This fall, some UGA number theorists (students, postdocs, junior and senior faculty) are meeting to work through the recent preprint of B. Mazur and K. Rubin, Ranks of twists of elliptic curves and Hilbert's 10th problem. This paper proves some extremely striking results on 2-Selmer ranks of elliptic curves in families of quadratic twists, with (e.g.) the following applications:

Corollary 1.9: For every number field K, there is an elliptic curve E over K with E(K) = 0.

Theorem 1.11: Assuming BSD, for every Galois extension L/K of number fields of prime degree, there exists an elliptic curve E over K such that the rank of E over K and the rank of E over L are both equal to one.

This last result, when combined with work of B. Poonen and A. Shlapentokh, yields:

Theorem 1.12: Assuming BSD, then for every number field K, Hilbert's 10th problem over the ring of integers of K has a negative answer.

Better yet, the technology employed by Mazur and Rubin is "middlebrow" compared to what is needed for most other breakthrough papers in 21st century elliptic curve theory (e.g. other papers by the authors). We wish to capitalize on this opportunity to learn an important new result with a relatively low startup cost.

Mazur and Rubin make use of theorems from the following papers:

J.W.S. Cassels, Arithmetic on curves of genus 1. VIII: On conjectures of Birch and Swinnerton-Dyer

T. Dokchitser and V. Dokchitser, Elliptic curves with all quadratic twists of positive rank

F. Gouvea and B. Mazur, The Square-Free Sieve and the Rank of Elliptic Curves

K. Kramer, Arithmetic of elliptic curves upon quadratic base extension

B. Mazur, Rational Points of Abelian Varieties with Values in Towers of Number Fields

B. Mazur and K. Rubin, Finding large Selmer rank via an arithmetic theory of local constants

L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres

J. Milne, Arithmetic Duality Theorems (big ups to Jim Milne for making this book available on his homepage!)

A. Wintner, On the prime number theorem

Other especially relevant citations:

S. Chang, On the arithmetic of twists of superelliptic curves

S. Chang, Quadratic twists of elliptic curves with small Selmer rank

D.R. Heath-Brown, The size of Selmer groups for the congruent number problem

K. Ono, Nonvanishing of quadratic twists of modular L-functions and applications to elliptic curve

K. Ono and C. Skinner, Nonvanishing of quadratic twists of modular L-functions

H.P.F. Swinnerton-Dyer, The effect of twisting on the 2-Selmer group
My MathSciNet review of Swinnerton-Dyer's paper: pdf

PROGRESS REPORT:

Week 0 (Friday, August 28th): Main speaker: Pete L. Clark

Discussion of some consequences of the main results -- Corollaries 1.8, 1.9 and 1.10. Definition of quadratic twists; invariance of rational 2-torsion under quadratic twists. Statement of Hilbert's 10th problem over Z and over other rings: positive results (Rumely) and negative results (Davis-Putnam-Robinson-Matiyasevich, Eisentrager, Shlapentokh, Poonen).

Week 1 (Friday, September 4th): Main speaker: Alex Rice

Review of Galois cohomology, the Kummer sequence of an isogeny, weak Mordell-Weil group, twist, principal homogeneous space, Weil-Chatelet group, Selmer group.

Week 2 (Friday, September 11th): Main speaker: Pete L. Clark

Further discussion of Section 1 of the paper. Introduction to the phenomenon of constant 2-Selmer parity (D&D).

Week 3 (Wednesday, September 23rd): Main speaker: Pete L. Clark

Beginning of Section 2: statement and proof of Cassels' Lemma on the image of the local Kummer map.

Week 4 (Friday, September 25th): Main speaker: Seyfi Turkelli (notes)

Review of root numbers of elliptic curves (assuming the conjectured analytic continuation and functional equation), connection with analytic rank. The D&D phenomenon.

Week 5 (Friday, October 2nd): Main Speaker: Nathan Walters

Week 6 (Friday, October 9th): Main Speaker: Jim Stankewicz

Week 7 (Friday, October 16th): Main Speaker: Jim Stankewicz

Week 8 (Friday, October 23rd): Main Speaker: Bob Rumely

Week 9 (Wednesday, October 28th): Main Speaker: Bob Rumely (notes)