Average size of 2-Selmer groups of elliptic curves, II
A natural question to ask is about the average order of Sel 2 (E(a, b)) with one variable, say b, varying over a certain interval. If one could show that the average order of the 2-Selmer groups of E(a, b) with b running over every interval of size between |a| 1−ε and |a| 1+ε was uniformly bounded as |a| → ∞, then (1.2) would be obviously true. Unfortunately, the average order of Sel 2 (E(a, b)) in this sense depends on a quite complicatedly, and it is not possible to get a uniform estimate for
... niform estimate for b running over that small interval. Moreover, even ignoring the uniformity, the expected boundedness may not always hold. For example, considering the Legendre curves E(1, u) and curves E(2, u) with 1 < |u| ≤ X, one finds out that, while the average order of Sel 2 (E(2, u)) is absolutely bounded, the average order of Sel 2 (E(1, u) ) is unbounded with an order of magnitude √ log X ! This seems mysterious, the secret, however, does not lie deep. The key difference between these two families of curves is that 1 is a square and 2 is not. For a given curve E(a, b), the order of Sel 2 (E(a, b) ) depends on the number of some special factorizations of a, b and a − b. Roughly speaking, the closer |a|, |b| and |a − b| are to squares, or the fewer prime divisors ab(a − b) has, a higher ratio the admissible factorizations occupy in all the factorizations. This heuristic explains why the average size of Sel 2 (E(1, u)) differs from that of Sel 2 (E(2, u)) significantly. In this paper, we shall show that this phenomenon persists for any fixed a, in accordance with whether |a| is a square or not. For a non-zero integer a, and a positive number X, let (1.3) S(a; X) := 1≤|b|≤X b =a Ann Arbor, MI 48109-1109, U.S.A.