Average Ranks of Elliptic Curves: Tension between Data and Conjecture

DSpace/Manakin Repository

Average Ranks of Elliptic Curves: Tension between Data and Conjecture

Citable link to this page

 

 
Title: Average Ranks of Elliptic Curves: Tension between Data and Conjecture
Author: Bektemirov, Baur; Mazur, Barry C.; Stein, William; Watkins, Mark

Note: Order does not necessarily reflect citation order of authors.

Citation: Bektemirov, Baur, Barry C. Mazur, William Stein, and Mark Watkins. 2007. Average ranks of elliptic curves: Tension between data and conjecture. Bulletin of the American Mathematical Society 44(2): 233-254.
Full Text & Related Files:
Abstract: Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control--thanks to ideas of Kolyvagin--the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today.

We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures, and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.
Published Version: doi:10.1090/S0273-0979-07-01138-X
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10355842
Downloads of this work:

Show full Dublin Core record

This item appears in the following Collection(s)

 
 

Search DASH


Advanced Search
 
 

Submitters