In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite-rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura-curve analogues of these results.

In an earlier paper the second author used the formal, algebraic properties of $2$-dimensional Shintani generating functions to construct a $1$-cocycle on ${\rm PGL}_2(\mathbb{Q})$. We aim to generalise these results by using such functions in dimension $n$ to obtain an $(n-1)$-cocycle on ${\rm PGL}_n(\mathbb{Q})$, presumably related to the Bernoulli and Eisenstein cocycles of R.~Sczech. By improving our methods we achieve this goal for $n=3$. For $n>3$ we encounter obstacles related to degenerate configurations of hyperplanes in $n$-space. Nevertheless, we obtain partial results closely connected to reciprocity laws for certain $n$-dimensional Dedekind sums. 1991 Mathematics Subject Classification: 11F20, 11F75.

The response of the photographic grain to light is a non-deterministic process which is as yet not completely understood. This response, as measured by the photographic density, is usually taken to be a function of the product of incident light intensity and exposure time interval duration, but at extreme values of either of these two quantities, this is no longer true. This latter effect is called reciprocity-law failure.

This paper discusses a probabilistic model, similar to a multiserver queue, for high-intensity reciprocity failure.