In this paper we first prove an isomorphism between certain spaces of Jacobi forms. Using this isomorphism, we study the mod $p$ theory of Hermitian Jacobi forms over $\mathbb{Q}(i)$. We then apply the mod $p$ theory of Hermitian Jacobi forms to characterize $U(p)$ congruences and to study Ramanujan-type congruences for Hermitian Jacobi forms and Hermitian modular forms of degree $2$ over $\mathbb{Q}(i)$.
The congruences for Jacobi sums of some lower orders has been treated by many authors in the literature. In this paper we establish the congruences for Jacobi sums of order 2l^2 with odd prime l. These congruences are useful to obtain algebraic and arithmetic characterizations for Jacobi sums of order 2l^2 .
We give two congruence properties of Hermitian modular forms of degree 2 over $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. The one is a congruence criterion for Hermitian modular forms which is generalization of Sturm's theorem. Another is the well-definedness of the $p$-adic weight for Hermitian modular forms.
In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) $S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ and the binomial coefficients ${3n\choose n}$ and ${4n\choose 2n}$. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving $S_n$ and we prove a cubic residuacity criterion in terms of sums of the binomia…
There are many families of functions on partitions, such as the shifted symmetric functions, for which the corresponding q-brackets are quasimodular forms. We extend these families so that the corresponding q-brackets are quasimodular for a congruence subgroup. Moreover, we find subspaces of these families for which the q-bracket is a modular form. These results follow from the properties of Taylor coefficients of strictly meromorphic quasi-Jacobi forms around rational lattice points.
Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case $t=(-1)^{d}/16$ of the former sum, where the congruences hold modulo $p^{5-d}$.
Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant θ_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq 1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all large enough n.
We consider the genus of $20$ classes of unimodular Hermitian lattices of rank $12$ over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global Arthur parameters for the associated automorphic representations, which recover the computed Hecke eigenvalues. Congruences between Hecke eigenspaces, combined with the assumed parameters, recover known congruences for classical modular forms, and support new instances of conjectured Eisenstein congruences for $\mathbb{U}(2,2)$ automorphic forms.
We introduce a certain differential (heat) operator on the space of Hermitian Jacobi forms of degree 1, show it's commutation with certain Hecke operators and use it to construct a lift of elliptic cusp forms to Hermitian Jacobi cusp forms. We construct Hermitian Jacobi forms as the image of the tensor product of two copies of Jacobi forms and also from differentiation of the variables. We determine the number of Fourier coefficients that determine a Hermitian Jacobi form and use it to embed a certain subspace of Hermitian Jacobi forms into a direct sum of modular forms for the full modular gr…
We give a simple and a more explicit proof of a mod $4$ congruence for a series involving the little $q$-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.
We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight $k$. We will conclude by giving numerical examples for each case.
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.
We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.
We study the Taylor expansion around the point $x=1$ of a classical modular form, the Jacobi theta constant $θ_3$. This leads naturally to a new sequence $(d(n))_{n=0}^\infty=1,1,-1,51,849,-26199,\ldots$ of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of $θ_3$. We prove several results about the numbers $d(n)$ and conjecture that they satisfy the congruence $d(n)\equiv (-1)^{n-1}\ (\textrm{mod }5)$ and other similar congruence relations.
We give a congruence for L-functions coming from affine additive exponential sums over a finite field. Precisely, we give a congruence for certain operators coming from Dwork's theory. This congruence is very similar to the congruence of Manin for the characteristic polynomial of the action of Frobenius on the Jacobian of a curve defined over a finite field.
We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2 and the discriminant kernels of special orthogonal groups SO 0 (2, n), n = 3, 4, 6. The proof is based on an application of linear algebra adapted to the number theoretical needs.
We determine the ring structure of Siegel modular forms of degree g modulo a prime p, extending Nagaoka's result in the case of degree g=2. We characterize U(p) congruences of Jacobi forms and Siegel modular forms, and surprisingly find different behaviors of Siegel modular forms of even and odd degrees.
Given an integral indefinite binary Hermitian form f over an imaginary quadratic number field, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most s by f, as s tends to infinity.
In a previous work we apply lattice point theorems on hyperbolic spaces obtaining asymptotic formulas for the number of integral representations of negative integers by quadratic and hermitian forms of signature (n,1) lying in Euclidean balls of increasing radius. This formula involves an error term that depends on the first nonzero eigenvalue of the Laplace-Beltrami operator on the corresponding congruence hyperbolic manifolds. The aim of this paper is to compare the error term obtained by experimental computations, with the error term mentioned above, for several choices of quadratic and her…
The determination of Jacobi sums, their congruences and cyclotomic numbers have been the object of attention for many years and there are large number of interesting results related to these in the literature. This survey aims at reviewing results concerning the diophantine systems for finding the cyclotomic numbers and coefficients of Jacobi sums and to indicate the current status of the problem.
The aim of this paper is to deal with congruences for Jacobi sums of order $2l^{2}$ over a finite field $\mathbb{F}_{q}, q=p^{r}$, $p^{r}\equiv 1\ (mod \ 2l^{2})$, where $l>3$ and $p$ are primes. Further, we also calculate Jacobi sums $J_{2l^{2}}(1,n)$ of order $2l^{2}$ in terms of Dickson-Hurwitz sums.
