Quadratic Chabauty for modular curves and modular forms of rank one

In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions 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 finiteness of this set for any modular curves $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ 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.

1 Introduction 1 2 The quadratic Chabauty condition (C) for a quotient 7 3 Proof of finiteness of the Chabauty-Kim set under (C) 11 4 Proof of (C) for X + 0 (N ) and X + ns (N ) 16

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 [Kim05], [Kim09] produces a tower of obstructions In [BDCKW18], it is conjectured that X(Q p ) n is finite for all n ≫ 0, but in general this is 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 X(Q p ) 2 when X is one of the modular curves X + ns (N ) or X + 0 (N ) (N a prime different from p), whose definition and properties we now recall briefly (more details are given in §4).
The curve X + 0 (N ) is the quotient of X 0 (N ) by the Atkin-Lehner involution w N . The curve X + ns (N ) is the quotient of X(N ) by the normalizer of a nonsplit Cartan subgroup. Determining the rational points of X + ns (N ) would resolve Serre's uniformity question [Ser72]: is there an N 0 such that, for all N > N 0 and all elliptic curves E defined over Q without complex multiplication, the mod N Galois representation is surjective? The Borel and normalizer of split Cartan subgroups of Serre's uniformity question have been given a positive answer respectively in [Maz77] and [BP11].
As is explained in §4, in contrast to X 0 (N ) and X + s (N ), for X = X + 0 (N ) or X = X + ns (N ), it is expected that X(Q p ) 1 is infinite. The main result of this paper is that we do obtain a finite set by refining the obstruction to 'depth two'.
• For all primes N for which one of the curves X above has genus 0 or 1, X(Q) is infinite so this is the sharpest finiteness result for X(Q p ) 2 one can expect.
• The only reason for the assumption that X + ns (N )(Q) is nonempty is that the definition of X(Q p ) 2 currently assumes that X has a rational point (if Serre's uniformity question has a positive answer, then there are infinitely many N for which X + ns (N )(Q) is empty). One can modify the definition of X(Q p ) 2 -for example in a similar manner to [Hai11] -to remove this assumption, and then X + ns (N )(Q p ) 2 will be finite whenever the genus of X + ns (N ) is greater than 1. As this involves several techniques not relevant to the proof of Theorem 1.1, we do not pursue this point in this paper.
• Finally, results of [BD19a], together with forthcoming work of Edixhoven and Parent, allow to deduce from our result an explicit bound (polynomial in N ) on the number of rational points on X + 0 (N ) and X + ns (N ), which we do in §3.1.
As alluded to above, one can often use finiteness of X(Q p ) n to determine X(Q) explicitly. The first motivation of this paper stems from the explicit determination of X + ns (13)(Q) (starting from the finiteness of X + ns (13)(Q p ) 2 ) in [BDM + 19]. The finiteness of X(Q p ) 2 has also been used recently to determine the rational points of X + 0 (N ) whenever it has genus two (in forthcoming work of Best-Bianchi-Triantifillou-Vonk) or genus three (in forthcoming work of Balakrishnan-Dogra-Müller-Tuitman-Vonk).
The proof of Theorem 1.1 proceeds along the lines of the so-called 'quadratic Chabauty method', which requires a precise inequality (namely (2)) in terms of invariants of the Jacobian J of X to hold (see §1.1). This inequality is expected to hold asymptotically for X = X + 0 (N ) or X = X + ns (N ) conditionally on Birch and Swinnerton-Dyer conjecture (see §4.1), but looks out of reach unconditionally for N in noncomputable range. There are thus two important steps obtained in the proof of Theorem 1.1: • For p a prime of good reduction of a smooth projective geometrically irreducible curve X over Q with X(Q) = ∅, X(Q p ) 2 is finite under the condition that a similar inequality to (2) holds not for J but a quotient abelian variety A of J, and under an additional hypothesis (H) on X, J, A.
The final input in the proof of Theorem 1.1 is the following Theorem. Theorem 1.3. For all M = N or N 2 with N prime, if the space S 2 (Γ 0 (M )) +,new is of dimension at least two, it contains two distinct normalised newforms f such that L ′ (f, 1) = 0.
As explained in Remark 5.2, this result of nonvanishing is in fact 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.

