Torsion units for some almost simple groups

Joe Gildea

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Displaying similar documents to “Torsion units for some almost simple groups”

The cuspidal torsion packet on hyperelliptic Fermat quotients

David Grant, Delphy Shaulis (2004)

Journal de Théorie des Nombres de Bordeaux

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Let $ℓ \geq 7$ be a prime, $C$ be the non-singular projective curve defined over $ℚ$ by the affine model $y (1 - y) = x^{ℓ}$ , $\infty$ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$ , and $φ : C \to J$ the albanese embedding with $\infty$ as base point. Let $\bar{ℚ}$ be an algebraic closure of $ℚ$ . Taking care of a case not covered in [], we show that $φ (C) \cap J_{tors} (\bar{ℚ})$ consists only of the image under $φ$ of the Weierstrass points of $C$ and the points $(x, y) = (0, 0)$ and $(0, 1)$ , where $J_{tors}$ denotes the torsion points of $J$ .

Torsion points in families of Drinfeld modules

Dragos Ghioca, Liang-Chung Hsia (2013)

Acta Arithmetica

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Let $Φ^{λ}$ be an algebraic family of Drinfeld modules defined over a field K of characteristic p, and let a,b ∈ K[λ]. Assume that neither a(λ) nor b(λ) is a torsion point for $Φ^{λ}$ for all λ. If there exist infinitely many λ ∈ K̅ such that both a(λ) and b(λ) are torsion points for $Φ^{λ}$ , then we show that for each λ ∈ K̅, a(λ) is torsion for $Φ^{λ}$ if and only if b(λ) is torsion for $Φ^{λ}$ . In the case a,b ∈ K, we prove in addition that a and b must be $̅_{p}$ -linearly dependent.

The Cohen-Lenstra heuristics, moments and $p^{j}$ -ranks of some groups

Christophe Delaunay, Frédéric Jouhet (2014)

Acta Arithmetica

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This article deals with the coherence of the model given by the Cohen-Lenstra heuristic philosophy for class groups and also for their generalizations to Tate-Shafarevich groups. More precisely, our first goal is to extend a previous result due to É. Fouvry and J. Klüners which proves that a conjecture provided by the Cohen-Lenstra philosophy implies another such conjecture. As a consequence of our work, we can deduce, for example, a conjecture for the probability laws of $p^{j}$ -ranks of...

A note on group algebras of $p$ -primary abelian groups

William Ullery (1995)

Commentationes Mathematicae Universitatis Carolinae

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Suppose $p$ is a prime number and $R$ is a commutative ring with unity of characteristic 0 in which $p$ is not a unit. Assume that $G$ and $H$ are $p$ -primary abelian groups such that the respective group algebras $R G$ and $R H$ are $R$ -isomorphic. Under certain restrictions on the ideal structure of $R$ , it is shown that $G$ and $H$ are isomorphic.

A note on Nakai’s conjecture for the ring $K [X ₁, . . ., X ₙ] / (a ₁ X ₁^{m} + \dots + a ₙ X ₙ^{m})$

Paulo Roberto Brumatti, Marcelo Oliveira Veloso (2011)

Colloquium Mathematicae

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$K (π, 1)$ conjecture for Artin groups

Luis Paris (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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The purpose of this paper is to put together a large amount of results on the $K (π, 1)$ conjecture for Artin groups, and to make them accessible to non-experts. Firstly, this is a survey, containing basic definitions, the main results, examples and an historical overview of the subject. But, it is also a reference text on the topic that contains proofs of a large part of the results on this question. Some proofs as well as few results are new. Furthermore, the text, being addressed to non-experts,...

A new proof of the $L^{p}$ -conjecture for locally compact abelian groups

David L. Johnson (1982)

Colloquium Mathematicae

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Torsion points on families of simple abelian surfaces and Pell's equation over polynomial rings (with an appendix by E. V. Flynn)

David Masser, Umberto Zannier (2015)

Journal of the European Mathematical Society

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In recent papers we proved a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples...

A characterization of Ext(G,ℤ) assuming (V = L)

Saharon Shelah, Lutz Strüngmann (2007)

Fundamenta Mathematicae

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We complete the characterization of Ext(G,ℤ) for any torsion-free abelian group G assuming Gödel’s axiom of constructibility plus there is no weakly compact cardinal. In particular, we prove in (V = L) that, for a singular cardinal ν of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence $(ν_{p} : p \in Π)$ of cardinals satisfying $ν_{p} \leq 2^{ν}$ (where Π is the set of all primes), there is a torsion-free abelian group G of size ν such that $ν_{p}$ equals the p-rank of Ext(G,ℤ)...

On the Davenport constant and group algebras

Daniel Smertnig (2010)

Colloquium Mathematicae

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For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence $S = g ₁ \cdot . . . \cdot g_{l}$ over G such that $(X^{g ₁} - a ₁) \cdot . . . \cdot (X^{g_{l}} - a_{l}) \neq 0 \in K [G]$ for all $a ₁, . . ., a_{l} \in K^{\times}$ . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for...

On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski (2012)

Colloquium Mathematicae

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Let $p_{i}$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n ² - 1 = p ₁^{α ₁} \dots p_{k}^{α_{k}}$ in positive integers n and α₁,..., $α_{k}$ . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).