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:
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,
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.
Handout on the unexpectedly hard-fought proof of Cassels' Lemma:
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)