Abelian varieties isogenous to a power of an elliptic curve over a Galois extension

Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\rm End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.Comment: 6 pages, added reference

A Local-global principle for isogenies of composite degree

Let $E$ be an elliptic curve over a number field $K$. If for almost all primes of $K$, the reduction of $E$ modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces $E$ to have a cyclic isogeny of that degree over $K$. Building upon the work of Sutherland, Anni, and Banwait-Cremona in the case of prime degree, we consider this question for cyclic isogenies of arbitrary degree.Comment: 30 page

Subspace configurations and low degree points on curves

This paper is devoted to understanding curves $X$ over a number field $k$ that possess infinitely many solutions in extensions of $k$ of degree at most $d$; such solutions are the titular low degree points. For $d=2,3$ it is known (by the work of Harris-Silverman and Abramovich-Harris) that such curves, after a base change to $\overline{k},$ admit a map of degree at most $d$ onto $\mathbb{P}^1$ or an elliptic curve. For $d \geqslant 4$ the analogous statement was shown to be false by Debarre and Fahlaoui. We prove that once the genus of $X$ is high enough, the low degree points still have geometric origin: they can be obtained as pullbacks of low degree points from a lower genus curve. We introduce a discrete-geometric invariant attached to such curves: a family of subspace configurations, with many interesting properties. This structure gives a natural alternative construction of curves with many low degree points, that were first discovered by Debarre and Fahlaoui. As an application of our methods, we obtain a classification of such curves over $k$ for $d=2,3$, and a classification over $\overline{k}$ for $d=4,5$

Soil amendment with activated charcoal can reduce dieldrin uptake by cucumbers

Organochlorine pesticides (OCP) were once applied world wide but have been banned meanwhile in most countries because of their ecotoxicity, bioaccumulation and persistence. However, residues can still be present in soils even many years after applications have been stopped and taken up by crop plants. OCP accumulation from bound residues was found to be a particular problem in Cucurbitaceae plants. Two soil surveys performed in 2002 and 2005 in Switzerland revealed that OCP residues were taken up by cucumbers grown in soils that have been converted to organic production in the meantime. Even if legal tolerance values are not exceeded, this is a serious economic problem for the farmers affected by contaminated crops, because consumers of organically grown crops are only willing to pay the higher prices for these than for conventional products because they are particularly concerned about health and environmental quality and therefore expect pristine food. One approach to address the problem would be to increase the capacity of affected soils to bind OCP residues in order to prevent their uptake by the crops. In this study, we wanted to test the potential use of activated charcoal (AC) for this purpose. In addition, we wanted to assess the possibility of using OCP sorption in soil by Tenax® beads as a predictor for the phytoavailability of these compounds to cucumbers. We performed two pot experiments in which the cash crop cucumber (Cucumis sativus L.) was grown in soil with bound residues of dieldrin (70 µg/kg), pentachloroaniline (<0.01 µg/kg) and p,p-DDE. The soil was taken from a field under organic farming in which these residues were found in the 2005 survey. In the first experiment, cucumbers were grown for 12 to 13 weeks (until fruits were ripe) in soil into which AC had been mixed at concentrations of 200, 400, and 800 mg/kg and in untreated controls. In the second experiment, Tenax® beads were added to the soil and cucumbers, grown with and without AC amendment (800 mg/kg soil), were harvested after 4, 8, 10, 11, 12, and 13 weeks. Dieldrin was the only pesticide detected in the sampled cucumbers and extracted from soil by the Tenax beads. Dieldrin concentrations in the cucumbers were significantly reduced in the treatments with 400 and 800 mg/kg AC. Also significantly less dieldrin was sorbed by Tenax from the soil amended with 800 mg/kg AC than from the untreated control soil. More dieldrin was found to be sorbed by Tenax in the last 3-4 weeks of the experiment, particularly in the control soil, but this trend was not significant. The correlation between the amounts of Tenax-sorbed dieldrin and dieldrin accumulation in the cucumber fruits was significant in control soil and 800 mg/kg AC soil. Hence, Tenax appeared to be suited for the assessment of dieldrin solubility in soil and of phytoavailability to cucumbers

Low degree points on curves

In this paper we investigate an arithmetic analogue of the gonality of a smooth projective curve $C$ over a number field $k$: the minimal $e$ such there are infinitely many points $P \in C(\bar{k})$ with $[k(P):k] \leq e$. Developing techniques that make use of an auxiliary smooth surface containing the curve, we show that this invariant can take any value subject to constraints imposed by the gonality. Building on work of Debarre--Klassen, we show that this invariant is equal to the gonality for all sufficiently ample curves on a surface $S$ with trivial irregularity.Comment: 14 pages, comments welcome

Stability of Tschirnhausen Bundles

Let $\alpha : X \to Y$ be a general degree $r$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $r$. We prove that the Tschirnhausen bundle of $\alpha$ is semistable if $g(Y) \geq 1$ and stable if $g(Y) \geq 2$

Stability of Normal Bundles of Space Curves

In this paper, we prove that the normal bundle of a general Brill-Noether space curve of degree $d$ and genus $g \geq 2$ is stable if and only if $(d,g) \not\in \{ (5,2), (6,4) \}$. When $g\leq1$ and the characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable. We show that this fails in characteristic $2$ for all rational curves of even degree.Comment: 24 pages, comments welcome

A transcendental Brauer-Manin obstruction to weak approximation on a Calabi-Yau threefold

In this paper we investigate the $\mathbb{Q}$-rational points of a class of simply connected Calabi-Yau threefolds, originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural 2-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation. Hosono and Takagi also showed that over $\mathbb{C}$ each of these Calabi-Yau threefolds $Y$ is derived equivalent to a Reye congruence Calabi-Yau threefold $X$. We show that these derived equivalences may also be constructed over $\mathbb{Q}$ and give sufficient conditions for $X$ to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi-Yau variety over $\mathbb{C}$