Local diophantine properties of modular curves of D-elliptic sheaves
نویسنده
چکیده
We study the existence of rational points on modular curves of D-elliptic sheaves over local fields and the structure of special fibres of these curves. We discuss some applications which include finding presentations for arithmetic groups arising from quaternion algebras, finding the equations of modular curves of D-elliptic sheaves, and constructing curves violating the Hasse principle.
منابع مشابه
Genus Formula for Modular Curves of D-elliptic Sheaves
We prove a genus formula for modular curves of D-elliptic sheaves. We use this formula to show that the reductions of modular curves of D-elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of D tends to infinity.
متن کاملModular Curves of D-elliptic Sheaves Are Asymptotically Optimal
We prove that the series of modular curves of D-elliptic sheaves with appropriate level structures attain the Drinfeld-Vladut bound over Fq2 .
متن کاملModular Varieties of D-elliptic Sheaves and the Weil-deligne Bound
We compare the asymptotic grows of the number of rational points on modular varieties of D-elliptic sheaves over finite fields to the grows of their Betti numbers as the degree of the level tends to infinity. This is a generalization to higher dimensions of a well-known result for modular curves. As a consequence of the main result, we also produce a new asymptotically optimal sequence of curves.
متن کاملASPECTS OF COMPLEX MULTIPLICATION Contents
1. Preview 2 Complex multiplication on elliptic curves over C 2 Traces of singular moduli 3 Class field theory 3 The Kronecker limit formula and Kronecker’s solution of Pell’s equation 4 Application to Diophantine equations (Villegas) 4 L-series and CM modular forms 5 Other topics 6 2. Complex Multiplication on Elliptic Curves over C 6 Elliptic Curves over C 6 Elliptic functions 7 Complex multi...
متن کاملIrregular Diophantine m-tuples and elliptic curves of high rank
A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them is one less than a perfect square. In this paper we characterize the notions of regular Diophantine quadruples and quintuples, introduced by Gibbs, by means of elliptic curves. Motivated by these characterizations, we find examples of elliptic curves over Q with torsion group Z/2Z × Z/2Z and ...
متن کامل