We define Jacobi forms with complex multiplication. Analogous to modular forms with complex multiplication, they are constructed from Hecke characters of the associated imaginary quadratic field. From this construction we obtain a Jacobi form which specializes to $η(τ)^{26}$ which we present to highlight an open question of Dyson and Serre. We give other examples and applications of Jacobi forms with complex multiplication including constructing theta blocks associated to elliptic curves with complex multiplication and new families of congruences and cranks for certain partition functions.
The Ekedahl-Oort type is a combinatorial invariant of a principally polarized abelian variety $A$ defined over an algebraically closed field of characteristic $p > 0$. It characterizes the $p$-torsion group scheme of $A$ up to isomorphism. Equivalently, it characterizes (the mod $p$ reduction of) the Dieudonné module of $A$ or the de Rham cohomology of $A$ as modules under the Frobenius and Vershiebung operators. There are very few results about which Ekedahl-Oort types occur for Jacobians of curves. In this paper, we consider the class of Hermitian curves, indexed by a prime power $q=p^n$, wh…
In this paper we outline the Hecke theory for Hermitian modular forms in the sense of Hel Braun for arbitrary class number of the attached imaginary-quadratic number field. The Hecke algebra turns out to be commutative. Its inert part has a structure analogous to the case of the Siegel modular group and coincides with the tensor product of its $p$-components for inert primes $p$. This leads to a characterization of the associated Siegel-Eisenstein series. The proof also involves Hecke theory for particular congruence subgroups.
For $K$ an imaginary quadratic field with discriminant $-D_K$ and associated quadratic Galois character $χ_K$, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level $D_K$ and nebentypus $χ_K$ via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group $U(2,2)$ split over $K$. We generalize this (under certain conditions on $K$ and $p$) to the case of $p$-oldforms of level $pD_K$ and character $χ_K$. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space o…
Let $Γ_{F,n}$ be the Hermitian modular group of degree $n>1$ in sense of Hel Braun with respect to an imaginary quadratic field $F$. Let $r$ be a natural number. There exists a multiplier system of weight $1/r$ (equivalently a Hermitian modular form of weight $k+1/r$, $k$ integral) on some congruence group if and only if $r=1$ or $2$. This follows from a much more general construction of Deligne [De] combining it with results of Hill [Hi], Prasad [P] and Prasad-Rapinchuk [PR]. As far as we know, the systems of weight $1/2$ have not yet been described explicitly. Remarkably Haowu Wang [Wa] gave…
Jacobi forms can be considered as vector valued modular forms, and Jacobi forms of critical weight correspond to vector valued modular forms of weight $\frac12$. Since the only modular forms of weight $\frac12$ on congruence subgroups of $\SL$ are theta series the theory of Jacobi forms of critical weight is intimately related to the theory of Weil representations of finite quadratic modules. This article explains this relation in detail, gives an account of various facts about Weil representations which are useful in this context, and it gives some applications of the theory developed herein …
Let ${\mathbf F}_q$ denote a finite field of characteristic $p$ and let $n$ be an effective divisor on the affine line over ${\mathbf F}_q$ and let $v$ be a point on the affine line outside $n$. In this paper, we get congruences between ${\mathbb Q}_l$-valued weight two $v$-old Drinfeld modular forms and $v$-new Drinfeld modular forms of level $vn$. In order to do this, we shall first construct a cokernel torsion-free injection from a full lattice in the space of $v$-old Drinfeld modular forms of level $vn$ into a full lattice in the space of all Drinfeld modular forms of level $vn$. To get th…
In this paper we show how Rank-Crank type PDE's (first found by Atkin and Garvan) occur naturally in the framework of non-holomorphic Jacobiforms and find an infinite family of such differential equations. As an application we show an infinite family of congruences for odd Durfee symbols, a partition statistic introduced by George Andrews.
We attempt to generalize a congruence property of elliptic modular forms proved by Sturm to that of Haupttypus of Siegel modular forms of degree 2 with level. Namely, we give an explicit bound of Fourier coefficients required to determine the congruence of modular forms. We give the analog of Sturm's theorem for Jacobi forms, which is required in the proof. In the case that Nebentypus of Siegel modular forms with prime level, in order to prove the congruence between two modular forms, we show that it suffices to check the congruence of the finitely many Fourier coefficients. Finally, we give t…
We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular curves in question correspond to the normaliser of a Cartan subgroup of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$. By computing explicit models for these double covers we find all pairs $(N, r)$ such that there exist infinitely many $j$-invariants of elliptic curves $E/\mathbb{Q}$ which are $N$-congruent with power $r$ to a quadratic twist of $E$. We also find an …
For an imaginary quadratic field $K$ of discriminant $-D$, let $χ= χ_K$ be the associated quadratic character. We will show that the space of special hermitian Jacobi forms of level $N$ is isomorphic to the space of plus forms of level $DN$ and nebentypus $χ$ (the hermitian analogue of Kohnen's plus space) for any integer $N$ prime to $D$. This generalizes the results of Krieg from $N = 1$ to arbitrary level. Combining this isomorphism with the recent work of Berger and Klosin and a modification of Ikeda's construction we prove the existence of a lift from the space of elliptic modular forms t…
It is shown that every weak Jacobi form of weight zero and index one on a congruence subgroup of the full Jacobi group can be decomposed into $N=4$ superconformal characters. Additionally, a simple expression for the mock modular form determining the superconformal character coefficients is obtained, as well as a universal completion structure. Along the way, a useful vector-valued mock modular form is also found and studied. These results are applied to analyze some Jacobi trace functions associated to super vertex operator algebras and a distinguished sector.