Chow-Heegner points and quadratic Chabauty
In general, X(Q p ) n cannot unconditionally be proved to be finite without some assumptions on the Jacobian of X (Kim showed that the Bloch-Kato conjectures imply that X(Q p ) n is finite for all n ≫ 0 [Kim09, Observation 2]). In the case n = 1 (which reduces to the classical set-up of Chabauty's method) it is known that a sufficient condition is that where rk(J) is the Mordell-Weil rank of J(Q). The simplest instance extending Chabauty's method when finiteness of X(Q p ) n can be proved for n > 1 is the following Lemma. To state the Lemma, define J := Jac(X), and the Picard number ρ(J) is defined to be the rank of the Néron-Severi group NS(J) := Pic(J)/ Pic 0 (J). By [Mil86,Proposition 17.2], this is the same as the dimension of the subspace denoted by End † (J) of End 0 (J) := End(J) ⊗ Q consisting of endomorphisms that are symmetric, i.e. fixed by the Rosati involution.
By Kolyvagin-Logachev type results due to Nekovář and Tian (see Proposition 7.1 and its Corollary 7.2), Theorem 1.3 implies that the Jacobians of X + 0 (N ) and X + ns (N ), which we will henceforth denote by J + 0 (N ) and J + ns (N ) respectively, do have Q-isogeny factors A satisfying rk(A) < dim(A) + ρ(A) − 1, but it seems unattainable to prove unconditionally such a result for the full Jacobian. To deduce Theorem 1.1, we thus need a 'quadratic Chabauty for quotients' result, analogous to the well-known fact that Chabauty's method also works under the relaxed condition rk(A) < dim(A), i.e. (1) for an isogeny factor A instead of J (in fact, for modular curves, Mazur-Kamienny's method refines this for factors A such that rk(A) = 0, see e.g. [Bak99]).
As explained below, in general such a result seems non-trivial. Fix a basepoint b ∈ X(Q), and let AJ : X → J be the corresponding Abel-Jacobi map. Let A, B be abelian varieties over Q, satisfying Hom Q (A, B) = 0, and suppose we have a surjection (π A , π B ) : J → A × B.
A slight modification denoted by AJ * of the pullback by AJ (which basically amounts to considering the restriction of AJ on symmetric line bundles, see §2.1) vanishes on Pic 0 (J), so it factors through NS(J) and AJ 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) with the rank of Ker(θ X,πA,πB ), which can be smaller than ρ(A) − 1.
In the case where rk(A) = dim(A) which we will focus on, we can simplify this condition in terms of nice correspondences, defined in §2.1. More precisely, (π A , π B ) induces an isomorphism End 0 (J) ∼ = End 0 (A)×End 0 (B), and X(Q p ) 2 is finite whenever there exists a nontrivial nice correspondence Z on X × X whose corresponding endomorphism of J is zero in End 0 (B), and whose corresponding Chow-Heegner point D Z (b) ∈ Pic 0 (X) is torsion when projected to B. Remark 1.7. Note that, since rk(Ker(θ X,πA,πB )) ≤ ρ(A) − 1, inequality (C) implies that A satisfies the naive analogue of Lemma 1.4 However, in general (C) is strictly stronger than (5) which is why we said in the introduction that an additional hypothesis (H) is required. In fact, the trivial lower bound on rk(Ker(θ X,πA,πB ) is ρ(A)−1−rk(B) and if the latter was positive, it would imply (2). This is why Proposition 1.8 looks quite particular to modular curves. Moreover, understanding the rank of Ker(θ X,πA,πB ) in general seems somewhat subtle -as becomes apparent in Example 2.8 and §3.2, this quantity is not an invariant of the pair (A, B), or even of the triple (X, A, B), and does not seem to behave so well functorially even under quite strong hypotheses. Finally, as explained in the first appendix, this quantity is also related to the Gross-Kudla-Schoen cycles constructed in [GS95].
The following proposition emphasises that in fact, the supplementary condition (H) can always be satisfied for our modular curves.
As will become apparent in the proof, in fact we have to take A the maximal isogeny factor of J such that rk(A) = dim A (at least analytically, see Corollary 7.2) and B its complement, otherwise we might not be able to ensure that the kernel of θ X,πA,πB is nontrivial. This idea relies heavily on the use of (traces of) Heegner points on the modular curves X 0 (N ), X ns (N ), which for good choices of them will generate A(Q) up to finite index, but will automatically be torsion in B(Q), both situations being ultimately by-products of the generalised Gross-Zagier formula (see section 4.2). Note that in this case the kernel of the theta morphism is not only nontrivial, but as large as it can be, which might indicate a deeper phenomenon at play.
The structure of the paper is as follows. In section 2, we give some reminders on Néron-Severi groups, Chow groups and correspondences, and describe the map θ X,πA,πB in terms of cycles. In section 3 we prove Proposition 1.6. In section 4, we prove Proposition 1.8 assuming Theorem 1.3, after some discussion on (C), and using generalised Gross-Zagier formulas. In section 5, we prove Theorem 1.3. Finally, for sake of clarity and by lack of easily available references in the literature, we gather in Appendix 6 results about the Chow-Heegner construction above and explain in Appendix 7 the proof of the Kolyvagin-Logachev type result needed to translate Theorem 1.3 into an algebraic rank result.

