Define the ECHO sequence {bn} recursively by (b 0 , b 1 , b 2 , b 3 ) = (1, 1, 2, 1) and for n ≥ 4,We relate this sequence {bn} to the coordinates of points on the elliptic curve E : y 2 + y = x 3 − 3x + 4. We use Galois representations attached to E to prove that the density of primes dividing a term in this sequence is equal to 179 336 . Furthermore, we describe an infinite family of elliptic curves whose Galois images match those of E.
Let C be a curve of genus at least three defined over a number field, and let r be the rank of the rational points of its Jacobian. Under mild hypotheses on r, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the number of rational points on C by a constant that depends only on its genus. Yet one expects an even stronger bound that depends favorably on r: when r is small, there should be fewer points on C. In a 2013 paper, Stoll established such a "rank-favorable" bound for hyperelliptic curves using Chabauty's method. In the present work we extend Stoll's results to superelliptic curves, noting in the process some differences that ought to inform uniformity conjectures for general curves. Our results have stark implications for bounding numbers of rational points, since r is expected to be small for "most" curves.
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