We propose a conjectural construction of determinants global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic Heegner via Čerednik–Drinfeld uniformization and definition classical Stark–Heegner points. In alignment with Nekovář Scholl's plectic conjectures, we expect non-triviality these to control Mordell–Weil group higher rank curves. provide some i...

results; out of 634 patients, 61 subjects (9.62%) had significant complications including: acute renal failure, alveolar hemorrhage, ards, clotting disorders, pancreatitis, gastrointestinal bleeding and intracranial bleeding; age, jaundice, renal and pulmonary involvement (oliguria and anuria), hypotension, leukocytosis, thrombocytopenia, hyponatremia, elevated bilirubin, alanine aminotransfera...

We study the structure of the Mordell–Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if T is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup T is empty, or it contains curves of rank 0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other...

We generalize Dirichlet’s S-unit theorem from the usual group of S-units of a number field K to the infinite rank group of all algebraic numbers having nontrivial valuations only on places lying over S. Specifically, we demonstrate that the group of algebraic S-units modulo torsion is a Q-vector space which, when normed by the Weil height, spans a hyperplane determined by the product formula, a...

is the Hecke L-series attached to the eigenform f . Hecke’s theory shows that L(f, s) has an Euler product expansion identical to (2), and also that it admits an integral representation as a Mellin transform of f . This extends L(f, s) analytically to the whole complex plane and shows that it satisfies a functional equation relating its values at s and 2− s. The modularity of E thus implies tha...

Given an L-function, one of the most important questions concerns its vanishing at the central point; for example, the Birch and Swinnerton-Dyer conjecture states that the order of vanishing there of an elliptic curve L-function equals the rank of the Mordell-Weil group. The Katz and Sarnak Density Conjecture states that this and other behavior is well-modeled by random matrix ensembles. This c...

We explain how to use results from Iwasawa theory to obtain information about p-parts of Tate-Shafarevich groups of specific elliptic curves over Q. Our method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when the rank of the Mordell-Weil group is g...

Let A be a simple Abelian variety of dimension g over , and let ` be the rank of the Mordell-Weil group A( ). Assume ` ≥ 1. A conjecture of Mazur asserts that the closure of A( ) into A( ) for the real topology contains the neutral component A( ) of the origin. This is known only under the extra hypothesis ` ≥ g − g + 1. We investigate here a quantitative refinement of this question: for each g...

We give an overview of some p-adic algorithms for computing with elliptic and hyperelliptic curves, starting with Kedlaya’s algorithm. While the original purpose of Kedlaya’s algorithm was to compute the zeta function of a hyperelliptic curve over a finite field, it has since been used in a number of applications. In particular, we describe how to use Kedlaya’s algorithm to compute Coleman inte...