We study natural families of $\bar {\partial } $-operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.

For a real quadratic field , let tk be the exact power of 2 dividing the class number hk of k and ηk the fundamental unit of k. The aim of this paper is to study tk and the value of Nk/ℚ(ηk). Various methods have been successfully applied to obtain results related to this topic. The idea of our work is to select a special circular unit ℰk of k and investigate C(k)=〈±ℰk 〉. We examine the indices [E(k):C(k)] and [C(k):CS (k)] , where E(k) is the group of units of k, and CS (k) is that of circular units of k defined by Sinnott. Then by using the Sinnott’s index formula [E(k):CS (k)]=hk, we obtain as much information about tk and Nk/ℚ (ηk) as possible.

We develop the Fredholm theory for Toeplitz operators, with symbols in the C*-algebra $D = [SO, SAP]_{n, n}$ generated by all slowly oscillating (SO) and semi-almost periodic (SAP) $n\times n$ matrix functions, on the Hardy spaces $H^p_n$ (with $1 < p < \infty$) over the upper half-plane. Using limit operator techniques, we get necessary Fredholm conditions for any operator in the Banach algebra ${\rm alg}(S, D)$ of singular integral operators with coefficients in $D$ on the space $[L^p (\mathbb{R})]_n$. Applying the Allan–Douglas local principle and the theory of Toeplitz operators with SAP matrix symbols, we also establish Fredholm criteria for Toeplitz operators with matrix symbols $g \in D$ on the space $H^p_n$. An index formula for Fredholm Toeplitz operators with matrix symbols in $D$ is obtained on the basis of an appropriate approximation of slowly oscillating components of the symbols.