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.

Let $q$, $h$, $a$, $b$ be integers with $q>0$. The classical and the homogeneous Dedekind sums are defined by $$s(h,q)=\sum _{j=1}^q\left(\left(\frac{j}{q}\right)\right)\left(\left(\frac{hj}{q}\right)\right),s(a,b,q)=\sum _{j=1}^q\left(\left(\frac{aj}{q}\right)\right)\left(\left(\frac{bj}{q}\right)\right),$$ respectively, where $$\left(\left(x\right)\right)=\left\{\begin{array}{cc}x-\left[x\right]-\frac{1}{2},\hfill & \text{if}x\text{is}\text{not}\text{an}\text{integer};\hfill \\ 0,\hfill & \text{if}x\text{is}\text{an}\text{integer}.\hfill \end{array}\right.$$ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$\begin{array}{c}\sum _{d\mid n}\sum _{r=1}^{d}s\left(\frac{n}{d}a+rq,dq\right)=\sigma \left(n\right)s(a,q),\\ \sum _{d\mid n}\sum _{r_1=1}^d\sum _{r_2=1}^{d}s\left(\frac{n}{d}a+r_{1}q,\frac{n}{d}b+r_{2}q,dq\right)=n\sigma \left(n\right)s(a,b,q),\end{array}$$ where $\sigma \left(n\right)={\sum }_{d\mid n}d$. In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.

We show that the modular functions j 1,N generate function fields of the modular curve X 1(N), N ∈ {7; 8; 9; 10; 12}, and apply them to construct ray class fields over imaginary quadratic fields.

In this paper we introduce a geometric formalism for studying modular forms of half-integral weight. We then use this formalism to define $p$-adic modular forms of half-integral weight and to construct $p$-adic Hecke operators.

Modular and quasimodular forms have played an important role in gravity and string theory. Eisenstein series have appeared systematically in the determination of spectrums and partition functions, in the description of non-perturbative effects, in higher-order corrections of scalar-field spaces, ...The latter often appear as gravitational instantons i.e. as special solutions of Einstein’s equations. In the present lecture notes we present a class of such solutions in four dimensions, obtained by...