Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion

Abstract

In 2016, Balakrishnan-Ho-Kaplan-Spicer-Stein-Weigandt [1] produced a database of elliptic curves over Q ordered by height in which they computed the rank, the size of the 2-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over Q whose rational torsion subgroup is isomorphic to Z/2Z × Z/8Z. Conditional on GRH and BSD, we compute the rank of 92% of the 202,461 curves with parameter height less than 10 3 . We also compute the size of the 2-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.

Tamagawa products of elliptic curves over $\mathbb{Q}$

“…There are 18 choices for t modulo 27 with t ≡ 0 (mod 3), and for each choice of t, there are 2 choices of s modulo 3. Together, these determine a 4 modulo 3 6 and a 6 modulo 3 7 Combining these observations, we obtain δ 3 (I 0 , 1) = 4/3 11 .…”

“…To place Theorem 1.1 in context, we recall that work by Klagsbrun and Lemke-Oliver [9], and of Chan, Hanselman and Li [6] shows that Tam(E) is unbounded in families of elliptic curves with prescribed Q-rational 2-torsion. Moreover, Figure A.14 of [4] suggests an "average Tamagawa product" of ≈ 1.82 .…”

“…In this situation, there are 6 choices for t modulo 9 so that t = 0 (mod 3), and 2 choices for s ≡ ±1 (mod 3). These choices together determine a 4 modulo 3 5 and a 6 modulo 3 6 . Therefore, the proportion of pairs (a 4 , a 6 ) satisfying these conditions is 4/3 10 Combining these observations, we obtain δ ′ 3 (II * , 1) = 2/3 9 .…”

“…For condition (ii), we have a 4 = −3t 2 , and a 6 ≡ 16−a 4 t−t 3 (mod 32), which implies that t ≡ 3 (mod 4). Therefore, by brute force we find that (a 4 , a 6 ) (mod 32) ∈ { (5,6), (13,30) ). Clearly, the proportion of curves satisfying (i) is 1/512.…”

The average analytic rank of elliptic curves with prescribed torsion

“…For some numerical data for 𝐺 = ℤ∕2ℤ × ℤ∕8ℤ, we refer to a result of Chan, Hanselman, and Li [5]. Young [23, §8] also computed bounds for average analytic rank for families of elliptic curves with some prescribed torsion 𝐺 under not only GRH for elliptic curve 𝐿-functions but also GRH for Dirichlet 𝐿-functions and some other assumptions.…”