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.