Darmon points on elliptic curves over totally real fields
Let F be a number field and let E be an elliptic curve over F of conductor an ideal N of F . We assume throughout that F is totally real: in that case, it is known, under minor hypotheses, that there is a newform f of weight 2 on Γ0(N) over F whose L-function coincides with that of E (see, e.g., [Zha01] or [Dar04, §7.4]). We fix a quadratic extension K/F . When K is totally complex, a classical theory produces a family of Heegner points of E, defined over ring class fields of K. The Galois action on them is given by a Shimura reciprocity law, and their heights relate to the derivative of the L-function of E over K at 1 (see, e.g., [Zha01] or [Dar04, §7.5]). We assume therefore from now on that K has at least one real place. The goal of the theory of Darmon points (earlier called Stark-Heegner points) is to extend to such K the construction of Heegner points. In [Dar01], Darmon started the theory in the case F = Q and K a totally real quadratic field. In [Gre09], Greenberg generalizes this work to give a conjectural construction of points in the case where F is arbitrary totally real quadratic field of narrow class number one, E is semistable, N 6= (1), the sign of the functional equation of E over K is −1, there is a prime dividing the conductor of E that is inert in K, and the discriminant of K is coprime to N . The techniques used are p-adic in nature. In [Gar11], Gartner generalizes the work of [Dar04, Chap 8] and [DL03] to give a construction of what he calls Darmon points in certain situations using archimedian techniques. The conditions under which Gartner’s construction works are a bit too technical to describe here, but we shall describe them in Section 2.1. All the constructions share a basic outline: one computes an archimedean or p-adic integral of the modular form associated to E/F , and plugs the resulting value into a Weierstrass of Tate parametrization of E to produce the Darmon point. Let O ⊆ K be an OF -order such that Disc(O/OF ) is coprime to N . In this article, we show that if the sign of the functional equation of E over K is −1, the discriminant of K is coprime to N , and the part of N divisible by primes that are inert in K is square-free, then one can apply either the construction of Gartner or the construction of Greenberg (after removing the assumption that F has narrow class number one, which we show how to do) to conjecturally associate to O a point that we call a Darmon point (actually there are choices in the construction, and so one gets several Darmon points). This point is intially defined over a transcendental extension of K, but we conjecture that the point is algebraic, defined over the narrow ring class field extension of K associated to the order O. This point comes with an action of the narrow class group of O, and we state a conjectural Shimura receprocity law for this action. In Section 2 we recall and slightly modify the constructions of Greenberg and Gartner. In Section 3 we show how one of the two constructions can be carried out under our hypotheses. We assume throughout this article that the reader is familiar with [Gre09] and [Gar11].
Points de Darmon et variétés de Shimura
Cette these s'interesse a la recherche de points rationnels sur les courbes elliptiques. Darmon et Logan ont propose une construction conjecturale de points rationnels sur des courbes elliptiques modulaires definies sur un corps de nombres totalement reel. Cette construction va au dela de la construction classique des points de Heegner. C'est sur la generalisation de ces travaux que porte cette these. Apres un premier chapitre de rappels concernant essentiellement les varietes de Shimura, on construit, dans le chapitre deux une forme differentielle dont l'ensemble des periodes est, sous une conjecture due a Yoshida, un reseau. On y definit aussi un ensemble de cycles dont la classe d'homologie est de torsion. A l'aide de ces donnees, on enonce au chapitre suivant une conjecture generalisant celle de Darmon et Logan. On s'interesse aussi aux proprietes de ces nouveaux points, principalement en lien avec les theoremes "classiques" de Gross-Zagier et Gross-Kohnen-Zagier. Le chapitre 4 tente de rendre holomorphes les operations du chapitre 2, et le chapitre 5 de les rendre plus explicites. Cette these comporte une annexe concernant les verifications informatiques de la conjecture de Darmon.
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...