Averages and moments associated to class numbers of imaginary quadratic fields

For any odd prime $\ell$, let $h_\ell(-d)$ denote the $\ell$-part of the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-d})$. Nontrivial pointwise upper bounds are known only for $\ell =3$; nontrivial upper bounds for averages of $h_\ell(-d)$ have previously been known only for $\ell =3,5$. In this paper we prove nontrivial upper bounds for the average of $h_\ell(-d)$ for all primes $\ell \geq 7$, as well as nontrivial upper bounds for certain higher moments for all primes $\ell \geq 3$.Comment: 26 pages; minor edits to exposition and notation, to agree with published versio

Simultaneous Integer Values of Pairs of Quadratic Forms

We prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.Comment: 63 page

Counting rational points on smooth cyclic covers

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space P^{n-1}. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n is at least 10, and surpass it for covers of degree 3 or higher when n > 10. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput's method

Azolium-Linked Cyclophanes: Effects of Structure, Solvent, and Counteranions on Solution Conformation Behavior

This paper describes the synthesis, structural characterization, and solution behavior of some xylyllinked imidazolium and benzimidazolium cyclophanes decorated with alkyl or alkoxy groups. The addition of alkyl/alkoxy chains to the cyclophanes allows for studies in chlorinated solvents, whereas previous solution studies of azolium cyclophanes have generally required highly polar solvents. The azolium cyclophanes may exist in a syn/syn conformation (azolium rings mutually syn, arene rings mutually syn) or a syn/anti conformation (azolium rings mutually syn, arene rings mutually anti). The preferred conformation is significantly affected by (i) binding of bromide (ion pairing) to the protons on theimidazolium or benzimidazolium rings, which occurs in solutions of bromide salts of the cyclophanes inchlorinated solvents, and (ii) the addition of alkoxy groups to the benzimidazolium cyclophanes. Thesestructural modifications have also led to cyclophanes that adopt conformations not previously identifiedfor similar azolium cyclophane analogues.Detailed 1H NMR studies for one cyclophane identified binding of bromide at two independent sites within the cyclophane