Analogy between arithmetic of elliptic curves and conics
英文摘要：
In this brief note we bring out the analogy between the arithmetic of elliptic curves and the Riemann zeta-function.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
相关论文
关于含一条圆锥曲线的两个二次曲面交的算术
On the arithmetic of intersections of two quadrics containing a conic
本文重录1993年未发表手稿，证明了Colliot-Thélène、Sansuc与Swinnerton-Dyer关于含圆锥曲线的两个二次曲面交算术性质的两个猜想。
math.NT
Arithmetic Progressions on Conic Sections
The set ${1, 25, 49}$ is a 3-term collection of integers which forms an arithmetic progression of perfect squares. We view the set ${(1,1), (5,25), (7,49)}$ as a 3-term collection of rational points on the parabola $y=x^2$ whose $y$-coordinates form an arithmetic progression. In this exposition, we provide a generalization to 3-term arithmetic progressions on arbitrary conic sections $\mathcal C$ with respect to a linear rational map $\ell: \mathcal C \to \mathbb P^1$. We explain how this construction is related to rational points on the universal elliptic curve $Y^2 + 4XY + 4kY = X^3 + kX^2$ …
math.NT
算术曲线约化中的非分裂圆锥曲线
Nonsplit conics in the reduction of an arithmetic curve
本文研究了代数函数域在离散赋值扩展下的剩余域性质，证明了非直纹剩余超越扩张的数量由函数域亏格界定，并给出了在实形式幂级数域上曲线函数域平方和问题的应用。
math.AGmath.NT
Conics - a Poor Man's Elliptic Curves
The aim of this article is to show that the arithmetic of Pell conics admits a description which is completely analogous to that of elliptic curves: there is a theory of 2-descent with associated Selmer and Tate-Shafarevich groups, and there should be an analog of the conjecture of Birch and Swinnerton-Dyer.
math.HOmath.NT
On the analogy between L-functions and Atiyah-Bott-Lefschetz trace formulas for foliated spaces
This paper is motivated by Deninger's programme. First we prove, using Alvarez Lopez-Kordyukov results, an Atiyah-Bott-Lefschetz trace formula for the cohomology groups associated to a ramified leafwise flat line bundle on a riemannian foliation. Then we argue, by precise computations, that cohomology groups associated to ramified leafwise flat vector bundles on a suitable foliated space (whose existence is still unknown) might be useful to study arithmetic L-functions via Atiyah-Bott-Lefschetz trace formulas.
math.NT
Geometric and arithmetic aspects of rational elliptic surfaces
PhD dissertation consists in three lines of investigation involving rational elliptic surfaces, namely 1) a study of conic bundles on these surfaces; 2) an investigation of the possible intersection numbers of two sections and 3) a theorem regarding the set of fibers whose Mordell-Weil ranks are at least 3 units higher than the generic rank of the elliptic fibration.
math.AGmath.NT
更多相关论文收起更多论文
低次数del Pezzo曲面的弱逼近
Weak approximation for del Pezzo surfaces of low degree
本研究受Denef工作启发，采用算术满射性方法，证明满足一般性假设的带两个圆锥纤维化的曲面成立弱弱逼近，且次数为1或2的带圆锥纤维化的一般del Pezzo曲面也满足该性质。
math.AGmath.NT
Applications of an Equivariant Etale Cohomology to Arithmetic Topology
A. Sikora has shown results which confirm the analogy between number fields and 3-manifolds. However, he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's formulas which leads us to clarify and to extend the dictionary of arithmetic topology.
math.AGmath.ATmath.NT
Motivic & Arithmetic probability of a semistable elliptic surface with a Weierstrass torsion section
We prove new sharp asymptotic for counting the semistable elliptic curves with two marked Weierstrass points at $\infty$ and $0$ and also the cases where $0$ is a 2-torsion or a 3-torsion marked Weierstrass point over $\mathbb{F}_q(t)$ by the bounded height of discriminant $Δ(X)$. We consider the motivic probabilities over any basefield $K$ with $\text{char}(K) \neq 2,3$ of picking a nonsingular semistable elliptic surface over $\mathbb{P}^{1}$ with two marked Weierstrass sections at $\infty$ and $0$ such that marked Weierstrass section at $0$ is 2-torsion or 3-torsion. In the end, we formulat…
math.AGmath.NT
Arithmetic of the moduli of semistable elliptic surfaces
We prove a new sharp asymptotic with the lower order term of zeroth order on $\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})$ for counting the semistable elliptic curves over $\mathbb{F}_q(t)$ by the bounded height of discriminant $Δ(X)$. The precise count is acquired by considering the moduli of nonsingular semistable elliptic fibrations over $\mathbb{P}^{1}$, also known as semistable elliptic surfaces, with $12n$ nodal singular fibers and a distinguished section. We establish a bijection of $K$-points between the moduli functor of semistable elliptic surfaces and the stack of morphisms $\mathcal…
math.AGmath.ATmath.NT
