Mirela Çiperiani

Solvable points on genus one curves

A genus one curve defined over Q which has points over Qp for all primes p may not have a rational point. It is natural to study the classes of Q-extensions over which all such curves obtain a global point. In this article, we show that every such genus one curve with semistable Jacobian has a point defined over a solvable extension of Q.

Weil–Châtelet divisible elements in Tate–Shafarevich groups I: The Bashmakov problem for elliptic curves over

For an abelian variety A over a number field k we discuss the maximal divisibile subgroup of H^1(k,A) and its intersection with the subgroup Sha(A/k). The results are most complete for elliptic curves over Q.For an abelian variety A over a number field k we discuss the divisibility in H^1(k,A) of elements of the subgroup Sha(A/k). The results are most complete for elliptic curves over Q.

-adic heights of Heegner points and Λ-adic regulators

Let E be an elliptic curve defined over Q. The aim of this paper is to make it possible to compute Heegner L-functions and anticyclotomic Λ-adic regulators of E, which were studied by Mazur-Rubin and Howard. We generalize results of Cohen and Watkins and thereby compute Heegner points of nonfundamental discriminant. We then prove a relationship between the denominator of a point of E defined over a number field and the leading coefficient of the minimal polynomial of its xcoordinate. Using this relationship, we recast earlier work of Mazur, Stein, and Tate to produce effective algorithms to compute p-adic heights of points of E defined over number fields. These methods enable us to give the first explicit examples of Heegner L-functions and anticyclotomic Λ-adic regulators.

Shadow Lines in the Arithmetic of Elliptic Curves

Let \(E/\mathbb{Q}\) be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field \(K/\mathbb{Q}\) satisfying the Heegner hypothesis for E we have a corresponding line in \(E(K) \otimes \mathbb{Q}_{p}\), known as a shadow line. When \(E/\mathbb{Q}\) has analytic rank 2 and E∕K has analytic rank 3, shadow lines are expected to lie in \(E(\mathbb{Q}) \otimes \mathbb{Q}_{p}\). If, in addition, p splits in \(K/\mathbb{Q}\), then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.