Related papers: Ineffective descent of genus one curves

The previous version of this paper relied on a paper by another author whose proof appears to be invalid in a fundamental way. In arXiv:1707.00649 the author, together with Jeff Yelton, came up with a new proof of almost identical results.…

This paper has been withdrawn by the author due to a critical error in the proof of Theorem A pointed out by Burkhard Wilking.

For any number field not containing $\QQ(i),$ we give an explicit construction to prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group is nontrivial.

The aim of this paper is to classify reduction types of algebraic curves. Reduction types capture the discrete invariants of fibres in one-dimensional families of curves, and they have been described in genus 1, 2 and 3. For fixed genus…

For a stable real bundle $E$ of rank $2$ and degree $1$ on a real genus $2$ curve, we describe the action of the real structure of the curve on the set of $4$ maximal line subbundles of degree $0$ of $E$. This describes the Galois action on…

The new result is the unirationality of the moduli space of curves of genus 14. The method applies to lower genus.

Using a geometric approach involving Riemann surface orbifolds, we provide lower bounds for the genus of an irreducible algebraic curve of the form $E_{A,B}:\, A(x)-B(y)=0$, where $A, B\in\mathbb C(z)$. We also investigate "series" of…

This short article concerns a method to obtain effectivity for the search of integral points on certain (sets of) curves of genus 2. More precisely, we wish to illustrate just an example of application of a criterion of Bilu, to derive…

Let $C$ be a smooth curve of genus $g \geq 1$ and let $C^{(2)}$ be its second symmetric product. In this note we prove that if $C$ is very general, then the blow-up of $C^{(2)}$ at a very general point has non-polyhedral pseudo-effective…

We prove the first nontrivial worst-case lower bounds for two closely related problems. First, $\Omega(n^{3/2})$ degree-1 reductions, series-parallel reductions, and $\Delta$Y transformations are required in the worst case to reduce an…

Igusa proved in 1958 that the polynomial determining the supersingularity of elliptic curve in Legendre form is separable. In this paper, we get an analogous result for curves of genus $2$ in Rosenhain form. More precisely we show that the…

For a knot diagram we introduce an operation which does not increase the genus of the diagram and does not change its representing knot type. We also describe a condition for this operation to certainly decrease the genus. The proof…

Working over imperfect fields, we give a comprehensive classification of genus-one curves that are regular but not geometrically regular, extending the known case of geometrically reduced curves. The description is given intrinsically, in…

We study two related problems on simple, un-directed graphs: Edge Deletion to Cactus and Spanning Tree to Cactus. Edge Deletion to Cactus has been known to be NP-hard on general graphs at least since 1988. We show improved exact algorithms…

A result of Crew implies that for an etale Galois p-cover of smooth proper varieties over a field of characteristic p, the alternating sum of the p-ranks of the cohomology groups behave like Euler characteristics in characteristic 0. That…

In the proofs of most cases of the Andr\'e-Oort conjecture, there are two different steps whose effectivity is unclear: the use of generalizations of Brauer-Siegel and the use of Pila-Wilkie. Only the case of curves in ${\bf C}^2$ is…

The existence of a finite global attractor for polynomial curve system has been known since the work of Belk et al. [4]. However, except in the hyperbolic case, the rate at which the pullback of a curve under a polynomial converges to the…

We introduce the notion of genus-one data for theories in (1+1)-dimensions with an anomalous finite group global symmetry. We outline the groups for which genus-one data is effective in detecting the anomaly, and also show that genus-one…

We show that when the genus and punctures of a surface are directly proportional by some rational number the minimal asymptotic translation length in the curve complex has behavior inverse to the square of the Euler characteristic. We also…

I give a new proof, in scheme-theoretic language, of Tate's old result on genus-change over nonperfect fields in characteristic p>0. Namely, for normal geometrically integral curves, the difference between arithmetic and geometric genus…