On the vanishing order of Jacobi forms at infinity

p-adic Aspects of Jacobi Forms

We use the relationship between Jacobi forms and vector‐valued modular forms to study the Fourier expansions of Jacobi forms of indexes p, p2, and pq for distinct odd primes p, q. Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector‐valued modular form. However, in the case of indexes of the form pq or p2, there are restrictions on which of the components will uniquely determine the form. Moreover, for indexes of the form p, this note gives an explicit reconstruction of the entire Jacobi form from a single associated vector‐valued modular form component. That is, we show how to start with a single associated vector component and use specific matrices from Sl2(ℤ) to find the other components and hence the entire Jacobi form. These results are used to discuss the possible modular forms of half‐integral weight associated to the Jacobi form for different subgroups.

We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface. Let S be a K3 surface and let Hilb(S) be the Hilbert scheme of d points of S. In case of elliptically fibered K3 surfaces S → P1, we calculate genus 0 Gromov-Witten invariants of Hilb(S), which count rational curves incident to two generic fibers of the induced Lagrangian fibration Hilb(S)→ Pd. The generating series of these invariants is the Fourier expansion of a product of Jacobi theta functions and modular forms, hence of a Jacobi form. The result is a generalization of the classical Yau-Zaslow formula which relates the number of rational curves on a K3 surface to the modular discriminant. We also prove results for genus 0 Gromov-Witten invariants of Hilb(S) for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov-Witten invariants of the Hilbert scheme of 2 points of P1 × E, where E is an elliptic curve. Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on Hilb(S) with respect to primitive classes. The conjecture is presented in terms of natural operators acting on the Fock space of S. We prove the conjecture in the first non-trivial case Hilb(S). As a corollary, the full genus 0 Gromov-Witten theory of Hilb(S) in primitive classes is governed by Jacobi forms. We state three applications of our results. First, in joint work with R. Pandharipande, a conjecture counting the number of maps from a fixed elliptic curve to Hilb(S) is presented. The result, summed over all d, is expressed in terms of the reciprocal of a Siegel modular form, the Igusa cusp form χ10. Second, we give a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through 2 general points. Third, we discuss a relationship between the Jacobi forms appearing in curve counting on Hilb(S) and the moduli space of holomorphic symplectic varieties.

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 ℚ ⁢ ( i ) {\mathbb{Q}(i)} . We then apply the mod p theory of Hermitian Jacobi forms to characterize U ⁢ ( p ) {U(p)} congruences and to study Ramanujan-type congruences for Hermitian Jacobi forms and Hermitian modular forms of degree 2 over ℚ ⁢ ( i ) {\mathbb{Q}(i)} .

We investigate $W(E_8)$-invariant Jacobi forms which are the Jacobi forms invariant under the action of the Weyl group of the root system $E_8$. This type of Jacobi forms has applications in mathematics and physics, but very little has been known about its structure. In this paper we show that the bigraded ring of weak $W(E_8)$-invariant Jacobi forms is not a polynomial algebra over $C$ and prove that every $W(E_8)$-invariant Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent holomorphic Jacobi forms introduced by Sakai with coefficients which are meromorphic $SL_2(Z)$ modular forms. The latter result implies that the graded ring of weak $W(E_8)$-invariant Jacobi forms of fixed index is a free module over the ring of $SL_2(Z)$ modular forms and the number of generators can be calculated by a generating series. We also determine and construct all generators of small index. These results extend Wirthm\{u}ller's theorem proved in 1992 to the last open case.

In this paper, we revisit an earlier conjecture by one of us that related conjugacy classes of M12 to Jacobi forms of weight zero and index one. We construct Jacobi forms for all conjugacy classes of M12 that are consistent with constraints from group theory as well as modularity. However, we obtain 1427 solutions that satisfy these constraints (to the order that we checked) and are unable to provide a unique Jacobi form. Nevertheless, as a consequence, we are able to provide a group theoretic proof of the evenness of the coefficients of all EOT Jacobi forms associated with conjugacy classes of M12:2⊂M24. We show that there exists no solution where the Jacobi forms (for order 4/8 elements of M12) transform with phases under the appropriate level. In the absence of a moonshine for M12, we show that there exist moonshines for two distinct L2(11) sub-groups of the M12. We construct Siegel modular forms for all L2(11) conjugacy classes and show that each of them arises as the denominator formula for a distinct Borcherds–Kac–Moody Lie superalgebra.

