Rank classification for twin-prime elliptic curves with p congruent to 1 modulo 8

Determine the Mordell–Weil rank of the elliptic curve E_p: y^2=x(x-2)(x-p), where p and p-2 are primes and p is congruent to 1 modulo 8, establishing whether the rank is 0 or 2.

Background

For the twin-prime elliptic curve E_p, the paper records the corrected version of a conjecture attributed to Beers and formulated by Joshi. The conjectural rank is 0 when p is congruent to 7 modulo 8, 1 when p is congruent to 3 or 5 modulo 8, and either 0 or 2 when p is congruent to 1 modulo 8.

The paper proves the relevant Selmer-corank and rank conclusions in the residue classes 3, 5, and 7 modulo 8 under its stated conditional Birch–Swinnerton-Dyer input. When p is congruent to 1 modulo 8, the 2-Selmer calculation permits several possibilities, including an unresolved odd-rank case associated with a potentially divisible 2-primary Tate–Shafarevich group. Thus the p congruent to 1 modulo 8 portion of the conjectural classification remains unresolved in the paper.