Geometric non-vanishing

We consider $L$-functions attached to representations of the Galois group of the function field of a curve over a finite field. Under mild tameness hypotheses, we prove non-vanishing results for twists of these $L$-functions by characters of order prime to the characteristic of the ground field and by certain representations with solvable image. We also allow local restrictions on the twisting representation at finitely many places. Our methods are geometric, and include the Riemann-Roch theorem, the cohomological interpretation of $L$-functions, and some monodromy calculations of Katz. As an application, we prove a result which allows one to deduce the conjecture of Birch and Swinnerton-Dyer for non-isotrivial elliptic curves over function fields whose $L$-function vanishes to order at most 1 from a suitable Gross-Zagier formula.Comment: 46 pages. New version corrects minor errors. To appear in Inventiones Mat

Review of oral appliances for treatment of sleep-disordered breathing

Between 1982 and 2006, there were 89 distinct publications dealing with oral appliance therapy involving a total of 3,027 patients, which reported results of sleep studies performed with and without the appliance. These studies, which constitute a very heterogeneous group in terms of methodology and patient population, are reviewed and the results summarized. This review focused on the following outcomes: sleep apnea (i.e. reduction in the apnea/hypopnea index or respiratory disturbance index), ability of oral appliances to reduce snoring, effect of oral appliances on daytime function, comparison of oral appliances with other treatments (continuous positive airway pressure and surgery), side effects, dental changes (overbite and overjet), and long-term compliance. We found that the success rate, defined as the ability of the oral appliances to reduce apnea/hypopnea index to less than 10, is 54%. The response rate, defined as at least 50% reduction in the initial apnea/hypopnea index (although it still remained above 10), is 21%. When only the results of randomized, crossover, placebo-controlled studies are considered, the success and response rates are 50% and 14%, respectively. Snoring was reduced by 45%. In the studies comparing oral appliances to continuous positive airway pressure (CPAP) or to uvulopalatopharyngoplasty (UPPP), an appliance reduced initial AHI by 42%, CPAP reduced it by 75%, and UPPP by 30%. The majority of patients prefer using oral appliance than CPAP. Use of oral appliances improves daytime function somewhat; the Epworth sleepiness score (ESS) dropped from 11.2 to 7.8 in 854 patients. A summary of the follow-up compliance data shows that at 30 months, 56–68% of patients continue to use oral appliance. Side effects are relatively minor but frequent. The most common ones are excessive salivation and teeth discomfort. Efficacy and side effects depend on the type of appliance, degree of protrusion, vertical opening, and other settings. We conclude that oral appliances, although not as effective as CPAP in reducing sleep apnea, snoring, and improving daytime function, have a definite role in the treatment of snoring and sleep apnea

Arithmetic and equidistribution of measures on the sphere

Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the Gaussian as the eigenvalue increases. For such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic L-functions associated to triples of eigenfunctions. Moreover we establish the conjecture for the third moments and give a conditional (on standard analytic conjectures about these arithmetic L-functions) proof of the equdistribution of mass conjecture.Comment: 18 pages, an appendix gives corrections to the article "On the central critical value of the triple product L-function" (In: Number Theory 1993-94, 1-46. Cambridge University Press 1996) by Siegfried Boecherer and Rainer Schulze-Pillot. Revised version (minor revisions, new abstract), paper to appear in Communications in Mathematical Physic