Action Of Hecke Operators Research Articles

Bianchi period polynomials: Hecke action and congruences

Let Γ=PSL2(O)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma ={{\\,\\mathrm{\ extrm{PSL}}\\,}}_2(\\mathcal {O})$$\\end{document} be a Bianchi group associated to one of the five Euclidean imaginary quadratic fields. We show that the space of weight k period polynomials for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} is “dual” to the space of weight k modular symbols for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}, reflecting the duality between the first and second cohomology groups of the arithmetic group Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. Using this result, we describe the action of Hecke operators on the space of period polynomials for Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} via the Heilbronn matrices. As in the classical case, spaces of Bianchi period polynomials are related to parabolic cohomology of Bianchi groups, and in turn, via the Eichler–Shimura–Harder Isomorphism, to spaces of Bianchi modular forms. In the second part of the paper, we numerically investigate congruences between level 1 Bianchi eigenforms via computer programs which implement the above mentioned Hecke action on spaces of Bianchi period polynomials. Computations with the Hecke action are used to indicate moduli of congruences between the underlying Bianchi forms; we then prove the congruences using the period polynomials. From this we find congruences between genuine Bianchi modular forms and both a base-change Bianchi form and an Eisenstein series. We believe these congruences are the first of their kind in the literature.

A note on some $p$-adic analytic Hecke actions

We show that the action of Hecke operators away from p on the space of ( p -adic) overconvergent modular forms is ( p -adically) locally analytic in a certain sense. As a corollary, the action of the Hecke algebra can be extended naturally to an action of rigid functions on its generic fiber. This directly determines the Hodge–Tate–Sen weights of Galois representation associated to an overconvergent eigenform and confirms a conjecture of Gouvêa.

Moments of ranks and cranks, and quotients of Eisenstein series and the Dedekind eta function

Atkin and Garvan introduced the functions Nk(n) and Mk(n), which denote the k-th moments of ranks and cranks in the theory of partitions. Let e2r(n) be the n-th Fourier coefficient of E2r(τ)/η(τ), where E2r(τ) is the classical Eisenstein series of weight 2r and η(τ) is the Dedekind eta function. Via the theory of quasi-modular forms, we find that for k≤5, Nk(n) and Mk(n) can be expressed using e2r(n) (0≤r≤k), p(n) and N2(n). For k>5, additional functions are required for such expressions. For r∈{2,3,4,5,7}, by studying the action of Hecke operators on E2r(τ)/η(τ), we provide explicit congruences modulo arbitrary powers of primes for e2r(n). Moreover, for ℓ∈{5,7,11,13} and any k≥1, we present uniform methods for finding nice representations for ∑n=0∞e2r(ℓkn+124)qn, which work for every r≥2. These representations allow us to prove congruences modulo powers of ℓ, and we have done so for e4(n) and e6(n) as examples. Based on the congruences satisfied by e2r(n), we establish congruences modulo arbitrary powers of ℓ for the moments and symmetrized moments of ranks and cranks as well as higher order spt-functions.

Hecke operators acting on optimal embeddings in indefinite quaternion algebras

We explore a natural action of Hecke operators acting on formal sums of optimal embeddings of real quadratic orders into Eichler orders. By associating an optimal embedding to its root geodesic on the corresponding Shimura curve, we can consider the signe

A supplement to “On the Fourier coefficients of Hilbert modular forms of half-integral weight over arbitrary algebraic number fields, Tsukuba J. Math. 37(2013), 1-11”

The purpose of this note is to prove that the Shintani lift of Hilbert modular forms over algebraic number fields commutes with the action of Hecke operators. We show our assertion using the commutativity of the Shimura lift with Hecke operators and some properties of adjoint mappings. This commutativity of the Shintani lift plays an essential role for the proof of the Waldspurger-type theorem concerning the Fourier coefficients of modular forms of half-integral weight over arbitrary algebraic number fields.

