Mod-p reducibility, the torsion subgroup, and the Shafarevich-Tate group
نویسنده
چکیده
Let E be an optimal elliptic curve overQ of prime conductorN . We show that if for an odd prime p, the mod p representation associated to E is reducible (in particular, if p divides the order of the torsion subgroup of E(Q)), then the p-primary component of the ShafarevichTate group of E is trivial. We also state a related result for more general abelian subvarieties of J0(N) and mention what to expect if N is not prime.
منابع مشابه
THE p-PART OF TATE-SHAFAREVICH GROUPS OF ELLIPTIC CURVES CAN BE ARBITRARILY LARGE
In this paper it is shown that for every prime p > 5 the dimension of the p-torsion in the Tate-Shafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K : Q] bounded by a constant depending only on p. From this we deduce that the dimension of the p-torsion in the Tate-Shafarevich group of A/Q can be arbitrarily large, where A is an a...
متن کامل787 The p - part of Tate - Shafarevich groups of elliptic curves can be arbitrarily large par REMKE KLOOSTERMAN
In this paper we show that for every prime p ~ 5 the dimension of the p-torsion in the Tate-Shafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K : Q] bounded by a constant depending only on p. From this we deduce that the dimension of the ptorsion in the Tate-Shafarevich group of A/Q can be arbitrarily large, where A is an abelia...
متن کاملDescent via isogeny on elliptic curves with large rational torsion subgroups
We outline PARI programs which assist with various algorithms related to descent via isogeny on elliptic curves. We describe, in this context, variations of standard inequalities which aid the computation of members of the Tate-Shafarevich group. We apply these techniques to several examples: in one case we use descent via 9-isogeny to determine the rank of an elliptic curve; in another case we...
متن کاملVisible Elements of the Shafarevich-tate Group
We study a subgroup of the Shafarevich-Tate group of an abelian variety known as the visible subgroup. We explain the geometric intuition behind this subgroup, prove its finiteness and describe several techniques for exhibiting visible elements. Two important results are proved one what we call the visualization theorem, which asserts that every element of the Shafarevich-Tate group of an abeli...
متن کاملSquareness in the special L-value and special L-values of twists
Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of N−1 12 . Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the s...
متن کامل