A Linear In Rank Bound For Integral Arithmetic Progressions On Elliptic Curves
Every known bound on the length N of an x-progression on an elliptic curve of rank r has the shape N ≤ C^{1+r}. We prove a bound that is linear in the rank for integral progressions, with an error depending only on the height of the model. Let y^2 = x^3 +a_2x^2 +a_4x+a_6 be an integral model and H_f = 1+|a2|+|a4|+|a6|. For every ε > 0, every integral x-progression of length N satisfies N ≤ (3/2 +ε)s+C_{ε} (1+log(H_f))^{1/2}, where s ≤ r is the rank of the span of the progression. In particular N ≤ (3/2 +o(1))s whenever log H_f = o(N^2). The proof has two halves. The local theory consists of…
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