Torsion points in families of Drinfeld modules

Dragos Ghioca; Liang-Chung Hsia

Displaying similar documents to “Torsion points in families of Drinfeld modules”

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...

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...

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...

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.

Structure of central torsion Iwasawa modules

Susan Howson (2002)

Bulletin de la Société Mathématique de France

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We describe an approach to determining, up to pseudoisomorphism, the structure of a central-torsion module over the Iwasawa algebra of a pro- $p$ , $p$ -adic, Lie group containing no element of order $p$ . The techniques employed follow classical methods used in the commutative case, but using Ore’s method of localisation. We then consider the properties of certain invariants which may prove useful in determining the structure of a module. Finally, we describe the case of pro- $p$ subgroups of ${GL}_{2} (ℤ_{p})$ ...

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 $τ$ -extending modules

Y. Talebi, R. Mohammadi (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we introduce the concept of $τ$ -extending modules by $τ$ -rational submodules and study some properties of such modules. It is shown that the set of all $τ$ -rational left ideals of $_{R} R$ is a Gabriel filter. An $R$ -module $M$ is called $τ$ -extending if every submodule of $M$ is $τ$ -rational in a direct summand of $M$ . It is proved that $M$ is $τ$ -extending if and only if $M = R e j_{M} E (R / τ (R)) \oplus N$ , such that $N$ is a $τ$ -extending submodule of $M$ . An example is given to show that the direct sum of $τ$ -extending modules need not...

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.

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...

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.

Recollements induced by good (co)silting dg-modules

Rongmin Zhu, Jiaqun Wei (2023)

Czechoslovak Mathematical Journal

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Let $U$ be a dg- $A$ -module, $B$ the endomorphism dg-algebra of $U$ . We know that if $U$ is a good silting object, then there exist a dg-algebra $C$ and a recollement among the derived categories $𝐃 (C, d)$ of $C$ , $𝐃 (B, d)$ of $B$ and $𝐃 (A, d)$ of $A$ . We investigate the condition under which the induced dg-algebra $C$ is weak nonpositive. In order to deal with both silting and cosilting dg-modules consistently, the notion of weak silting dg-modules is introduced. Thus, similar results for good cosilting dg-modules are obtained....

Coherence relative to a weak torsion class

Zhanmin Zhu (2018)

Czechoslovak Mathematical Journal

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Let $R$ be a ring. A subclass $𝒯$ of left $R$ -modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let $𝒯$ be a weak torsion class of left $R$ -modules and $n$ a positive integer. Then a left $R$ -module $M$ is called $𝒯$ -finitely generated if there exists a finitely generated submodule $N$ such that $M / N \in 𝒯$ ; a left $R$ -module $A$ is called $(𝒯, n)$ -presented if there exists an exact sequence of left $R$ -modules $0 ⟶ K_{n - 1} ⟶ F_{n - 1} ⟶ \dots ⟶ F_{1} ⟶ F_{0} ⟶ M ⟶ 0$ such that $F_{0}, \dots, F_{n - 1}$ are finitely generated free and $K_{n - 1}$ is $𝒯$ -finitely generated;...

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...

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 $m$ .

Local cohomology, cofiniteness and homological functors of modules

Kamal Bahmanpour (2022)

Czechoslovak Mathematical Journal

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Let $I$ be an ideal of a commutative Noetherian ring $R$ . It is shown that the $R$ -modules $H_{I}^{j} (M)$ are $I$ -cofinite for all finitely generated $R$ -modules $M$ and all $j \in ℕ_{0}$ if and only if the $R$ -modules ${Ext}_{R}^{i} (N, H_{I}^{j} (M))$ and ${Tor}_{i}^{R} (N, H_{I}^{j} (M))$ are $I$ -cofinite for all finitely generated $R$ -modules $M$ , $N$ and all integers $i, j \in ℕ_{0}$ .

A descent map for curves with totally degenerate semi-stable reduction

Shahed Sharif (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $K$ be a local field of residue characteristic $p$ . Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to- $p$ rational torsion subgroup on the Jacobian of $C$ . We also determine divisibility of line bundles on $C$ , including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of $C$ .

Cominimaxness of local cohomology modules

Moharram Aghapournahr (2019)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ . Let $t \in ℕ_{0}$ be an integer and $M$ an $R$ -module such that ${Ext}_{R}^{i} (R / I, M)$ is minimax for all $i \leq t + 1$ . We prove that if $H_{I}^{i} (M)$ is ${FD}_{\leq 1}$ (or weakly Laskerian) for all $i < t$ , then the $R$ -modules $H_{I}^{i} (M)$ are $I$ -cominimax for all $i < t$ and ${Ext}_{R}^{i} (R / I, H_{I}^{t} (M))$ is minimax for $i = 0, 1$ . Let $N$ be a finitely generated $R$ -module. We prove that ${Ext}_{R}^{j} (N, H_{I}^{i} (M))$ and ${Tor}_{j}^{R} (N, H_{I}^{i} (M))$ are $I$ -cominimax for all $i$ and $j$ whenever $M$ is minimax and $H_{I}^{i} (M)$ is ${FD}_{\leq 1}$ (or weakly Laskerian) for all $i$ .

A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot (2012)

Bulletin de la Société Mathématique de France

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If $X$ is a smooth scheme over a perfect field of characteristic $p$ , and if $𝒟_{X}^{(\infty)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of $𝒟_{X}^{(\infty)}$ on an $𝒪_{X}$ -module $ℰ$ is equivalent to giving an infinite sequence of $𝒪_{X}$ -modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characteristic $p$ , endowed with Frobenius liftings. We also show that it...