We use constructions of surfaces as abelian covers to write down exceptional collections of maximal length for every surface X in certain families of surfaces of general type with pg = 0 and K X = 3, 4, 5, 6, 8. We also compute the algebra of derived endomorphisms for an appropriately chosen exceptional collection, and the Hochschild cohomology of the corresponding quasiphantom category. As a c...

Given an integer r, we consider the problem of enumerating the inert prime ideals p of Q(i) for which a given elliptic curve E has trace of Frobenius at p equal to r. We prove that on average the number of such prime ideals up to x is asymptotic to cr log log x where cr is an explicit constant computed in terms of an Euler product. This result is in accordance with the standard heuristics. This...

We discuss the theory of certain partially ordered sets that capture the structure of commutation classes of words in monoids. As a first application, it follows readily that counting words in commutation classes is #P-complete. We then apply the partially ordered sets to Coxeter groups. Some results are a proof that enumerating the reduced words of elements of Coxeter groups is #P-complete, a ...