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].
Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields
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The honesty of thinking : reflections on critical thinking in Nietzsche's middle period and the later Heidegger
This dissertation engages with contemporary interpretations of Nietzsche and Heidegger on the issue of self-knowing with respect to the notions of honesty and authenticity. Accounting for the two philosophers' developing conceptions of these notions allows a response to interpreters who conceive the activity of self-knowing as a primarily personal problem. The alternative accounts proposed take as a point of departure transitional texts that reveal both thinkers to be engaged in processes of revision. The reading of honesty in Chapters 1 and 2 revolves around Nietzsche's groundwork on prejudice in Morgenrothe (1880-81), where he first problematizes the moral-historical forces entailed in actuating the 'will to truth'. The reading of authenticity in Chapters 3 and 4 revolves around Heidegger's lectures on what motivates one's thinking in Was heist Denken? (1951-52). The lectures call into question his previous formal suppositions on what calls forth one's 'will-to-have-a-conscience', in an interpretation of Parmenides on the issue of thought's linguistic determination, discussed further in the context of Unterwegs zur Sprache (1950-59). Chapter 5 shows how Heidegger's confrontation with Nietzsche contributed to his ongoing revisions to the notion of authenticity, and to the attending conceptions of critique and its authority. Particular attention is given to the specific purposes to which distinct Nietzschean foils are put near the confrontation's beginning--in Heidegger's lectures on Nietzsche's second Unzeitgemasse Betrachtung (1938), and in the monograph entitled Besinnung (1939) which they prepare--and near its end, in the interpretation of Also Sprach Zarathustra (1883-85) presented in the first half of Was heist Denken? Chapter 6 recapitulates the developments traced from the vantage point of the retrospective texts Die Zollikoner Seminare (1959-72) and the fifth Book of Die frohliche Wissenschaft (1887). Closing remarks are made in relation to recent empirical research on the socio-environmental structures involved in determining self-identity.