Peter Hr Green

Elliptic Curves and Class Fields of Real Quadratic Fields: Algorithms and Evidence

The article [Darmon 02] proposes a conjectural p-adic analytic construction of points on (modular) elliptic curves, points which are defined over ring class fields of real quadratic fields. These points are related to classical Heegner points in the same way as Stark units to circular or elliptic units. For this reason they are called “Stark-Heegner points,” following a terminology introduced in [Darmon 98]. If K is a real quadratic field, the Stark-Heegner points attached to K are conjectured to satisfy an analogue of the Shimura reciprocity law, so that they can in principle be used to find explicit generators for the ring class fields of K. It is also expected that their heights can be expressed in terms of derivatives of the Rankin L-series attached to E and K, in analogy with the Gross-Zagier formula . The main goal of this paper is to describe algorithms for calculating Stark-Heegner points and supply numerical evidence for the Shimura reciprocity and Gross-Zagier conjectures, focussing primarily on elliptic curves of prime conductor.

Periods and Points Attached to Quadratic Algebras

Φ : H/Γ0(N) −→ E(C), (0–1) where H is the Poincare upper half-plane and Γ0(N) is Hecke’s congruence group of level N . Fix a quadratic field K; when it is imaginary, the theory of complex multiplication combined with (0–1) yields the construction of a remarkable collection of points on E defined over certain ring class fields of K. These are the Heegner points recalled in Section 1 whose study is one of the themes of this proceedings volume. When p is a prime that divides N exactly, and E a factor of the Jacobian of a Shimura curve attached to a quaternion algebra ramified at p, the uniformisation Φ of (0–1) admits a p-adic analogue based on theorems of Jacquet–Langlands and Cerednik–Drinfeld, in which H is replaced by Drinfeld’s p-adic upper half plane Hp := Cp − Qp, and Γ0(N) by an appropriate {p}-arithmetic subgroup

Gluten Introduction, Breastfeeding, and Celiac Disease: Back to the Drawing Board.

This commentary by the leadership of the North American Society for the Study of Celiac Disease (NASSCD) concerns recent research findings regarding infant feeding practices. Celiac disease has increased markedly in recent decades, and seroprevalence studies indicate that this is a true rise, rather than one due to increased awareness and testing. Prior studies have suggested that infant feeding practices and timing of initial gluten exposure are central to the development of celiac disease. Two recent multicenter randomized trials tested strategies of early or delayed gluten introduction in infants, and neither strategy appeared to influence celiac disease risk. These studies also found that breastfeeding did not protect against the development of celiac disease. While disappointing, these results should spur the study of wider environmental risk factors beyond infant feeding, such as intrauterine and perinatal exposures as well as environmental influences later in life, including drug exposure, microbial infections, and the microbiome. Given that celiac disease can develop at any age, it is imperative to study these proposed triggers so as to elucidate the loss of tolerance to gluten and to develop future intervention strategies.