Displaying similar documents to “Arithmetic differential equations in several variables”

On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields

Matteo Longo (2006)

Annales de l’institut Fourier

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Let E / F be a modular elliptic curve defined over a totally real number field F and let φ be its associated eigenform. This paper presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of E over suitable quadratic imaginary extensions K / F . In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when [ F : ℚ ] is even and φ not new at any prime.

On a conjecture of Watkins

Neil Dummigan (2006)

Journal de Théorie des Nombres de Bordeaux

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Watkins has conjectured that if R is the rank of the group of rational points of an elliptic curve E over the rationals, then 2 R divides the modular parametrisation degree. We show, for a certain class of E , chosen to make things as easy as possible, that this divisibility would follow from the statement that a certain 2 -adic deformation ring is isomorphic to a certain Hecke ring, and is a complete intersection. However, we show also that the method developed by Taylor, Wiles and others,...

Quasi-modular forms attached to elliptic curves, I

Hossein Movasati (2012)

Annales mathématiques Blaise Pascal

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In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out...

Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks

Masafumi Yoshino, Todor Gramchev (2008)

Annales de l’institut Fourier

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We study the simultaneous linearizability of d –actions (and the corresponding d -dimensional Lie algebras) defined by commuting singular vector fields in ℂ n fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of d vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators...

Anticyclotomic Iwasawa theory of CM elliptic curves

Adebisi Agboola, Benjamin Howard (2006)

Annales de l’institut Fourier

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We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Z p -extension of the CM field, where p is a prime of good, ordinary reduction for E . When the complex L -function of E vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the p -power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg show that it is not a torsion...