form using linguistic if-then rules developed from the fundamental physical metallurgy concept. Coordination among the distributed agents in the form of sharing knowledge has been attempted to divide the whole mechanism of TRIP in separate modules to make the system fault tolerant. The model is validated by data collected from published literature and found to be sound enough, considering the limitations of a rule-based model.

Let π be a regular algebraic cuspidal automorphic representation of GL 2 over an imaginary quadratic number field K, and let ℓ be a prime number. Assuming the central character of π is invariant under the non-trivial automorphism of K, it is shown that there is a continuous irreducible ℓ-adic representation ρ of Gal(K/K) such that L(s, ρv) = L(s, πv) whenever v is a prime of K outside an explicit finite set.

Euler proved the following recurrence for p(n), the number of partitions of an integer n : (1) p(n) + ∞ X k=1 (−1) k (p(n − ω(k)) + p(n − ω(−k))) = 0 for ω(k) = 3k 2 +k 2. Using the Jacobi Triple Product identity we show analogues of Euler's recurrence formula for common restricted partition functions. Moreover following Kolberg, these recurrences allow us to determine that these partition functions are both even and odd infinitely often. Using the theory of modular forms, these recurrences may be viewed as infinite product identities involving Dedekind's η-function. Specifically, if the generating function for an arithmetical function is a modular form, then one often obtains analogous recurrence formulas; in particular here we get recurrence relations involving the number of t-core partitions, the number of representations of sums of squares, certain divisor functions, the number of points in finite fields on certain elliptic curves with complex multiplication, the Ramanujan τ −function and some appropriate analogs. In some cases recurrences hold for almost all n, and in others these recurrences hold for all n where the equality is replaced by a congruence mod m for any fixed integer m. These new recurrences are consequences of some of the theory of modular forms as developed by Deligne, Ribet, Serre, and Swinnerton-Dyer. p(n) ∼ 1 4n √ 3 e π √ 2n 3

We propose a program for counting microstates of four-dimensional BPS black holes in N >= 2 supergravities with symmetric-space valued scalars by exploiting the symmetries of timelike reduction to three dimensions. Inspired by the equivalence between the four dimensional attractor flow and geodesic flow on the three-dimensional scalar manifold, we radially quantize stationary, spherically symmetric BPS geometries. Connections between the topological string amplitude, attractor wave function, the Ooguri-Strominger-Vafa conjecture and the theory of automorphic forms suggest that black hole degeneracies are counted by Fourier coefficients of modular forms for the three-dimensional U-duality group, associated to special "unipotent" representations which appear in the supersymmetric Hilbert space of the quantum attractor flow.

The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H 3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP 2 and IP 1  ×  IP 1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold \({{\mathbb {C}^3} / {\mathbb {Z}_3}}\) .

Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the level, with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an L-invariant L_FM(f). The first goal of this paper is building a suitable p-adic integration theory that allows us

The name "K3 surfaces" was coined by A. Weil in 1957 when he formulated a research programme for these surfaces and their moduli. Since then, irreducible holomorphic symplectic manifolds have been introduced as a higher dimensional analogue of K3 surfaces. In this paper we present a review of this theory starting from the definition of K3 surfaces and going as far as the global Torelli theorem for irreducible holomorphic symplectic manifolds as recently proved by M. Verbitsky. For many years the last open question of Weil's programme was that of the geometric type of the moduli spaces of polarised K3 surfaces. We explain how this problem has been solved. Our method uses algebraic geometry, modular forms and Borcherds automorphic products. We collect and discuss the relevant facts from the theory of modular forms with respect to the orthogonal group O(2,n). We also give a detailed description of quasi pull-back of automorphic Borcherds products. This part contains previ...

Phage K is a polyvalent phage of the Myoviridae family which is active against a wide range of staphylococci. Phage genome sequencing revealed a linear DNA genome of 127,395 bp, which carries 118 putative open reading frames. The genome is organized in a modular form, encoding modules for lysis, structural proteins, DNA replication, and transcription. Interestingly, the structural module shows high homology to the structural module from Listeria phage A511, suggesting intergenus horizontal transfer. In addition, phage K exhibits the potential to encode proteins necessary for its own replisome, including DNA ligase, primase, helicase, polymerase, RNase H, and DNA binding proteins. Phage K has a complete absence of GATC sites, making it insensitive to restriction enzymes which cleave this sequence. Three introns (lys-I1, pol-I2, and pol-I3) encoding putative endonucleases were located in the genome. Two of these (pol-I2 and pol-I3) were found to interrupt the DNA polymerase gene, while the other (lys-I1) interrupts the lysin gene. Two of the introns encode putative proteins with homology to HNH endonucleases, whereas the other encodes a 270-amino-acid protein which contains two zinc fingers (CX 2 CX 22 CX 2 C and CX 2 CX 23 CX 2 C). The availability of the genome of this highly virulent phage, which is active against infective staphylococci, should provide new insights into the biology and evolution of large broad-spectrum polyvalent phages.