We prove in this paper an uniform surjectivity result for Galois representations associated with non-CM Q-curves over imaginary quadratic fields, using various tools for the proof, such as Mazur's method, isogeny theorems, Runge's method and analytic estimates of sums of L-functions.

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.

Abstract. We prove that for d ∈ { , , , , } and K a quadratic (or rational) eld of discriminant D and Dirichlet character χ, if a prime p is large enough compared to D, there is a newform f ∈ S (Γ (d p )) with sign (+ ) with respect to the Atkin-Lehner involution w p such that L( f ⊗ χ, ) = . is result is obtained through an estimate of a weighted sum of twists of L-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the L-functions L( f ⊗ χ, ⋅) and a Petersson trace formula restricted to Atkin-Lehner eigenspaces. An application of this nonvanishing theorem will be given in terms of existence of rank zero quotients of some twisted jacobians, which generalises a result of Darmon and Merel.

Runge's method is a tool to figure out integral points on algebraic curves effectively in terms of height. This method has been generalised to varieties of any dimension, unfortunately its conditions of application are often too restrictive. In this paper, we provide a further generalisation intended to be more flexible while still effective, and exemplify its applicability by giving finiteness results for integral points on some Siegel modular varieties. As a special case, we obtain an explicit finiteness result for integral points on the Siegel modular variety A2(2).

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