Stefano Vigni

On rigid analytic uniformizations of Jacobians of Shimura curves

The main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the Jacobian of a Shimura curve over

Torsion points on elliptic curves over function fields and a theorem of Igusa

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A note on control theorems for quaternionic hida families of modular forms

at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem of \v{C}erednik and Drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant. As a corollary, we offer a proof of a conjecture formulated by M. Greenberg in his paper on Stark-Heegner points and quaternionic Shimura curves, thus making Greenbergs construction of local points on elliptic curves over

Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes

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On ring class eigenspaces of Mordell–Weil groups of elliptic curves over global function fields

unconditional.

An irreducibility criterion for group representations, with arithmetic applications

Abstract If F is a global function field of characteristic p > 3 , we employ Tates theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F.

Special values of L-functions and the arithmetic of Darmon points

We extend a result of Greenberg and Stevens on the interpolation of modular symbols in Hida families to the context of non-split rational quaternion algebras. Both the definite case and the indefinite case are considered.

Quaternion algebras, Heegner points and the arithmetic of Hida families

We extend to the supersingular case the

On the vanishing of Selmer groups for elliptic curves over ring class fields

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