Quadratic Chabauty for modular curves and modular forms of rank one
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Mathematische Annalen
Quadratic Chabauty for modular curves and modular forms of rank one Netan Dogra1
· Samuel Le Fourn2
Received: 15 November 2019 / Revised: 26 August 2020 / Accepted: 26 October 2020 © The Author(s) 2020
Abstract In this paper, we provide refined sufficient conditions for the quadratic Chabauty method on a curve X to produce an effective finite set of points containing the rational points X (Q), with the condition on the rank of the Jacobian of X replaced by condition on the rank of a quotient of the Jacobian plus an associated space of Chow–Heegner points. We then apply this condition to prove the effective finiteness of X (Q) for any + (N ) of genus at least 2 with N prime. The proof modular curve X = X 0+ (N ) or X ns relies on the existence of a quotient of their Jacobians whose Mordell–Weil rank is equal to its dimension (and at least 2), which is proven via analytic estimates for orders of vanishing of L-functions of modular forms, thanks to a Kolyvagin–Logachev type result. Mathematics Subject Classification 11G18 · 14G05 · 11G30
1 Introduction The Chabauty–Kim method is a method for determining the set X (Q) of rational points of a curve X over Q of genus bigger than 1. The idea is to locate X (Q) inside X (Q p ) by finding an obstruction to a p-adic point being global. The method developed in [39,40] produces a tower of obstructions
X (Q p ) ⊃ X (Q p )1 ⊃ X (Q p )2 ⊃ . . . ⊃ X (Q)
Communicated by Wei Zhang.
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Netan Dogra [email protected]
1
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK
2
Université Grenoble Alpes, CNRS, Saint-Martin-d’Héres, IF 38000, France
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N. Dogra, S. Le Fourn
In [5], it is conjectured that X (Q p )n = X (Q) for all n 0, and in [40] it is proved that standard conjectures in arithmetic geometry imply X (Q p )n is finite for all n 0, but in general these results are not known. The first obstruction set X (Q p )1 is the one produced by Chabauty’s method. In situations when X (Q p )1 is finite, it can often be used to determine X (Q). The main results of this paper concern the finiteness of the Chabauty–Kim set + (N ) or X + (N ) (N a prime different X (Q p )2 when X is one of the modular curves X ns 0 from p), whose definition and properties we now recall briefly (more details are given in Sect. 4). The curve X 0+ (N ) is the quotient of X 0 (N ) by the Atkin–Lehner involution w N . The + (N ) is the quotient of X (N ) by the normalizer of a nonsplit Cartan subgroup. curve X ns + (N ) would resolve Serre’s uniformity question Determining the rational points of X ns [58, §4.3]: is there an N0 such that, for all N > N0 and all elliptic curves E defined over Q without complex multiplication, the mod N Galois representation ρ E,N : Gal(Q/Q) → Aut(E[N ]) is surjective? The Borel and normalizer of split Cartan subgroups of Serre’s uniformity question have been given a positive answer respectively in the celebrated papers [50] and [11,12]. Mazur’s proof may, very crudely, be described as having two stages. 1. Constru
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