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

Saharon Shelah; Lutz Strüngmann

Displaying similar documents to “A characterization of Ext(G,ℤ) assuming (V = L)”

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.

On the torsion of the Jacobians of the hyperelliptic curves y² = xⁿ + a and y² = x(xⁿ+a)

Tomasz Jędrzejak (2016)

Acta Arithmetica

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Consider two families of hyperelliptic curves (over ℚ), $C^{n, a} : y ² = x ⁿ + a$ and $C_{n, a} : y ² = x (x ⁿ + a)$ , and their respective Jacobians $J^{n, a}$ , $J_{n, a}$ . We give a partial characterization of the torsion part of $J^{n, a} (ℚ)$ and $J_{n, a} (ℚ)$ . More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8, a} (ℚ)$ . Namely, we show that $J_{8, a} {(ℚ)}_{t o r s} = J_{8, a} (ℚ) [2]$ . In addition, we characterize the torsion parts of $J_{p, a} (ℚ)$ , where p is an odd...

The torsion subgroup of a family of elliptic curves over the maximal abelian extension of $ℚ$

Jerome Tomagan Dimabayao (2020)

Czechoslovak Mathematical Journal

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We determine explicitly the structure of the torsion group over the maximal abelian extension of $ℚ$ and over the maximal $p$ -cyclotomic extensions of $ℚ$ for the family of rational elliptic curves given by $y^{2} = x^{3} + B$ , where $B$ is an integer.

A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$

Tomasz Jędrzejak (2016)

Banach Center Publications

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This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q, p, a} : y^{q} = x^{p} + a$ , and its Jacobians $J_{q, p, a}$ , where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3, 5, a} (ℚ)$ (resp. $J_{q, p, a} (ℚ)$ ). The main tools are computations of the zeta function of $C_{3, 5, a}$ (resp. $C_{q, p, a}$ ) over $_{l}$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l...

Ordinals in topological groups

Raushan Z. Buzyakova (2007)

Fundamenta Mathematicae

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We show that if an uncountable regular cardinal τ and τ + 1 embed in a topological group G as closed subspaces then G is not normal. We also prove that an uncountable regular cardinal cannot be embedded in a torsion free Abelian group that is hereditarily normal. These results are corollaries to our main results about ordinals in topological groups. To state the main results, let τ be an uncountable regular cardinal and G a T₁ topological group. We prove, among others, the following...

Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Tomasz Jędrzejak (2013)

Acta Arithmetica

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Consider the families of curves $C^{n, A} : y ² = x ⁿ + A x$ and $C_{n, A} : y ² = x ⁿ + A$ where A is a nonzero rational. Let $J^{n, A}$ and $J_{n, A}$ denote their respective Jacobian varieties. The torsion points of $C^{3, A} (ℚ)$ and $C_{3, A} (ℚ)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7, A} (ℚ)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7, A} (ℚ) [2]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3, A}$ (A ≠ 4) and $J^{5, A}$ . We...

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...

The importance of rational extensions

Frans Loonstra (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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The rational completion $\bar{M}$ of an $R$ -module $M$ can be characterized as a $\tau_{M}$ -injective hull of $M$ with respect to a (hereditary) torsion functor $\tau_{M}$ depending on $M$ . Properties of a torsion functor depending on an $R$ -module $M$ are studied.

Prescribing endomorphism algebras of $ℵ_{n}$ -free modules

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2014)

Journal of the European Mathematical Society

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...

The importance of rational extensions

Frans Loonstra (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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On special torsion-free groups

Antonio Machí (1969)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si studia la classe dei gruppi privi di torsione che godono della seguente proprietà: dati comunque due elementi $x$ e $y$ esiste un intero positivo $n=n(x,y)$ tale che $x^{n}y=yx^{n}$ . Si dà una condizione sufficiente perché tali gruppi siano abeliani. Si congettura, infine, che detti gruppi non possano essere semplici.

On a family of elliptic curves of rank at least 2

Kalyan Chakraborty, Richa Sharma (2022)

Czechoslovak Mathematical Journal

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Let $C_{m} : y^{2} = x^{3} - m^{2} x + p^{2} q^{2}$ be a family of elliptic curves over $ℚ$ , where $m$ is a positive integer and $p$ , $q$ are distinct odd primes. We study the torsion part and the rank of $C_{m} (ℚ)$ . More specifically, we prove that the torsion subgroup of $C_{m} (ℚ)$ is trivial and the $ℚ$ -rank of this family is at least 2, whenever $m \neg \equiv 0 (mod 3)$ , $m \neg \equiv 0 (mod 4)$ and $m \equiv 2 (mod 64)$ with neither $p$ nor $q$ dividing $m$ .

On prolongation of connections

Włodzimierz M. Mikulski (2010)

Annales Polonici Mathematici

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Let Y → M be a fibred manifold with m-dimensional base and n-dimensional fibres. Let r, m,n be positive integers. We present a construction $B^{r}$ of rth order holonomic connections $B^{r} (Γ, \nabla) : Y \to J^{r} Y$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M. Then we prove that any construction B of rth order holonomic connections $B (Γ, \nabla) : Y \to J^{r} Y$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M is equal...

Number of solutions in a box of a linear equation in an Abelian group

Maciej Zakarczemny (2016)

Colloquium Mathematicae

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For every finite Abelian group Γ and for all $g, a ₁, . . ., a_{k} \in Γ$ , if there exists a solution of the equation $\sum_{i = 1}^{k} a_{i} x_{i} = g$ in non-negative integers $x_{i} \leq b_{i}$ , where $b_{i}$ are positive integers, then the number of such solutions is estimated from below in the best possible way.

