Displaying similar documents to “A note on the torsion of the Jacobians of superelliptic curves y q = x p + a

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

Tomasz Jędrzejak (2016)

Acta Arithmetica

Similarity:

Consider two families of hyperelliptic curves (over ℚ), C n , a : y ² = x + a and C n , a : y ² = x ( x + a ) , and their respective Jacobians J n , a , J n , a . We give a partial characterization of the torsion part of J n , a ( ) and J n , a ( ) . 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 ( ) . Namely, we show that J 8 , a ( ) t o r s = J 8 , a ( ) [ 2 ] . In addition, we characterize the torsion parts of J p , a ( ) , where p is an odd...

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

Tomasz Jędrzejak (2013)

Acta Arithmetica

Similarity:

Consider the families of curves C n , A : y ² = x + A x and C n , A : y ² = x + 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 ( ) and C 3 , A ( ) are well known. We show that for any nonzero rational A the torsion subgroup of J 7 , A ( ) is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to J 7 , A ( ) [ 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...

On a family of elliptic curves of rank at least 2

Kalyan Chakraborty, Richa Sharma (2022)

Czechoslovak Mathematical Journal

Similarity:

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

The importance of rational extensions

Frans Loonstra (1988)

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

Similarity:

The rational completion M ¯ of an R -module M can be characterized as a τ M -injective hull of M with respect to a (hereditary) torsion functor τ M depending on M . Properties of a torsion functor depending on an R -module M are studied.

An effective proof of the hyperelliptic Shafarevich conjecture

Rafael von Känel (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let C be a hyperelliptic curve of genus g 1 over a number field K with good reduction outside a finite set of places S of K . We prove that C has a Weierstrass model over the ring of integers of K with height effectively bounded only in terms of g , S and K . In particular, we obtain that for any given number field K , finite set of places S of K and integer g 1 one can in principle determine the set of K -isomorphism classes of hyperelliptic curves over K of genus g with good reduction outside...

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

Similarity:

We show a p -parity result in a D 2 p n -extension of number fields L / K ( p 5 ) for the twist 1 η τ : W ( E / K , 1 η τ ) = ( - 1 ) 1 η 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...

Elliptic curves with ( [ 3 ] ) = ( ζ 3 ) and counterexamples to local-global divisibility by 9

Laura Paladino (2010)

Journal de Théorie des Nombres de Bordeaux

Similarity:

We give a family h , β of elliptic curves, depending on two nonzero rational parameters β and h , such that the following statement holds: let be an elliptic curve and let [ 3 ] be its 3-torsion subgroup. This group verifies ( [ 3 ] ) = ( ζ 3 ) if and only if belongs to h , β . Furthermore, we consider the problem of the local-global divisibility by 9 for points of elliptic curves. The number 9 is one of the few exceptional powers of primes, for which an answer...

An arithmetic Riemann-Roch theorem for pointed stable curves

Gérard Freixas Montplet (2009)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let ( 𝒪 , Σ , F ) be an arithmetic ring of Krull dimension at most 1, 𝒮 = Spec 𝒪 and ( π : 𝒳 𝒮 ; σ 1 , ... , σ n ) an n -pointed stable curve of genus g . Write 𝒰 = 𝒳 j σ j ( 𝒮 ) . The invertible sheaf ω 𝒳 / 𝒮 ( σ 1 + + σ n ) inherits a hermitian structure · hyp from the dual of the hyperbolic metric on the Riemann surface 𝒰 . In this article we prove an arithmetic Riemann-Roch type theorem that computes the arithmetic self-intersection of ω 𝒳 / 𝒮 ( σ 1 + ... + σ n ) hyp . The theorem is applied to modular curves X ( Γ ) , Γ = Γ 0 ( p ) or Γ 1 ( p ) , p 11 prime, with sections given by the cusps. We show Z ' ( Y ( Γ ) , 1 ) e a π b Γ 2 ( 1 / 2 ) c L ( 0 , Γ ) , with p 11 m o d 12 when Γ = Γ 0 ( p ) . Here Z ( Y ( Γ ) , s ) is the Selberg...

On the birational gonalities of smooth curves

E. Ballico (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let C be a smooth curve of genus g . For each positive integer r the birational r -gonality s r ( C ) of C is the minimal integer t such that there is L Pic t ( C ) with h 0 ( C , L ) = r + 1 . Fix an integer r 3 . In this paper we prove the existence of an integer g r such that for every integer g g r there is a smooth curve C of genus g with s r + 1 ( C ) / ( r + 1 ) > s r ( C ) / r , i.e. in the sequence of all birational gonalities of C at least one of the slope inequalities fails.

Purity of level m stratifications

Marc-Hubert Nicole, Adrian Vasiu, Torsten Wedhorn (2010)

Annales scientifiques de l'École Normale Supérieure

Similarity:

Let k be a field of characteristic p > 0 . Let D m be a BT m over k (i.e., an m -truncated Barsotti–Tate group over k ). Let S be a k -scheme and let X be a BT m over S . Let S D m ( X ) be the subscheme of S which describes the locus where X is locally for the fppf topology isomorphic to D m . If p 5 , we show that S D m ( X ) is pure in S , i.e. the immersion S D m ( X ) S is affine. For p { 2 , 3 } , we prove purity if D m satisfies a certain technical property depending only on its p -torsion D m [ p ] . For p 5 , we apply the developed techniques to show that...

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Given an integer n 3 , let u 1 , ... , u n be pairwise coprime integers 2 , 𝒟 a family of nonempty proper subsets of { 1 , ... , n } with “enough” elements, and ε a function 𝒟 { ± 1 } . Does there exist at least one prime q such that q divides i I u i - ε ( I ) for some I 𝒟 , but it does not divide u 1 u n ? We answer this question in the positive when the u i are prime powers and ε and 𝒟 are subjected to certain restrictions. We use the result to prove that, if ε 0 { ± 1 } and A is a set of three or more primes that contains all prime divisors of any...