On the Tate-shafarevich Group of Elliptic Curves over Q
نویسندگان
چکیده
Let E be an elliptic curve over Q. Using Iwasawa theory, we give what seems to be the first general upper bound for the order of vanishing of the p-adic L-function at s = 0, and the Zp-corank of the Tate-Shafarevich group for all sufficiently large good ordinary primes p.
منابع مشابه
Visualizing elements of order three in the Shafarevich-Tate group
1. Introduction. If we wish to write the equations of curves of genus 1 that give elements of the Shafarevich-Tate group of an elliptic curve over a number field K, a choice of ways is open to us. For example, if the element in question is of order 3 the curve of genus 1 corresponding to it occurs as a smooth plane cubic curve over K. In a recent article [C-M] we raised the question of when one...
متن کاملOn the Tate-shafarevich Groups of Certain Elliptic Curves
The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.
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 کاملTHE 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 کاملTate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves
Let E/Q be a fixed elliptic curve over Q which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula for the number of primes p ≤ x such that the reduction of E modulo p has a trivial Tate-Shafarevich group. Recent results of A. C. Cojocaru and C. David lead to a better error term. We introduce a new arg...
متن کامل