Quadratic Chabauty for modular curves and modular forms of rank one

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 $$X({\mathbb {Q}})$$ 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 $$X({\mathbb {Q}})$$ X ( Q ) for any modular curve $$X=X_0^+(N)$$ X = X 0 + ( N ) or $$X_\mathrm{{ns}}^+(N)$$ X 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.

Quadratic Chabauty for modular curves: Algorithms and examples

“…For modular curves, by Gross-Zagier-Kolyvagin-Logachev this can be verified by checking that the associated eigenforms have analytic rank one (see e.g. [DF,§7]). For hyperelliptic curves, it is sometimes simpler to carry out a two-descent.…”

“…The assumption on the image of J(Q) in H 0 (X Q p , Ω 1 ) ∨ is innocuous: if r = g and this assumption does not hold, one could apply the Chabauty-Coleman method. For the Atkin-Lehner quotients X + 0 (N), the weak Birch-Swinnerton conjecture implies J(Q) always generates H 0 (X Q p , Ω 1 ) ∨ (see [DF,Lemma 7]).…”

Quadratic Chabauty for modular curves: algorithms and examples

“…One place where increasing precision may not work is if the -adic logarithm does not induce an isomorphism , even though the rank of is . For the Atkin–Lehner quotients , the weak Birch–Swinnerton-Dyer conjecture implies that always generates (see [DLF21, Lemma 7]). In general, if and the Zariski closure of is , then a conjecture of Waldschmidt [Wal11, Conjecture 1] (an analogue of the Leopoldt conjecture for abelian varieties) implies that the -adic logarithm is always an isomorphism.…”

p-adic adelic metrics and Quadratic Chabauty I

“…For this, we modify the construction of the divisor D Z (b) in [DF21, Section 2.2]. The divisor D Z (b) of [DF21] corresponds to the diagonal cycle D(b, b) in our notation. 8.1.…”

“…The splitting is straightforward -since π 2 • i 1,b and π 1 • i 2,z are constant maps, we see that ker(i * 1,b ) ⊕ ker(i * 2,z ) is a natural complement to the image of π * 1 (Pic(X)) ⊕ π * 2 (Pic(X)) in Pic(X × X). This gives a natural isomorphism, which by similar abuse of notation as in [DF21], we call ψ −1 z ψ −1 z : End(J)…”