We look at genera of even unimodular lattices of rank $12$ over the ring of integers of $\mathbb{Q}(\sqrt{5})$ and of rank $8$ over the ring of integers of $\mathbb{Q}(\sqrt{3})$, using Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find instances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank $12$ over the Eisenstein integers, even and unimodular over $\mathbb…
Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Appendix, V. Emery computes these volumes using Prasad's general formula. We use hyperbolic geometry in dimension 5 to describ…
Since their introduction by Andrews, generalized Frobenius partitions have interested a number of authors, many of whom have worked out explicit formulas for their generating functions in specific cases. This has uncovered interesting combinatorial structure and led to proofs of a number of congruences. In this paper, we show how Andrews' generating functions can be cast in the framework of Eichler and Zagier's Jacobi forms. This reformulation allows us to compute explicit formulas for the generalized Frobenius partition generating functions (and in fact provides formulas for further functions…
Let $λ$ be an integer, and $f(z)=\sum_{n\gg-\infty} a(n)q^n$ be a weakly holomorphic modular form of weight $λ+\frac 12$ on $Γ_0(4)$ with integral coefficients. Let $\ell\geq 5$ be a prime. Assume that the constant term $a(0)$ is not zero modulo $\ell$. Further, assume that, for some positive integer $m$, the Fourier expansion of $(f|U_{\ell^m})(z) = \sum_{n=0}^\infty b(n)q^n$ has the form \[ (f|U_{\ell^m})(z) \equiv b(0) + \sum_{i=1}^{t}\sum_{n=1}^{\infty} b(d_i n^2) q^{d_i n^2} \pmod{\ell}, \] where $d_1, \ldots, d_t$ are square-free positive integers, and the operator $U_\ell$ on formal pow…
We show how Andrews' generating functions for generalized Frobenius partitions can be understood within the theory of Eichler and Zagier as specific coefficients of certain Jacobi forms. This reformulation leads to a recursive process which yields explicit formulas for the generalized Frobenius partition generating functions in terms of infinite $q$-products. In particular, we show that specific examples of our result easily reestablish previously known formulas, and we describe new congruences, both conjectural and proven, in additional cases. The modular structure of Jacobi forms indicates t…
Let p be a prime and let a be a positive integer. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ for all d=0,...,p^a, where m is any integer not divisible by p. For example, we show that if $p\not=2,5$ then $$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}}p (mod p),$$ where F_n is the n-th Fibonacci number and (-) is the Jacobi symbol. We also prove that if p>3 then $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}k={8/9} p^2B_{p-3} (mod p^3),$$ where B_n denotes the n-th Bernoulli number.
A criterion for the validity of the Riemann hypothesis reduced the problem to the search for a certain estimate, for a hermitian form associated by means of the Weyl symbolic calculus of operators to a distribution in the plane of an arithmetic nature. One can reduce the question further to an algebraic question. After completion of the calculation of the hermitian form obtained, this attempt does not seem to lead to a genuinely new method of analysis of the conjecture. A better cooperation between usual analysis and congruence arithmetic may be called for, and some possible hints are given at…
Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $π^2$, $π^{-2}$ …
Let $k$ be a subfield of $\mathbb{C}$ which contains all $2$-power roots of unity, and let $K = k(α_{1}, α_{2}, ... , α_{2g + 1})$, where the $α_{i}$'s are independent and transcendental over $k$, and $g$ is a positive integer. We investigate the image of the $2$-adic Galois action associated to the Jacobian $J$ of the hyperelliptic curve over $K$ given by $y^{2} = \prod_{i = 1}^{2g + 1} (x - α_{i})$. Our main result states that the image of Galois in $\mathrm{Sp}(T_{2}(J))$ coincides with the principal congruence subgroup $Γ(2) \lhd \mathrm{Sp}(T_{2}(J))$. As an application, we find generator…
We find that the first Hurwitz triplet possesses two distinct arithmetic structures. As Shimura curves $X_1$, $X_2$, $X_3$, whose levels are with norm 13. As non-congruence modular curves $Y_1$, $Y_2$, $Y_3$, whose levels are 7. Both of them are defined over ${\Bbb Q}(\cos \frac{2 π}{7})$. However, for the third non-congruence modular curve $Y_3$, there exist an "exotic" duality between the associated non-congruence modular forms and the Hilbert modular forms, both of them are related to ${\Bbb Q}(e^{\frac{2 πi}{13}})$! Our results have relations and applications to modular equations of degree…