Notation and conventions
Unless stated otherwise, we adopt the following conventions in this paper.
• X is a smooth projective geometrically irreducible curve of genus ≥ 2 over Q. J is the Jacobian of X and AJ : X → J is the Albanese morphism with a fixed base point b ∈ X(Q). The notation AJ * refers to twice the pullback on symmetric line bundles of X to Pic(X) (see (13)), and then factors through NS(J) (this is not the same as just the pullback AJ * from Pic(J) to Pic(X), which does not vanish on Pic 0 (J)).
• For any n and any S ⊂ {1, · · · , n}, the morphism When there is no ambiguity on b we denote it simply by i S . Similarly, the morphism π S : X → X #S (7) denotes the projection of (x 1 , · · · , x n ) on the coordinates belonging to S.
• Morphisms between algebraic varieties over Q and their structures (line bundles, divisors, etc) are assumed to be defined over Q.
• For a smooth projective algebraic variety Y over Q, NS(Y ) is the Néron-Severi group of Y , and ρ(Y ) := rk NS(J) is the Picard number of J (see §2.1).
• For any abelian variety A over Q (in particular for J), rk(A) is the rank of the finite type Z-module A(Q) and End 0 (A) := (End Q A) ⊗ Q.
• N is a prime number (the level of our modular curves) and M = N or N 2 . • X 0 (N ) (resp. X + s (N ), X + ns (N )) is the modular curve quotient of X 0 (N ) corresponding to the Borel structure (resp. quotient of X(N ) corresponding to the normaliser of split Cartan, normaliser of nonsplit Cartan), X + 0 (N ) is the quotient of X 0 (N ) by the Atkin-Lehner w N . Accordingly, J 0 (N ), J + s (N ), J + ns (N ), J + 0 (N ) denote their respective Jacobians (see §4).
• For X a variety over a field K ⊂ C, H k (X, Z) refers to the singular cohomology of X(C).
• Given a unipotent group U , the central series filtration of U is defined by U (1) = U and U (i+1) = [U, U (i) ], and gr i (U ) := U (i) /U (i+1) (in particular gr 1 (U ) = U ab ). If a group G acts continuously on U , then G acts on the set of normal subgroups of U , and we say that a quotient U/H is G-stable if the normal subgroup H is stabilised by G. In this case there is a unique G-action on U/H making the surjection G-equivariant.
• p is a prime number different from N which will be used (except in appendix 7) only in the context of p-adic numbers.

