A p-arithmetic subgroup of $\mathbf {SL}_2(\mathbb {Q})$ like the Ihara group $\Gamma := \mathbf {SL}_2(\mathbb {Z}[1/p])$ acts by Möbius transformations on the Poincaré upper half plane $\mathcal H$ and on Drinfeld’s p-adic upper half plane ${\mathcal H_p := \mathbb {P}_1(\mathbb {C}_p)-\mathbb {P}_1(\mathbb {Q}_p)}$. The diagonal action of $\Gamma $ on the product is discrete, and the quotient $\Gamma \backslash (\mathcal H_p\times \mathcal H)$ can be envisaged as a ‘mock Hilbert modular surface’. According to a striking prediction of Neková$\check {\text {r}}$ and Scholl, the CM points on genuine Hilbert modular surfaces should give rise to ‘plectic Heegner points’ that encode nontrivial regulators attached, notably, to elliptic curves of rank two over real quadratic fields. This article develops the analogy between Hilbert modular surfaces and their mock counterparts, with the aim of transposing the plectic philosophy to the mock Hilbert setting, where the analogous plectic invariants are expected to lie in the alternating square of the Mordell–Weil group of certain elliptic curves of rank two over $\mathbb {Q}$.

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let ${\it\varrho}_{\flat }$ and ${\it\varrho}_{\sharp }$ be odd two-dimensional Artin representations for which ${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$ is self-dual. The progress on modularity achieved in recent decades ensures the existence of normalized eigenforms $f$, $g$, and $h$ of respective weights two, one, and one, giving rise to $E$, ${\it\varrho}_{\flat }$, and ${\it\varrho}_{\sharp }$ via the constructions of Eichler and Shimura, and of Deligne and Serre. This article examines certain $p$-adic iterated integrals attached to the triple $(f,g,h)$, which are $p$-adic avatars of the leading term of the Hasse–Weil–Artin $L$-series $L(E,{\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp },s)$ when it has a double zero at the centre. A formula is proposed for these iterated integrals, involving the formal group logarithms of global points on $E$—referred to as Starkpoints—which are defined over the number field cut out by ${\it\varrho}_{\flat }\otimes {\it\varrho}_{\sharp }$. This formula can be viewed as an elliptic curve analogue of Stark’s conjecture on units attached to weight-one forms. It is proved when $g$ and $h$ are binary theta series attached to a common imaginary quadratic field in which $p$ splits, by relating the arithmetic quantities that arise in it to elliptic units and Heegner points. Fast algorithms for computing $p$-adic iterated integrals based on Katz expansions of overconvergent modular forms are then exploited to gather numerical evidence in more exotic scenarios, encompassing Mordell–Weil groups over cyclotomic fields, ring class fields of real quadratic fields (a setting which may shed light on the theory of Stark–Heegner points attached to Shintani-type cycles on ${\mathcal{H}}_{p}\times {\mathcal{H}}$), and extensions of $\mathbb{Q}$ with Galois group a central extension of the dihedral group $D_{2n}$ or of one of the exceptional subgroups $A_{4}$, $S_{4}$, and $A_{5}$ of $\mathbf{PGL}_{2}(\mathbb{C})$.

Let $E$ be an elliptic curve, and let ${{L}_{n}}$ be the Kummer extension generated by a primitive ${{p}^{n}}$ -th root of unity and a ${{p}^{n}}$ -th root of $a$ for a fixed $a\,\in \,{{\mathbb{Q}}^{\times }}\,-\,\left\{ \pm 1 \right\}$ . A detailed case study by Coates, Fukaya, Kato and Sujatha and $V$ . Dokchitser has led these authors to predict unbounded and strikingly regular growth for the rank of $E$ over ${{L}_{n}}$ in certain cases. The aim of this note is to explain how some of these predictions might be accounted for by Heegner points arising from a varying collection of Shimura curve parametrisations.

