Displaying similar documents to “Some remarks on almost rational torsion points”

A note on the ramification of torsion points lying on curves of genus at least two

Damian Rössler (2010)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let C be a curve of genus g ≥ 2 defined over the fraction field K of a complete discrete valuation ring R with algebraically closed residue field. Suppose that char ( K ) = 0 and that the characteristic p of the residue field is not 2 . Suppose that the Jacobian Jac ( C ) has semi-stable reduction over R . Embed C in Jac ( C ) using a K -rational point. We show that the coordinates of the torsion points lying on C lie in the unique tamely ramified quadratic extension of the field generated over K by the coordinates of...

On rational torsion points of central ℚ -curves

Fumio Sairaiji, Takuya Yamauchi (2008)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let E be a central ℚ -curve over a polyquadratic field k . In this article we give an upper bound for prime divisors of the order of the k -rational torsion subgroup E t o r s ( k ) (see Theorems 1.1 and 1.2). The notion of central ℚ -curves is a generalization of that of elliptic curves over ℚ . Our result is a generalization of Theorem 2 of Mazur [], and it is a precision of the upper bounds of Merel [] and Oesterlé [].

On a dynamical Brauer–Manin obstruction

Liang-Chung Hsia, Joseph Silverman (2009)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let ϕ : X → X be a morphism of a variety defined over a number field  K , let  V ⊂ X be a K -subvariety, and let  𝒪 ϕ ( P ) = { ϕ n ( P ) : n ≥ 0 } be the orbit of a point  P ∈ X ( K ) . We describe a local-global principle for the intersection  V ∩ 𝒪 ϕ ( P ) . This principle may be viewed as a dynamical analog of the Brauer–Manin obstruction. We show that the rational points of  V ( K ) are Brauer–Manin unobstructed for power maps on  ℙ 2 in two cases: (1)  V is a translate of a torus. (2)  V is a line and  P has a preperiodic coordinate. A key tool in the proofs is the...

Torsion and Tamagawa numbers

Dino Lorenzini (2011)

Annales de l’institut Fourier

Similarity:

Let K be a number field, and let A / K be an abelian variety. Let c denote the product of the Tamagawa numbers of A / K , and let A ( K ) tors denote the finite torsion subgroup of A ( K ) . The quotient c / | A ( K ) tors | is a factor appearing in the leading term of the L -function of A / K in the conjecture of Birch and Swinnerton-Dyer. We investigate in this article possible cancellations in this ratio. Precise results are obtained for elliptic curves over ℚ or quadratic extensions K / ℚ , and for abelian surfaces A / ℚ . The smallest possible...

Invariants and coinvariants of semilocal units modulo elliptic units

Stéphane Viguié (2012)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let p be a prime number, and let k be an imaginary quadratic number field in which p decomposes into two primes 𝔭 and 𝔭 ¯ . Let k ∞ be the unique ℤ p -extension of k which is unramified outside of 𝔭 , and let K ∞ be a finite extension of k ∞ , abelian over k . Let 𝒰 ∞ / 𝒞 ∞ be the projective limit of principal semi-local units modulo elliptic units. We prove that the various modules of invariants and coinvariants of 𝒰 ∞ / 𝒞 ∞ are finite. Our approach uses distributions and the p -adic L -function, as defined in []. ...