REMARKS ON HILBERT’S TENTH PROBLEM AND THE IWASAWA THEORY OF ELLIPTIC CURVES

Iwasawa Theory for Elliptic Curves

The topics that we will discuss have their origin in Mazur’s synthesis of the theory of elliptic curves and Iwasawa’s theory of ZZp-extensions in the early 1970s. We first recall some results from Iwasawa’s theory. Suppose that F is a finite extension of Q and that F∞ is a Galois extension of F such that Gal(F∞/F ) ∼= ZZp, the additive group of p-adic integers, where p is any prime. Equivalentl...

Anticyclotomic Iwasawa Theory of Cm Elliptic Curves

We study the Iwasawa theory of a CM elliptic curve E in the anticyclotomic Zp-extension of the CM field, where p is a prime of good, ordinary reduction for E. When the complex L-function of E vanishes to even order, Rubin’s proof the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the p-power Selmer group over the anticyclotomic extension is a torsion Iwasawa ...

Hilbert’s Tenth Problem and Elliptic Curves

Lectures on the Iwasawa Theory of Elliptic Curves

These are a very preliminary (and incomplete!) version of the author’s lectures for the 2018 Arizona Winter School on Iwasawa Theory.

Descent on Elliptic Curves and Hilbert’s Tenth Problem

Descent via an isogeny on an elliptic curve is used to construct two subrings of the field of rational numbers, which are complementary in a strong sense, and for which Hilbert’s Tenth Problem is undecidable. This method further develops that of Poonen, who used elliptic divisibility sequences to obtain undecidability results for some large subrings of the rational numbers. 1. Hilbert’s Tenth P...