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

Christophe Delaunay; Frédéric Jouhet

Displaying similar documents to “The Cohen-Lenstra heuristics, moments and $p^{j}$ -ranks of some groups”

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...

$K (π, 1)$ 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,...

Some examples of 5 and 7 descent for elliptic curves over $Q$

Tom Fisher (2001)

Journal of the European Mathematical Society

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We perform descent calculations for the families of elliptic curves over $Q$ with a rational point of order $n = 5$ or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over $Q$ may become arbitrarily large.

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

David L. Johnson (1982)

Colloquium Mathematicae

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Deformation theory and finite simple quotients of triangle groups I

Michael Larsen, Alexander Lubotzky, Claude Marion (2014)

Journal of the European Mathematical Society

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Let $2 \leq a \leq b \leq c \in ℕ$ with $μ = 1 / a + 1 / b + 1 / c < 1$ and let $T = T_{a, b, c} = 〈 x, y, z : x^{a} = y^{b} = z^{c} = x y z = 1 〉$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$ ? (Classically, for $(a, b, c) = (2, 3, 7)$ and more recently also for general $(a, b, c)$ .) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially...

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).

On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman–De Bruijn function

Ke-Pao Lin, Xue Luo, Stephen S.-T. Yau, Huaiqing Zuo (2014)

Journal of the European Mathematical Society

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It is well known that getting the estimate of integral points in right-angled simplices is equivalent to getting the estimate of Dickman-De Bruijn function $ψ (x, y)$ which is the number of positive integers $\leq x$ and free of prime factors $> y$ . Motivating from the Yau Geometry Conjecture, the third author formulated the Number Theoretic Conjecture which gives a sharp polynomial upper estimate that counts the number of positive integral points in n-dimensional ( $n \geq 3$ ) real right-angled simplices. In this...

Characterization of the alternating groups by their order and one conjugacy class length

Alireza Khalili Asboei, Reza Mohammadyari (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group, and let $N (G)$ be the set of conjugacy class sizes of $G$ . By Thompson’s conjecture, if $L$ is a finite non-abelian simple group, $G$ is a finite group with a trivial center, and $N (G) = N (L)$ , then $L$ and $G$ are isomorphic. Recently, Chen et al. contributed interestingly to Thompson’s conjecture under a weak condition. They only used the group order and one or two special conjugacy class sizes of simple groups and characterized successfully sporadic simple groups (see Li’s PhD dissertation)....

Greenberg’s conjecture for $ℤ_{p}^{d}$ -extensions

Andrea Bandini (2003)

Acta Arithmetica

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On coarse embeddability into $ℓ_{p}$ -spaces and a conjecture of Dranishnikov

Piotr W. Nowak (2006)

Fundamenta Mathematicae

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We show that the Hilbert space is coarsely embeddable into any $ℓ_{p}$ for 1 ≤ p ≤ ∞. It follows that coarse embeddability into ℓ₂ and into $ℓ_{p}$ are equivalent for 1 ≤ p < 2.

Homotopy invariance of higher signatures and $3$ -manifold groups

Michel Matthey, Hervé Oyono-Oyono, Wolfgang Pitsch (2008)

Bulletin de la Société Mathématique de France

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For closed oriented manifolds, we establish oriented homotopy invariance of higher signatures that come from the fundamental group of a large class of orientable $3$ -manifolds, including the “piecewise geometric” ones in the sense of Thurston. In particular, this class, that will be carefully described, is the class of all orientable $3$ -manifolds if the Thurston Geometrization Conjecture is true. In fact, for this type of groups, we show that the Baum-Connes Conjecture With Coefficients...

A problem of Kollár and Larsen on finite linear groups and crepant resolutions

Robert Guralnick, Pham Tiep (2012)

Journal of the European Mathematical Society

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The notion of age of elements of complex linear groups was introduced by M. Reid and is of importance in algebraic geometry, in particular in the study of crepant resolutions and of quotients of Calabi–Yau varieties. In this paper, we solve a problem raised by J. Kollár and M. Larsen on the structure of finite irreducible linear groups generated by elements of age $\leq 1$ . More generally, we bound the dimension of finite irreducible linear groups generated by elements of bounded deviation....

Recent progress on the Jacobian Conjecture

Michiel de Bondt, Arno van den Essen (2005)

Annales Polonici Mathematici

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We describe some recent developments concerning the Jacobian Conjecture (JC). First we describe Drużkowski’s result in [6] which asserts that it suffices to study the JC for Drużkowski mappings of the form $x + {(A x)}^{* 3}$ with A² = 0. Then we describe the authors’ result of [2] which asserts that it suffices to study the JC for so-called gradient mappings, i.e. mappings of the form x - ∇f, with $f \in k^{[n]}$ homogeneous of degree 4. Using this result we explain Zhao’s reformulation of the JC which asserts the...

