Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least . The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.
Let be a curve of genus defined over the fraction field of a complete discrete valuation ring with algebraically closed residue field. Suppose that and that the characteristic of the residue field is not . Suppose that the Jacobian has semi-stable reduction over . Embed in using a -rational point. We show that the coordinates of the torsion points lying on lie in the unique tamely ramified quadratic extension of the field generated over by the coordinates of the -torsion...
Let be a modular elliptic curve, and let be an imaginary quadratic field. We show that the -Selmer group of over certain finite anticyclotomic extensions of , modulo the universal norms, is annihilated by the «characteristic ideal» of the universal norms modulo the Heegner points. We also extend this result to the anticyclotomic -extension of . This refines in the current contest a result of [1].
We construct analogs of the classical Δ-function for quotients of the upper half plane 𝓗 by certain arithmetic triangle groups Γ coming from quaternion division algebras B. We also establish a relative integrality result concerning modular functions of the form Δ(αz)/Δ(z) for α in B⁺. We give two explicit examples at the end.