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({mathbb {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({mathbb {Q}}) for any modular curve X=X_0^+(N) or X_mathrm{{ns}}^+(N) of genus at least 2 with N prime. The proof 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.
Highlights
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 ‘quadratic Chabauty for quotients’ result that we prove in this paper says that we can replace J with A, but the price we pay is that we replace ρ(J ) − 1 with the rank of Ker(θX,πA,πB ), which can be smaller than ρ( A) − 1
The trivial lower bound on rk(Ker(θX,πA,πB ) is ρ( A)−1−rk(B) and if the latter was positive, it would imply (2). This is why Proposition 2 looks quite particular to modular curves
Summary
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. )(Q), is empty in these cases ( the large genera of such curves mean that in practice such curves are currently beyond the scope of existing computational methods for other reasons) As this involves several techniques not relevant to the proof of Theorem 1, we do not pursue this point in this paper. As explained in Remark 8, this result of nonvanishing is quite weak compared to known or expected asymptotic estimates (giving a positive linear proportion of nonvanishing values) so the main difficulty in the proof of Theorem 2 lies in making such estimates effective enough to prove the result except for small enough N so that the remaining cases can be checked algorithmically
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