Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.Comment: v5: Some typos correcte

Equidistribution of Heegner Points and Ternary Quadratic Forms

We prove new equidistribution results for Galois orbits of Heegner points with respect to reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize one of the equidistribution theorems established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system

Linear spaces on the intersection of cubic hypersurfaces

Upper bounds for the number of variables necessary to imply the existence of an m -dimensional linear variety on the intersection of r cubic hypersurfaces over local and global fields are given.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41633/1/605_2006_Article_BF02349626.pd

'Unkonventionelle Medizinische Richtungen' (UMR) und 'Unkonventionelle Methoden der Krebsbekaempfung' (UMK) Abschlussbericht der Projektbegleitung

In recent years, methods of complementary medicine and naturopathy have become increasingly important within the health system. An ever-increasing number of both patients and practitioners regularly use these methods, a trend which can be observed in many European and non-European countries. Against this backdrop, questions about the scientific evaluation of these methods are receiving increasing attention. Thus stimulated, a field of research is currently developing within unconventional medicine. State research support in Germany is endeavouring to stimulate the development of research in this field through specific grant awards, and to establish high quality research standards. This is being implemented through the funding areas: 'Unconventional Methods of Combatting Cancer' and 'Unconventional Medical Approaches', the former having already gathered 13 years experience in funding research projects, while the latter started funding in 1994. A project group at the University of Witten/Herdecke organized these funding areas. (orig.)SIGLEAvailable from TIB Hannover: F97B2161 / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekBundesministerium fuer Bildung, Wissenschaft, Forschung und Technologie, Bonn (Germany)DEGerman