Average Rank of Elliptic Curves [after Manjul Bhargava and Arul Shankar]
نویسندگان
چکیده
The abelian group E(Q) of rational points on E is finitely generated [Mor22]. Hence E(Q) ' Z ⊕ T for some nonnegative integer r (the rank) and some finite abelian group T (the torsion subgroup). The torsion subgroup is well understood, thanks to B. Mazur [Maz77], but the rank remains a mystery. Already in 1901, H. Poincaré [Poi01, p. 173] asked what is the range of possibilities for the minimum number of generators of E(Q), but it is not known even whether r is bounded. There are algorithms that compute r successfully in practice, given integers A and B of moderate size, but to know that the algorithms terminate in general, it seems that one needs a conjecture: either the finiteness of the Shafarevich–Tate group X (or of its p-primary part for some prime p), or the Birch and Swinnerton-Dyer conjecture that r equals the analytic rank ran := ords=1 L(E, s) [BSD65]. The main results of Bhargava and Shankar (Section 1.4) concern the average value of r as E ranges over all elliptic curves over Q.
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