The Jacobi group is the semi-direct product of the symplectic group and the Heisenberg group. The Jacobi group is an important object in the framework of quantum mechanics, geometric quantization and optics. In this paper, we study the Weil representations of the Jacobi group and their properties. We also provide their applications to the theory of automorphic forms on the Jacobi group and repr...

We describe how to prove the Mordell-Weil theorem for Jacobians of hyperelliptic curves over Q and how to compute the rank and generators for the Mordell-Weil group.

We say a lattice Λ is rigid if it its isometry group acts irreducibly on its ambient Euclidean space. We say Λ is Mordell-Weil if there exists an abelian variety A over a number field K such that A(K)/A(K)tor, regarded as a lattice by means of its height pairing, contains at least one copy of Λ. We prove that every rigid lattice is Mordell-Weil. In particular, we show that the Leech lattice can...

By the Mordell-Weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎There is no known algorithm for finding the rank of this group‎. ‎This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

For every prime p we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil-Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q which is not divisible by 4 in the Weil-Châtelet group. This gives a negative answer to a question of Cassels.

In this article we develop a broad generalization of the classical Bost-Connes system, where roots of unit are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unit, Weil restriction, algebraic numbers, Weil numbers, CM fields, germs, completion of Weil numbers, etc. Making use of the Tannakian formalism, we categorify th...

Let E/F be a finite and Galois extension of non-archimedean local fields. G connected reductive group defined over E let M:=RE/FG the F obtained by Weil restriction scalars. We investigate depth, enhanced Langlands correspondence, in transition from G(E) to M(F). obtain depth-comparison formula for Weil-restricted groups.