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### A certain Dirichlet series attached to Siegel modular forms of degree two.

Inventiones mathematicae

### A decomposition theorem for Jacobi forms.

Mathematische Annalen

### A generalization of the reciprocity law of multiple Dedekind sums

Annales de l’institut Fourier

Various multiple Dedekind sums were introduced by B.C.Berndt, L.Carlitz, S.Egami, D.Zagier and A.Bayad.In this paper, noticing the Jacobi form in Bayad , the cotangent function in Zagier , Egami’s result on cotangent functions  and their reciprocity laws, we study a special case of the Jacobi forms in Bayad  and deduce a generalization of Egami’s result on cotangent functions and a generalization of Zagier’s result. Further, we consider their reciprocity laws.

### A geometrical approach to the theory of Jacobi forms

Compositio Mathematica

### A trace formula for Jacobi forms.

Journal für die reine und angewandte Mathematik

### An arithmetic theory of Jacobi forms in higher dimensions.

Journal für die reine und angewandte Mathematik

### Asymptotic formulae for partition ranks

Acta Arithmetica

Using an extension of Wright's version of the circle method, we obtain asymptotic formulae for partition ranks similar to formulae for partition cranks which where conjectured by F. Dyson and recently proved by the first author and K. Bringmann.

### Asymptotic formulas for the coefficients of certain automorphic functions

Acta Arithmetica

We derive asymptotic formulas for the coefficients of certain classes of weakly holomorphic Jacobi forms and weakly holomorphic modular forms (not necessarily of integral weight) without using the circle method. Then two applications of these formulas are given. Namely, we estimate the growth of the Fourier coefficients of two important weak Jacobi forms of index 1 and non-positive weights and obtain an asymptotic formula for the Fourier coefficients of the modular functions ${\theta }^{k}/{\eta }^{l}$ for all integers k,l...

### Automorphic forms on Os + 2,2(R) and infinte products.

Inventiones mathematicae

### Beiträge zur Theorie der hyperelliptischen Sigmafunctionen

Mathematische Annalen

### Chern numbers for singular varieties and elliptic homology.

Annals of Mathematics. Second Series

### Class numbers, Jacobi forms and Siegel-Eisenstein series of weight 2 on Sp2.

Mathematische Zeitschrift

### Cohen-Kuznetsov liftings of quasimodular forms

Acta Arithmetica

Jacobi-like forms for a discrete subgroup Γ of SL(2,ℝ) are formal power series which generalize Jacobi forms, and they correspond to certain sequences of modular forms for Γ. Given a modular form f, a Jacobi-like form can be constructed by using constant multiples of derivatives of f as coefficients, which is known as the Cohen-Kuznetsov lifting of f. We extend Cohen-Kuznetsov liftings to quasimodular forms by determining an explicit formula for a Jacobi-like form associated to a quasimodular form....

### Congruences for Siegel modular forms

Annales de l’institut Fourier

We employ recent results on Jacobi forms to investigate congruences and filtrations of Siegel modular forms of degree $2$. In particular, we determine when an analog of Atkin’s $U\left(p\right)$-operator applied to a Siegel modular form of degree $2$ is nonzero modulo a prime $p$. Furthermore, we discuss explicit examples to illustrate our results.

### Construction of Jacobi forms.

Mathematische Zeitschrift

### Dedekind sums involving Jacobi modular forms and special values of Barnes zeta functions

Annales de l’institut Fourier

In this paper we study three new shifted sums of Apostol-Dedekind-Rademacher type. The first sums are written in terms of Jacobi modular forms, and the second sums in terms of cotangent and the third sums are expressed in terms of special values of the Barnes multiple zeta functions. These sums generalize the classical Dedekind-Rademacher sums. The main aim of this paper is to state and prove the Dedekind reciprocity laws satisfied by these new sums. As an application of our Dedekind reciprocity...

### Formes de jacobi et formule de Weber $p$-adique

Journal de théorie des nombres de Bordeaux

Dans ce texte, on construit sur un corps local de caractéristique strictement positive, un analogue $p$-adique aux formes de Jacobi méromorphes complexes ${D}_{L}\left(z;\varphi \right)$, étudiées dans  et . Le théorème principal établit que les formes de Jacobi $p$-adiques obtenues satisfont deux relations de distribution et d’inversion additives. L’analogue $p$-adique à une formule de Weber généralisée est prouvé comme corollaire du théorème principal.

### Fourier coefficients of Jacobi forms over Cayley numbers.

Revista Matemática Iberoamericana

In this paper we shall compute explicitly the Fourier coefficients of the Eisenstein seriesEk,m(z,w) = 1/2 ∑(c,d)=1 (cz + d)-k ∑t∈o exp {2πim((az + b/cz +d)N(t)) + σ(t,(w/cz +d) - (cN(w)/cz + d)}which is a Jacobi form of weight k and index m defined on H1 x CC, the product of the upper half-plane and Cayley numbers over the complex field C. The coefficient of e2πi(nz + σ(t,w)) with nm &gt; N(t) has the form-2(k - 4)/Bk-4 ∏p SpHere Sp is an elementary factor which depends only on νp(m), νp(t),...

### Generalized Jacobi forms and abelian schemes over arithmetic varieties.

Collectanea Mathematica

We generalize Jacobi forms of an arbitrary degree and construct torus bundles over abelian schemes whose sections can be identified with such generalized Jacobi forms.

### Harmonic weak Maass-modular grids in higher level cases

Acta Arithmetica

We extend Guerzhoy's Maass-modular grids on the full modular group SL₂(ℤ) to congruence subgroups Γ₀(N) and Γ₀⁺(p).

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