By deriving Bol's type result, we show how to construct a Jacobi form with different weight from the given Jacobi form. We also show how this analogous theory related to the classical theory of Bol[1] by using the theta-series expansion of the Jacobi form. Furthermore, the partial converse of Bol's result involving the periods of modular forms reveals a connection between Jacobi forms and periods of modular forms.

We study a necessary and sufficient condition for Jacobi integrals of weight \(-r+\frac{j}{2}\), r∈ℤ≥0, and index ℳ(j) on ℋ×ℂj to have a dual Jacobi form of weight \(r+\frac{j}{2}+2\) and index ℳ(j). Such a meromorphic Jacobi integral with a dual Jacobi form is called a mock Jacobi form whose concept was first introduced by Zagier in Seminaire Bourbaki, 60eme annee, 2006–2007, N° 986. In fact, we show the map \(L^{r+1}_{\mathcal{M}^{(j)}}\) from the space of mock Jacobi forms to that of Jacobi forms is surjective by constructing the corresponding inverse image via Eichler integral of vector valued modular forms which are coming from the theta decomposition of Jacobi forms. We discuss Lerch sums as a typical example.

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito-Kurokawa conjecture. Later Skoruppa introduced skew-holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between Jacobi forms and skew-holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on and the space of skew-holomorphic Jacobi cusp forms on with the same half-integral weight to the Eichler cohomology group of with a coefficient module coming from polynomials.

We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain Seiberg-Witten curves of type E_6 and E_7 for the E-string theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthm\"uller.

We define Jacobi forms over a totally real algebraic number field K and construct examples by first embedding the group and the space into the symplectic group and the symplectic upper half space respectively. Then symplectic modular forms are created and Jacobi forms arise by taking the appropriate Fourier coefficients. Also some known relations of Jacobi forms to vector valued modular forms over rational numbers are extended to totally real fields.

Multilinear Operators on Siegel Modular Forms of Genus 1 and 2

Jacobi forms with CM and applications

Doi and Naganuma (see [6]) constructed a lifting map from elliptic modular forms to Hilbert modular forms in the case of a real quadratic field with narrow class number one. A Converse Theorem for Hilbert modular forms was one of their basic tools. This gives rise to the question of constructing a lifting map in the case of Jacobi forms. Here we do the first step in this direction and prove a Converse Theorem for Hilbert-Jacobi forms. Studying the connection between functions that satisfy certain transformation laws and the functional equation of their associated L-functions has value on its own and a long history. In a celebrated paper (see [9]), Hecke showed that the automorphy of a cusp form with respect to SL2(Z) is equivalent to the functional equation of its associated L-functions. That only one functional equation is needed is in a way atypical and highly depends on the fact that SL2(Z) is generated by the matrices ( 1 1 0 1 ) and ( 0 −1 1 0 ). This situation already changes if one considers cusp forms with respect to a subgroup of SL2(Z) which have a character. In this case the functional equation of twists is required (see [18]). Hecke’s work has inspired an astonishing number of people and a lot of generalizations of his “Converse Theorem” have been made, e.g. generalizations to Hilbert modular forms as mentioned above (see [6]), Siegel modular forms (see [1], [10]) or Jacobi forms (see [14],[15]). Maass showed an analogue of Hecke’s result for nonholomorphic modular forms (see [13]). He proved that these correspond to certain L-functions in quadratic fields. An outstanding generalization of a Converse Theorem for GL(n) was done by Jacquet and Langlands for n = 2 (see [11]), Jacquet, Piatetski-Shapiro, and Shalika for n = 3 (see [12]) and Cogdell and Piatetski-Shapiro for general n (see [5]). In this paper, we prove a Converse Theorem for Hilbert-Jacobi cusp forms over a totally real number field K of degree g := [K : Q] with discriminant DK and narrow class number 1. The case g = 1, i.e., Jacobi forms over Q as considered by Eichler and Zagier (see [7]), is treated in two interesting papers by Martin (see [14] and [15]). To describe our result, we consider functions φ(τ, z) from H × C into C that have a Fourier expansion with certain conditions on the Fourier coefficients (see (3.4),(3.5), and (3.6)). We show that φ is a Hilbert-Jacobi cusp form (for the definition see Section 2) if

Jacobi Forms of Several Variables and the Maaß Space