2022/06/22 by Betts, L. Alexander, Corwin, David, Leonhardt, Marius
#11G35 #14H25 #FOS: Mathematics #Number Theory (math.NT) #Primary: 14G05 #Secondary: 14H30
paper · doi:10.48550/arxiv.2206.11085
Conditionally on the Tate--Shafarevich and Bloch--Kato Conjectures, we give an explicit upper bound on the size of the p-adic Chabauty--Kim locus, and hence on the number of rational points, of a smooth projective curve X/ℚ of genus g≥2 in terms of p, g, the Mordell--Weil rank r of its Jacobian, and the reduction types of X at bad primes. This is achieved using the effective Chabauty--Kim method, generalising bounds found by Coleman and Balakrishnan--Dogra using the abelian and quadratic Chabauty methods.