Average ranks of elliptic curves: Tension between data and conjecture

Abstract

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.

Ranks of Elliptic Curves Over Function Fields

“…There is a widely held belief that one half of all elliptic curves have infinitely many rational points, but all experimental data which has been collected so far suggests that the fraction is actually two thirds [2, Introduction, Items (1) and (2)]. The final purpose of this paper is to present experimental evidence supporting the widely held belief.…”

“…For any k 0, one constructs a set which generates the quotient in (12) 2 where w 2 ∈ Z k+2 . Repeating the process we find u 2 ∈ S(k) W and v 2 ∈ B k+2 such that w = (u 1 + pu 2 ) + (v 1 + pv 2 ) + p 2 w 2 for some w 2 ∈ Z k+2 .…”

Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks

“…In [15], Cremona describes algorithms for listing all elliptic curves of given conductor and collects arithmetic data for these curves; these algorithms have now produced an exhaustive list of curves of conductor at most 380 000 in an ongoing project [16]. A large currently available database of elliptic curves is due to Stein and Watkins [2,29], which includes 136 832 795 curves over Q of conductor up to 10 8 and a table of 11 378 911 elliptic curves over Q of prime conductor up to 10 10 , extending earlier tables of this type by Brumer and McGuinness [14].…”

“…If the rank is not determined at this stage, then we compute the Cassels-Tate pairing on elements of S 2 (E)/E(Q) [2]. Recall that the Cassels-Tate pairing Γ is an alternating bilinear pairing on X(E) taking values in Q/Z; if X(E) is finite, then it is non-degenerate.…”

Global Duality, Signature Calculus and the Discrete Logarithm Problem

“…In fact, assuming the Riemann hypothesis for all of the curves E d , HeathBrown [14] has proved that at least 1/4 of all the E d with the sign in the functional equation of the L-function being +1 will have rank 0 and at least 3/4 of all the E d with the sign being -1 will have rank 1 (see [14], Theorem 4). In the algorithm, we will first liftẼ/F p to E/Q that has rank at least one by construction, and we will make the heuristic assumption that E(Q) is of rank exactly one and therefore the sign of the functional equation is -1 (see [1], §3 for why this is considered to be reasonable). Using ( [27], Theorem 7.2), Heath-Brown's result just mentioned, and taking sufficiently many random d, we can heuristically arrange for the sign of the functional equation of E d to be equal to +1 and for E d (Q) to have rank 0.…”