这是K写的一封信。里贝特呼叫J F. 1987年11月在梅斯特雷。这封信证明了三个公式，已aptured梅斯特和Oesterlé。这些公式涉及到作为某些模曲线的雅可比行列式的最优导数出现的椭圆曲线的分量组，以及分量组与雅可比行列式的复曲面分量的特征群之间的关系。(The“复曲面部件”和部件组以正特性计算。
英文摘要：
This is the text of a letter written by K. Ribet to J.-F. Mestre in November, 1987. The letter proves three formulas that had been conjectured by Mestre and Oesterlé. These formulas concern component groups of elliptic curves that occur as optimal quotients of certain Jacobians of modular curves, and relations between the component groups and the character group of the toric component of the Jacobian. (The "toric component" and component groups are computed in positive characteristic.)
We introduce a procedure for constructing a generalized elliptic curve from a genus-one twisted curve, and we use this procedure to define an explicit, modular isomorphism between two compactifications of moduli stacks of elliptic curves with level structure.
We prove that all elliptic curves over quadratic fields with a subgroup isomorphic to $C_{16}$, as well as all elliptic curves over cubic fields with a subgroup isomorphic to $C_2\times C_{14}$, are base changes of elliptic curves defined over $\Q$. We obtain these results by studying geometric properties of modular curves and maps between modular curves, and then obtaining a modular description of these curves and maps.
We discuss a non-computational elementary approach to a well-known criterion of divisibility by 2 in the group of rational points on an elliptic curve.
In this paper, we study the finiteness problem of torsion points on an elliptic curve whose coordinates satisfy some multiplicative dependence relations. In particular, we prove that on an elliptic curve defined over a number field there are only finitely many torsion points whose coordinates are multiplicatively dependent. Moreover, we produce an effective result when the elliptic curve is defined over the rational numbers or has complex multiplication.
For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup.
We give an asymptotic formula for the number of elliptic curves over $\mathbb{Q}$ with bounded Faltings height. Silverman has shown that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in $\mathbb{R}^2$.
We show that the canonical-lift construction for ordinary elliptic curves over perfect fields of characteristic $p>0$ extends uniquely to arbitrary families of ordinary elliptic curves, even over $p$-adic formal schemes. In particular, the universal ordinary elliptic curve has a canonical lift. The existence statement is largely a formal consequence of the universal property of Witt vectors applied to the moduli space of ordinary elliptic curves, at least with enough level structure. As an application, we show how this point of view allows for more formal proofs of recent results of Finotti an…
In this note we investigate the $p$-degree function of elliptic curves over the field $\mathbb{Q}_p$ of $p$-adic numbers. The $p$-degree measures the least complexity of a non-zero $p$-torsion point on an elliptic curve. We prove some properties of this function and compute it explicitly in some special cases.
This article is a contribution to the project of classifying the torsion growth of elliptic curve upon base-change. In this article we treat the case of elliptic curve defined over the rationals with complex multiplication. For this particular case, we give a description of the possible torsion growth over cubic fields and a completely explicit description of this growth in terms of some invariants attached to a given elliptic curve.
This paper is a follow up of arXiv:1702.02255 [math.NT]. We construct explicitly versal families of elliptic curves with rational points of order 4, 6, 8, 10, 12 respectively.
A cycle of elliptic curves is a list of elliptic curves over finite fields such that the number of points on one curve is equal to the size of the field of definition of the next, in a cyclic way. We study cycles of elliptic curves in which every curve is pairing-friendly. These have recently found notable applications in pairing-based cryptography, for instance in improving the scalability of distributed ledger technologies. We construct a new cycle of length 4 consisting of MNT curves, and characterize all the possibilities for cycles consisting of MNT curves. We rule out cycles of length 2 …
We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L-equivalence and derived equivalence. The proof of the L-equivalence for curves is based on Kuznetsov's Homological Projective Duality for Gr(2,5), and L-equivalence is extended from genus one curves to elliptic surfaces using the Ogg--Shafarevich theory of twisting…
From the product of two elliptic curves, Shioda and Inose constructed an elliptic $K3$ surface having two $\mathrm{II}^*$ fibers. Its Mordell-Weil lattice structure depends on the morphisms between the two elliptic curves. In this paper, we give a method of writing down generators of the Mordell-Weil lattice of such elliptic surfaces when two elliptic curves are $3$-isogenous. In particular, we obtain a basis of the Mordell-Weil lattice for the singular $K3$ surfaces $X_{[3,3,3]}$, $X_{[3,2,3]}$ and $X_{[3,0,3]}$.
We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree $p$, resp. $d$, onto smooth elliptic curves, with particular attention to the case $p$ prime.
In this paper, we develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque-Enriquez-Etingof and Levin-Racinet. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.
In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the Néron--Severi group. We use the geometric method introduced by Oguiso and moreover we provide a geometric construction of the fibrations classified. If the non-symplectic involution fixes at least one curve of genus 1, we relate all the elliptic fibrations on the K3 surface with either elliptic fibrations or generalized conic bundles on rational elliptic surfaces. This description allows us to write the Weierstrass equations of the elliptic fibr…
The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.
We derive an efficient algorithm to find solutions to Euler's concordant form problem and rational points on elliptic curves associated with this problem.
Given a principally polarized abelian variety $(A,Θ)$, we give a characterization of all elliptic curves that lie on $A$ in terms of intersection numbers of divisor classes in its Néron-Severi group.
In this work we describe a construction of semistable fibrations over an elliptic curve with one unique singular fibre and we give effective examples using monodromy of curves.
Let $E$ be a smooth elliptic curve over $\mathbb{C}$. For $E$ embedded into $\mathbb{P}^2$ as Hesse cubic and $V$ an Ulrich bundle on $E$ we derive a explicit presentation of $V$ using Moore matrices and theta functions.