The strength of the projective Martin conjecture

C. T. Chong, Wei Wang, Liang Yu (2010)

Fundamenta Mathematicae

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We show that Martin’s conjecture on Π¹₁ functions uniformly $\leq_{T}$ -order preserving on a cone implies Π¹₁ Turing Determinacy over ZF + DC. In addition, it is also proved that for n ≥ 0, this conjecture for uniformly degree invariant $Π ¹_{2 n + 1}$ functions is equivalent over ZFC to $Σ ¹_{2 n + 2}$ -Axiom of Determinacy. As a corollary, the consistency of the conjecture for uniformly degree invariant Π¹₁ functions implies the consistency of the existence of a Woodin cardinal.

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

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We prove that Bloch’s conjecture is true for surfaces with $p_{g} = 0$ obtained as $0$ -sets $X_{σ}$ of a section $σ$ of a very ample vector bundle on a variety $X$ with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as $0$ on holomorphic $2$ -forms of $X_{σ}$ , then it acts as $0$ on $0$ -cycles of degree $0$ of $X_{σ}$ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general $X_{σ}$ ...

The density of representation degrees

Martin Liebeck, Dan Segal, Aner Shalev (2012)

Journal of the European Mathematical Society

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For a group $G$ and a positive real number $x$ , define $d_{G} (x)$ to be the number of integers less than $x$ which are dimensions of irreducible complex representations of $G$ . We study the asymptotics of $d_{G} (x)$ for algebraic groups, arithmetic groups and finitely generated linear groups. In particular we prove an “alternative” for finitely generated linear groups $G$ in characteristic zero, showing that either there exists $α > 0$ such that $d_{G} (x) > x^{α}$ for all large $x$ , or $G$ is virtually abelian (in which case $d_{G} (x)$ is bounded). ...

Invariance of the parity conjecture for $p$ -Selmer groups of elliptic curves in a $D_{2 p^{n}}$ -extension

Thomas de La Rochefoucauld (2011)

Bulletin de la Société Mathématique de France

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We show a $p$ -parity result in a $D_{2 p^{n}}$ -extension of number fields $L / K$ ( $p \geq 5$ ) for the twist $1 \oplus η \oplus τ$ : $W (E / K, 1 \oplus η \oplus τ) = {(- 1)}^{〈1 \oplus η \oplus τ, X_{p} (E / L)〉}$ , where $E$ is an elliptic curve over $K$ , $η$ and $τ$ are respectively the quadratic character and an irreductible representation of degree $2$ of $Gal (L / K) = D_{2 p^{n}}$ , and $X_{p} (E / L)$ is the $p$ -Selmer group. The main novelty is that we use a congruence result between $ε_{0}$ -factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction above 2 and 3). We also give applications to the $p$ -parity conjecture...

Regularity of sets with constant intrinsic normal in a class of Carnot groups

Marco Marchi (2014)

Annales de l’institut Fourier

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In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type $☆$ ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $☆$ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano...

On a conjecture of Dekking : The sum of digits of even numbers

Iurie Boreico, Daniel El-Baz, Thomas Stoll (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $q \geq 2$ and denote by $s_{q}$ the sum-of-digits function in base $q$ . For $j = 0, 1, \dots, q - 1$ consider $# {0 \leq n < N : s_{q} (2 n) \equiv j (mod q)} .$ In 1983, F. M. Dekking conjectured that this quantity is greater than $N / q$ and, respectively, less than $N / q$ for infinitely many $N$ , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

On NIP and invariant measures

Ehud Hrushovski, Anand Pillay (2011)

Journal of the European Mathematical Society

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We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if $p = tp (b / A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd (A)$ , (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ ...

A basis of ℤₘ, II

Min Tang, Yong-Gao Chen (2007)

Colloquium Mathematicae

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Given a set A ⊂ ℕ let $σ_{A} (n)$ denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, $σ_{A} (n)$ is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and $σ_{A} (n ̅) \leq 5120$ for all n̅ ∈ ℤₘ.

Displaying similar documents to “The Cohen-Lenstra heuristics, moments and p j -ranks of some groups”

Displaying similar documents to “The Cohen-Lenstra heuristics, moments and $p^{j}$ -ranks of some groups”