arXiv 2201.023832022-01-07math.NT

The regulator dominates the rank

Fabien Pazuki

英文摘要：

After noticing that the regulator of a number field dominates the rank of its group of units, we bound from below the regulator of the Mordell-Weil group of elliptic curves over global function fields of characteristic $p\geq5$. The lower bound is an increasing function of the rank and of the height.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

Regulators of rank one quadratic twists

We investigate the regulators of elliptic curves with rank 1 in some families of quadratic twists of a fixed elliptic curve. In particular, we formulate some conjectures on the average size of these regulators. We also describe an efficient algorithm to compute explicitly some of the invariants of an odd quadratic twist of an elliptic curve (regulator, order of the Tate-Shafarevich group, etc.) and we discuss the numerical data that we obtain and compare it with our predictions.

2008-02-06math.NT

Class Number and Regulator Computation in Purely Cubic Function Fields of Unit Rank Two

We describe and give computational results of a procedure to compute the divisor class number and regulator of most purely cubic function fields of unit rank 2. Our implementation is an improvement to Pollard's Kangaroo method in infrastructures, using distribution results of class numbers as well as information on the congruence class of the divisor class number, and an adaptation that efficiently navigates these torus-shaped infrastructures. Moreover, this is the first time that an efficient "square-root" algorithm has been applied to the infrastructure of a global field of unit rank 2. With…

2010-01-22math.NT

Heights, ranks and regulators of abelian varieties

We lower bound the Faltings height of an abelian variety over a number field by the sum of its injectivity diameter and the norm of its bad reduction primes. It leads to an unconditional bound on the rank of Mordell-Weil groups. Assuming the height conjecture of Lang and Silverman, we then obtain a Northcott property for the regulator on the set of simple abelian varieties defined over a fixed number field, of fixed dimension $g$, bounded rank and with dense rational points over a number field. We remove the simplicity assumption in the principally polarized case by giving a refined version of…

2016-10-06math.NT

Regulators of elliptic curves

We study the regulator of the Mordell-Weil group of elliptic curves over number fields, functions fields of characteristic zero or function fields of characteristic $p>0$. We prove a new Northcott property for the regulator of elliptic curves of rank at least $4$ defined over a number field. In the function field case, we obtain that non-trivial families of elliptic curves with bounded regulators and bounded ranks are limited families. We conclude by asking new questions.

2019-02-27math.NT

关于 Leopoldt 和 Gross 的调节映射的秩

On the rank of Leopoldt's and Gross's regulator maps

Universität Paderborn（帕德博恩大学）

本文推广了 Waldschmidt 对 Leopoldt 缺陷的界，并证明了 Gross 缺陷的界，同时证明了 Gross 有限性猜想的新情况，并展示了 Gross p 进调节的秩。

2023-02-16math.NT

函数域算术中的调节器

Regulators in the Arithmetic of Function Fields

Sorbonne Université（索邦大学）；Université Paris Cité（巴黎西岱大学）；CNRS, IMJ-PRG（法国国家科学研究中心，巴黎数学研究所）

本文研究函数域算术中Anderson A-动机的调节器概念，证明了A-动机上同调的有限性，并在特定重量条件下显示调节器源和目标的维度相同，但发现调节器像可能不具有满秩，影响了Beilinson猜想的类比。

2026-09-17math.AGmath.NT

更多相关论文收起更多论文

A case of the Rodriguez Villegas conjecture

Let L be a number field and let E be any subgroup of the units O_L^* of L. If rank(E) = 1, Lehmer's conjecture predicts that the height of any non-torsion element of E is bounded below by an absolute positive constant. If rank(E) = rank(O_L^*), Zimmert proved a lower bound on the regulator of E which grows exponentially with [L:Q]. Fernando Rodriguez Villegas made a conjecture in 2002 that "interpolates" between these two extremes of rank. Here we prove a high-rank case of this conjecture. Namely, it holds if L contains a subfield K for which [L:K] >> [K:Q] and E contains the kernel of the nor…

2019-03-04math.NT

$p$-adic $L$-函数的二阶导数与秩二CM椭圆曲线的Shafarevich--Tate群

Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves

本文为秩二CM椭圆曲线证明了分圆型判据：在显式排除的素数外，p-adic L函数二阶Taylor系数为单位当且仅当分圆调节子为单位且Sha消失，并解决了Coates等人遗留的p=577情形。

2026-09-09math.NT

The Fried average entropy and slow entropy for actions of higher rank abelian groups

We consider two numerical entropy--type invariants for actions of $\Zk$, invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case, i.e. $\Zk,\,k\ge 2$ actions on $k+1$-dimensional manifolds. In this case we show that for a fixed dimension (or, equivalently, rank) each of the invariants determines the other and their values are closely related to regulators in algebraic number fields. In particular, in contrast with the classical case of $\mathbb Z$ actions the entropies of ergodic maximal…

2014-07-16math.DSmath.NT

Families of Thue equations associated with a rank one subgroup of the unit group of a number field

Twisting a binary form $F_0(X,Y)\in{\mathbb{Z}}[X,Y]$ of degree $d\ge 3$ by powers $\upsilon^a$ ($a\in{\mathbb{Z}}$) of an algebraic unit $\upsilon$ gives rise to a binary form $F_a(X,Y)\in{\mathbb{Z}}[X,Y]$. More precisely, when $K$ is a number field of degree $d$, $σ_1,σ_2,\dots,σ_d$ the embeddings of $K$ into $\mathbb{C}$, $α$ a nonzero element in $K$, $a_0\in{\mathbb{Z}}$, $a_0>0$ and $$ F_0(X,Y)=a_0\displaystyle\prod_{i=1}^d (X-σ_i(α) Y), $$ then for $a\in{\mathbb{Z}}$ we set $$ F_a(X,Y)=\displaystyle a_0\prod_{i=1}^d (X-σ_i(α\upsilon^a) Y). $$ Given $m\ge 0$, our main result is an effect…

2017-01-05math.NT

Remarks on Euclidean Minima

The Euclidean minimum $M(K)$ of a number field $K$ is an important numerical invariant that indicates whether $K$ is norm-Euclidean. When $K$ is a non-CM field of unit rank 2 or higher, Cerri showed $M(K)$, as the supremum in the Euclidean spectrum $Spec(K)$, is isolated and attained and can be computed in finite time. We extend Cerri's works by applying recent dynamical results of Lindenstrauss and Wang. In particular, the following facts are proved: (1) For any number field $K$ of unit rank 3 or higher, $M(K)$ is isolated and attained and Cerri's algorithm computes $M(K)$ in finite time. (2)…

2012-07-21math.DSmath.NT