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.
Highlights
Suppose you are given an algebraic curve C defined, let us say, as the locus of zeroes of a polynomial f (x, y) in two variables with rational coefficients
Suppose you are told that C has at least one rational point; i.e., there is a pair of rational numbers (a, b) such that f (a, b) = 0
If we are to try to extract an actual number between 0 and 1 that will describe “the” probability that a curve of genus 1 possessing at least one rational point has infinitely many, we have to be precise about exactly which curves we want to count and how we propose to “sort” them
Summary
Suppose you are given an algebraic curve C defined, let us say, as the locus of zeroes of a polynomial f (x, y) in two variables with rational coefficients. As a refinement to Goldfeld’s conjecture, Peter Sarnak gave a heuristic that predicts that among the first D members of such a quadratic twist family (essentially arranged in order of increasing conductor) the number of those with even parity and infinitely many rational points is caught between D3/4− and D3/4+ for any positive and D sufficiently large. Fields gave impetus to the random matrix theory calculations of Keating and Snaith [KeSn00] regarding moments of L-functions and their value distribution This was combined with a discretization process by Conrey, Keating, Rubinstein, and Snaith in [CKRS02] to give a more precise guess for the (asymptotic) number of even parity curves with infinitely many rational points in a given quadratic twist family. We would be more than delighted to see unconditional results of this precision established for questions such as the one motivating this survey article
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