Reflecting character and pseudocharacter

Lucia R. Junqueira, Alberto M. E. Levi (2015)

Commentationes Mathematicae Universitatis Carolinae

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We say that a cardinal function $φ$ reflects an infinite cardinal $κ$ , if given a topological space $X$ with $φ (X) \geq κ$ , there exists $Y \in {[X]}^{\leq κ}$ with $φ (Y) \geq κ$ . We investigate some problems, discussed by Hodel and Vaughan in Reflection theorems for cardinal functions, Topology Appl. 100 (2000), 47–66, and Juhász in Cardinal functions and reflection, Topology Atlas Preprint no. 445, 2000, related to the reflection for the cardinal functions character and pseudocharacter. Among other results, we present some new equivalences...

On the vector-valued Fourier transform and compatibility of operators

In Sook Park (2005)

Studia Mathematica

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Let be a locally compact abelian group and let 1 < p ≤ 2. ’ is the dual group of , and p’ the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform $F^{}$ if $F^{} \otimes T : L_{p} () \otimes X \to L_{p^{'}} (^{'}) \otimes Y$ admits a continuous extension $[F^{}, T] : [L_{p} (), X] \to [L_{p^{'}} (^{'}), Y]$ . Let $ℱ T_{p}^{}$ denote the collection of such T’s. We show that $ℱ T_{p}^{ℝ \times} = ℱ T_{p}^{ℤ \times} = ℱ T_{p}^{ℤ ⁿ \times}$ for any and positive integer n. Moreover, if the factor group of by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then...

On ordinals accessible by infinitary languages

Saharon Shelah, Pauli Väisänen, Jouko Väänänen (2005)

Fundamenta Mathematicae

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Let λ be an infinite cardinal number. The ordinal number δ(λ) is the least ordinal γ such that if ϕ is any sentence of $L_{λ ⁺ ω}$ , with a unary predicate D and a binary predicate ≺, and ϕ has a model ℳ with $⟨ D^{ℳ}, ≺^{ℳ} ⟩$ a well-ordering of type ≥ γ, then ϕ has a model ℳ ’ where $⟨ D^{ℳ^{'}}, ≺^{ℳ^{'}} ⟩$ is non-well-ordered. One of the interesting properties of this number is that the Hanf number of $L_{λ ⁺ ω}$ is exactly $ℶ_{δ (λ)}$ . It was proved in [BK71] that if ℵ₀ < λ < κ $a r e r e g u l a r c a r d i n a l n u m b e r s, t h e n t h e r e i s a f o r c i n g e x t e n s i o n, p r e s e r v i n g c o f i n a l i t i e s, s u c h t h a t i n t h e e x t e n s i o n$ 2λ = κ $a n d δ (λ) < λ ⁺ ⁺ . W e i m p r o v e t h i s r e s u l t b y p r o v i n g t h e f o l l o w i n g : S u p p o s e ℵ ₀ < λ < θ \leq κ a r e c a r d i n a l n u m b e r s s u c h t h a t$ ∙ $λ^{< λ} = λ$ ; ∙ cf(θ) ≥ λ⁺ and $μ^{λ} < θ$ whenever μ < θ; ∙ $κ^{λ} = κ$ . Then there...

How to construct a Hovey triple from two cotorsion pairs

James Gillespie (2015)

Fundamenta Mathematicae

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Let be an abelian category, or more generally a weakly idempotent complete exact category, and suppose we have two complete hereditary cotorsion pairs $(, \tilde{})$ and $(\tilde{},)$ in satisfying $\tilde{} \subseteq$ and $\cap \tilde{} = \tilde{} \cap$ . We show how to construct a (necessarily unique) abelian model structure on with (resp. $\tilde{}$ ) as the class of cofibrant (resp. trivially cofibrant) objects, and (resp. $\tilde{}$ ) as the class of fibrant (resp. trivially fibrant) objects.

$ℵ_{k}$ -free separable groups with prescribed endomorphism ring

Daniel Herden, Héctor Gabriel Salazar Pedroza (2015)

Fundamenta Mathematicae

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We will consider unital rings A with free additive group, and want to construct (in ZFC) for each natural number k a family of $ℵ_{k}$ -free A-modules G which are separable as abelian groups with special decompositions. Recall that an A-module G is $ℵ_{k}$ -free if every subset of size $< ℵ_{k}$ is contained in a free submodule (we will refine this in Definition 3.2); and it is separable as an abelian group if any finite subset of G is contained in a free direct summand of G. Despite the fact that such a...

Self-small products of abelian groups

Josef Dvořák, Jan Žemlička (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $A$ and $B$ be two abelian groups. The group $A$ is called $B$ -small if the covariant functor $Hom (A, -)$ commutes with all direct sums $B^{(κ)}$ and $A$ is self-small provided it is $A$ -small. The paper characterizes self-small products applying developed closure properties of the classes of relatively small groups. As a consequence, self-small products of finitely generated abelian groups are described.

Pairs of square-free values of the type $n^{2} + 1$ , $n^{2} + 2$

Stoyan Dimitrov (2021)

Czechoslovak Mathematical Journal

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We show that there exist infinitely many consecutive square-free numbers of the form $n^{2} + 1$ , $n^{2} + 2$ . We also establish an asymptotic formula for the number of such square-free pairs when $n$ does not exceed given sufficiently large positive number.