math.AG — Algebraic Geometry
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 exact point counts, the root-class lemma, pinned values, and a forbidden range of primes reaching N^2/49 by a shift-and-square argument. It forces log |b| ≫ N^2, where b is the common difference. The spectral cap is proved in an exact form. Splitting the Neron–Tate pairing into local kernels of positive type and summing the regularised diagonal terms over all places gives an identity in which the bad-prime corrections cancel. Combined with a uniform archimedean separation lemma, this yields λ_{max}(G) ≤ (3/4) logX + (1/4) ψ(N − 1) + (5/4) log(H_f )+ 2, where X =max|x_i|. Over function fields we prove an exact colouring formula for the Gram matrix of a progression of integral sections, which gives N ≤ min(6χ, (3/2)s). We also classify the 3-term Gram types on rational elliptic surfaces and show that all of them occur, and we show that all sign data are realised on elliptic K3 surfaces. The main bound is vacuous for every known example (all have N ≤14); its interest is that it is linear rather than exponential in the rank.