HECKE OPERATORS AND THE COHERENT COHOMOLOGY OF SHIMURA VARIETIES

AbstractWe consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

Symmetric orbifold theories from little string residues

We study a class of Little String Theories (LSTs) of A type, described by $N$ parallel M5-branes spread out on a circle and which in the low energy regime engineer supersymmetric gauge theories with $U(N)$ gauge group. The BPS states in this setting correspond to M2-branes stretched between the M5-branes. Generalising an observation made in arXiv:1706.04425, we provide evidence that the BPS counting functions of special subsectors of the latter exhibit a Hecke structure in the Nekrasov-Shatashvili (NS) limit, i.e. the different orders in an instanton expansion of the supersymmetric gauge theory are related through the action of Hecke operators. We extract $N$ distinct such reduced BPS counting functions from the full free energy of the LST with the help of contour integrals with respect to the gauge parameters of the $U(N)$ gauge group. Physically, the states captured by these functions correspond to configurations where the same number of M2-branes is stretched between some of these neighbouring M5-branes, while the remaining M5-branes are collapsed on top of each other and a particular singular contribution is extracted. The Hecke structures suggest that these BPS states form the spectra of symmetric orbifold CFTs. We furthermore show that to leading instanton order (in the NS-limit) the reduced BPS counting functions factorise into simpler building blocks. These building blocks are the expansion coefficients of the free energy for $N=1$ and the expansion of a particular function, which governs the counting of BPS states of a single M5-brane with single M2-branes ending on it on either side. To higher orders in the instanton expansion, we observe new elements appearing in this decomposition, whose coefficients are related through a holomorphic anomaly equation.

Hecke-Langlands Duality and Witten’s Gravitational Moonshine

The purpose of this research is to give a dual description of conformal blocks of d=2 rational CFT (conformal field theory) in terms of Hecke eigenforms and eigensheaves. In particular, partition functions, conformal characters and lattice theta functions may be reconstructed from the action of Hecke operators. This method can be applied to: 1) rings of integers of Galois number fields equipped with the trace (or anti-trace) form; 2) root lattices of affine Kac-Moody algebras and WZW-models; 3) minimal models of Belavin-Polyakov-Zamolodchikov and related d=2 spin-chain/lattice models; 4) vertex algebras of Leech and Niemeier lattices and others. We also use the original Witten’s idea to construct the 3-dimensional quantum gravity as the AdS/CFT-dual of c=24 Monster vertex algebra of Frenkel-Lepowsky- Meurman. Concerning the geometric Langlands duality, we use results of Beilinson-Drinfeld, Frenkel-Ben-Zvi, Gukov-Kapustin-Witten and many others (cf. references). The main new result in this paper is the construction of number-theoretical lattice vertex superalgebras in Section 5 and applications to conformal field theories and quantum gravity.

The Hecke algebra action and the Rezk logarithm on Morava E-theory of height 2

Given a one-dimensional formal group of height 2, let E be the Morava E-theory spectrum associated to its universal deformation over the Lubin-Tate ring. By computing with moduli spaces of elliptic curves, we give an explicitation for an algebra of Hecke operators acting on E-cohomology. This leads to a vanishing result for Rezk's logarithmic cohomology operation on the units of E. It identifies a family of elements in the kernel with meromorphic modular forms whose Serre derivative is zero. Our calculation finds a connection to logarithms of modular units. In particular, we work out an action of Hecke operators on certain "logarithmic" q-series, in the sense of Knopp and Mason, that agrees with our vanishing result and extends the classical Hecke action on modular forms.

Universal Fourier expansions of Bianchi modular forms

We generalize Merel’s work on universal Fourier expansions to Bianchi modular forms over Euclidean imaginary quadratic fields, under the assumption of the nondegeneracy of a pairing between Bianchi modular forms and Bianchi modular symbols. Among the key inputs is a computation of the action of Hecke operators on Manin symbols, building upon the Heilbronn–Merel matrices constructed by Mohamed.