Let and be modular forms of half-integral weight k+1/2 and integral weight 2k respectively that are associated to each other under the Shimura–Kohnen correspondence. For suitable fundamental discriminants D, a theorem of Waldspurger relates the coefficient c(D) to the central critical value L(f,D,k) of the Hecke L-series of f twisted by the quadratic Dirichlet character of conductor D. This paper establishes a similar kind of relationship for central critical derivatives in the special case k=1, where f is of weight 2. The role of c(D) in our main theorem is played by the first derivative in the weight direction of the Dth Fourier coefficient of a p-adic family of half-integral weight modular forms. This family arises naturally, and is related under the Shimura correspondence to the Hida family interpolating f in weight 2. The proof of our main theorem rests on a variant of the Gross–Kohnen–Zagier formula for Stark–Heegner points attached to real quadratic fields, which may be of some independent interest. We also formulate a more general conjectural formula of Gross–Kohnen–Zagier type for Stark–Heegner points, and present numerical evidence for it in settings that seem inaccessible to our methods of proof based on p-adic deformations of modular forms.

Plasma concentrations of vitamin E and carotenoids are governed by several factors, including genetic factors. Single nucleotide polymorphisms (SNP) in some genes involved in lipid metabolism have recently been associated with fasting plasma concentrations of these fat-soluble micronutrients. To further investigate the role of genetic factors that modulate the plasma concentrations of these micronutrients, we assessed whether SNP in five candidate genes (apo C-III, CETP, hepatic lipase, I-FABP and MTP) were associated with the plasma concentrations of these micronutrients. Fasting plasma vitamin E and carotenoid concentrations were measured in 129 French Caucasian subjects (forty-eight males and eighty-one females). Candidate SNP were genotyped by PCR amplification followed by restriction fragment length polymorphisms. Plasma γ-tocopherol, α-carotene and β-carotene concentrations were significantly different (P < 0·05) in subjects who carried different SNP variants in hepatic lipase. Plasma α-tocopherol concentrations were significantly different in subjects who had different SNP variants in apo C-III and cholesteryl ester transfer protein (CETP). Plasma lycopene concentrations were significantly different (P < 0·05) in women who had different SNP variants in intestinal fatty acid binding protein (I-FABP). Finally, there was no effect of SNP variants in microsomal TAG transfer protein upon the plasma concentrations of these micronutrients. Most of the observed differences remained significant after the plasma micronutrients were adjusted for plasma TAG and cholesterol. These results suggest that apo C-III, CETP and hepatic lipase play a role in determining the plasma concentrations of tocopherols while hepatic lipase and I-FABP may modulate plasma concentrations of carotenoids.

The Main Conjecture of Iwasawa theory for an elliptic curve $E$ over $\mathbb{Q}$ and the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$ was studied in \cite{bertolini_darmon}, in the case where $p$ is a prime of ordinary reduction for $E$. Analogous results are formulated, and proved, in the case where $p$ is a prime of supersingular reduction. The foundational study of supersingular main conjectures carried out by Perrin-Riou, Pollack, Kurihara, Kobayashi and Iovita and Pollack are required to handle this case in which many of the simplifying features of the ordinary setting break down.

Soient $r$ et $p$ deux nombres premiers distincts, soit $K\,=\,\mathbb{Q}(\cos \,\frac{2\pi }{r})$ , et soit $\mathbb{F}$ le corps résiduel de $K$ en une place au-dessus de $p$ . Lorsque l’image de $(2\,-\,2\,\cos \,\frac{2\pi }{r})$ dans $\mathbb{F}$ n’est pas un carré, nous donnons une construction géométrique d’une extension réguliere de $K\left( t \right)$ de groupe de Galois $\text{PS}{{\text{L}}_{2}}(\mathbb{F})$ . Cette extension correspond à un revêtement de ${{\mathbb{P}}^{1}}/k$ de « signature $\left( r,\,p,\,p \right)$ » au sens de [3, sec. 6.3], et son existence est prédite par le critère de rigidité de Belyi, Fried, Thompson et Matzat. Sa construction s’obtient en tordant la representation galoisienne associée aux points d’ordre $p$ d’une famille de variétés abéliennes à multiplications réelles par $K$ découverte par Tautz, Top et Verberkmoes [6]. Ces variétés abéliennes sont définies sur un corps quadratique, et sont isogènes à leur conjugué galoisien. Notre construction généralise une méthode de Shih [4], [5], que l’on retrouve quand $r\,=\,2$ et $r\,=\,3$ .

We formulate a conjecture analogous to Gross' refinement of the Stark conjectures on special values of abelian L-series at s = 0. Some evidence for the conjecture can be obtained, thanks to the fundamental ideas of F. Thaine.