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].
STARK-HEEGNER POINTS AND THE COHOMOLOGY OF QUATERNIONIC SHIMURA VARIETIES
Let F be a totally real field of narrow class number one, and let E/F be a modular, semistable elliptic curve of conductor N � (1) .L etK/F be a non-CM quadratic extension with (Disc K, N) = 1 such that the sign in the functional equation of L(E/K, s) is −1. Suppose further that there is a prime p|N that is inert in K .W e describe a p-adic construction of points on E which we conjecture to be rational over ring class fields of K/F and satisfy a Shimura reciprocity law. These points are expected to behave like classical Heegner points and can be viewed as new instances of the Stark-Heegner point construction of [5]. The key idea in our construction is a reinterpretation of Darmon’s theory of modular symbols and mixed period integrals in terms of group cohomology.
Decidability of some intuitionistic predicate theories
Suppose T is a first order intuitionistic theory (more precisely, a theory of Heyting's predicate calculus, e.g., abelian groups, one unary function, dense linear order, etc.) presented to us by a set of axioms (denoted also by) T. Question. Is T decidable? One knows that if the classical counterpart of T (i.e., take the same axioms but with the classical predicate calculus as the underlying logic) is not decidable, then T cannot be decidable. The problem remains for theories whose classical counterpart is decidable. In [8], sufficient conditions for undecidability were given, and several intuitionistic theories such as abelian groups and unary functions (both with decidable equality) were shown to be undecidable. In this note we show decidability results (see Theorems 1 and 2 below), and compare these results with the undecidability results previously obtained. The method we use is the reduction-method, described fully in [12] and widely applied in [3], which is applied here roughly as follows: Let T be a given theory of Heyting's predicate calculus. We know that Heyting's predicate calculus is complete for the Kripke-model type of semantics. We choose a class M of Kripke models for which T is complete, i.e., all axioms of T are valid in any model of the class and whenever φ is not a theorem of T, φ is false in some model of M.