Counting zeros in quaternion algebras using Jacobi forms

We use the theory of Jacobi forms to study the number of elements in a maximal order of a definite quaternion algebra over the field of rational numbers whose characteristic polynomial equals a given polynomial. A certain weighted average of such numbers equals (up to some trivial factors) the Hurwitz class number H ( 4 n − r 2 ) H(4n-r^2) . As a consequence we obtain new proofs for Eichler’s trace formula and for formulas for the class and type number of definite quaternion algebras. As a secondary result we derive explicit formulas for Jacobi Eisenstein series of weight 2 2 on Γ 0 ( N ) \Gamma _0(N) and for the action of Hecke operators on Jacobi theta series associated to maximal orders of definite quaternion algebras.

Hyper-Algebras of Vector-Valued Modular Forms

We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series.

ON KOHNEN PLUS-SPACE OF JACOBI FORMS OF HALF INTEGRAL WEIGHT OF MATRIX INDEX

We introduce a plus-space of Jacobi forms, which is a certain subspace of Jacobi forms of half-integral weight of matrix index. This is an analogue to the Kohnen plus-space in the framework of Jacobi forms. We shall show a linear isomorphism between the plus-space of Jacobi forms and the space of Jacobi forms of integral weight of certain matrix index. Moreover, we shall show that this linear isomorphism is compatible with the action of Hecke operators of both spaces. This result is a kind of generalization of Eichler-Zagier-Ibukiyama correspondence, which is an isomorphism between the generalized plus-space of Siegel modular forms of general degree and Jacobi forms of index $1$ of general degree.

Hecke eigenvalues and relations for Siegel Eisenstein series of arbitrary degree, level, and character

We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalize the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalize the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.

Galois representations attached to tensor products of arithmetic cohomology

We compute the action of Hecke operators on tensor products of cohomology classes of lower congruence subgroups of SL(n,Z) in trivial weight. We use this computation to prove that if each representation in a collection of Galois representations is attached to a cohomology class of a lower congruence subgroup in trivial weight, then a sum of certain twists of the representations consistent with the main conjectures of [5,10] is also attached to such a cohomology class.

P-Adic properties of coefficients of certain half-integral weight modular forms

In this paper, we study the parallel cases of Zagier's and Folsom–Ono's grids of weakly holomorphic (resp. weakly holomorphic and mock modular) forms of weights 3/2 and 1/2, investigating their p-adic properties under the action of Hecke operators.

Homological stability for completed homology

We prove that the completed homology groups of \(\mathrm {GL}_n(\mathbf {Z})\) in fixed degree stabilize as N goes to infinity. We also prove that the action of Hecke operators on stable cohomology is trivial, in a precisely defined sense.

Rational period functions in higher level cases

Extending Knopp's results [9,10] we investigate examples and properties of rational period functions in higher level cases, including location of poles and behavior under the action of Hecke operators. More precisely, we prove that a rational period function may have poles only at 0 or at real quadratic irrationalities. Moreover by applying the action of Hecke operators we prove that for positive odd integer k and p∈{2,3}, the space of all rational period functions of weight 2k for Γ0+(p) is infinite dimensional.

On the action of Hecke operators on Drinfeld modular forms

In this paper we determine the explicit structure of the semisimple part of the Hecke algebra that acts on Drinfeld modular forms of full level modulo T. We show that modulo T the Hecke algebra has a non-zero semisimple part. In contrast, a well-known theorem of Serre asserts that for classical modular forms the action of Tℓ for any odd prime ℓ is nilpotent modulo 2. After proving the result for Drinfeld modular forms modulo T, we use computations of the Hecke action modulo T to show that certain powers of the Drinfeld modular form h cannot be eigenforms. Finally, we pose a question a positive answer to which will mean that the Hecke algebra that acts on Drinfeld modular forms of full level is not smooth for large weights, which again contrasts the classical situation.

A WEIGHT INDEPENDENCE RESULT FOR QUATERNIONIC HECKE ALGEBRAS

Let p be a prime and B be a quaternion algebra indefinite over Q and ramified at p. We consider the space of quaternionic modular forms of weight k and level p∞, endowed with the action of Hecke operators. By using cohomological methods, we show that the p-adic topological Hecke algebra does not depend on the weight k. This result provides a quaternionic version of a theorem proved by Hida for classical modular forms; we discuss the relationship of our result to Hida's theorem in terms of Jacquet–Langlands correspondence.