Acknowledgements
The authors wish to thank heartily Samir Siksek, who initiated this project and contributed to its progression, but declined to be listed as a co-author. He also graciously authorised us to include his original argument from his preprint [Sik17], which is found in paragraph 4.1. We would also like to thank Daniel Kohen and Jan Vonk for helpful discussions. Definition 2.1. For any geometrically smooth and irreducible projective variety Y over Q and any k ≤ dim Y : • The Chow group CH k (Y ) is the group of cycles of Y of codimension k up to rational equivalence.
is the cycle map, and CH k 0 (Y ) := Ker(c k ) is its subgroup of homologically trivial cycles (in Y (C)).
We can also define a geometric étale cycle map [Del77,Cycle] c l,ét and an absolute étale cycle map By the Artin comparison theorem we have Ker( l c l,ét k ) = CH k 0 (Y ). The étale Abel-Jacobi morphism is a homomorphism which may be defined using the Leray spectral sequence or (equivalently but more directly) by realising the extension class of a homologically trivial cycle Z inside H 2k−1 ((X − Z) Q , Q p (k)) (see Jannsen [Jan90,II.9] or Nekovar [Nek93, 5.1]). By Poincaré duality, we may equivalently think of the target of AJé t as being In particular, when Y = X is a curve, and for k = 1, the target of AJé t is where J is the Jacobian of X and V p (J) = T p (J) ⊗ Zp Q p . Let us now review the basic definitions of correspondences.
Definition 2.2. For two curves X 1 , X 2 as before: • A correspondence Z on X 1 , X 2 is a divisor of Div(X 1 × X 2 ), prime if the underlying divisor is. It is called fibral if its prime components are horizontal or vertical divisors.
is invariant under this involution, if the Rosati involution acts as −1 on a trace zero endomorphism φ, then its image in J under θ X,b will be 2-torsion. Hence up to 2-torsion we may equivalently think of θ X,b as a morphism End † (J) tr=0 . By [Mum86,IV.20], the morphism after composition by the inverse of a natural principal polarisation on J given by a theta divisor. Hence if we define to be the composite of (14) (restricted to Ker(NS(J) → NS(X))) with θ X,b , then θ X,b and θ X,b are the same up to two-torsion, in the sense that the minus one eigenspace of the Rosati involution lies in the kernel of θ X,b . Remark 2.3. In [BDM + 19], an element of Pic(X × X) whose image under (11) lies in End † (J) tr=0 is referred to as a 'nice correspondence'. Remark 2.4. This definition of the homomorphism NS(J) → End(J) is the same as applying the composition f • (AJ (2) ) * up to sign. Indeed, via the natural morphisms J ∼ = Pic 0 (J) and Pic 0 (X) ∼ = J, the inverse J → J of the principal polarisation given by a theta divisor on J is equal to −AJ * from Pic 0 (J) to Pic 0 (X) ([BL04, Proposition 11.3.5]). Now, in terms of line bundles, by definition, given a line bundle L on X × X, the endomorphism of Pic(X) associated to it is given on points by x → i * 2 (x)(L) with notation (6). Now, •AJ on X so for a line bundle L on Pic(J) and x, y points of X the endomorphism associated to L = (AJ (2) ) * L sends which gives the equality up to −1.

Chow-Heegner points and diagonal cycles
We recall an equivalent version of the morphism θ X,b , which appears in [DR14] and [BD18b]. As our discussion applies in fairly broad generality, we take X to be a smooth geometrically irreducible projective curve over a field K of characteristic zero. Fix b ∈ X(K), and S ⊂ {1, . . . n}, let X S denote the image of X under the closed immersion i S (b) defined in (6). For any Z ∈ Div(X × X), We refer to D Z (b) and C Z (b) as Chow-Heegner points, following [DRS12]. The map Z → D Z (b) factors through Pic(X × X), and has the following relation to θ X,b . The projection We define Z t ∈ CH 1 (X × X) to be the pull-back of Z under the involution Lemma 2.5. We have where f Z is the endomorphism of J associated to Z as in Definition 2.2.
By definition of the correspondences, we then have which proves the equality for as the degrees are then equal.
Definition 2.6. Given a surjective homomorphism π B : J → B of abelian varieties, we obtain a homomorphism By Lemma 2.5 and (16), if the endomorphism of J associated to L via the morphism NS(X × X) → End(J) has image contained in Ker(π B ), then the image of [L] in B is independent of the choice of basepoint. In particular, if we have a surjection (π A , π B ) : J → A × B, and Hom(A, B) = 0, then we obtain a homomorphism independent of b, which we will denote by θ X,πA,πB : Ker(d πA ) → B Remark 2.7. This construction also has a direct description in terms of line bundles, although this is not the one we use to calculate θ X,πA,πB in examples. Given a line bundle L on A whose pull-back to X via AJ * • π * A has degree zero, we may also consider the projection of AJ * • π * A (L) to B. Variants of this construction are studied in the thesis of Michael Daub [Dau13]. By (13), we have the identity [Dau13, Proposition 3.3.3] θ X,πA,πB = [2] • π B • AJ * • π * A , in particular the right-hand side does vanish on Pic 0 (A) [Dau13, Proposition 3.3.2]. Example 2.8. Note that θ X,πA,πB is not an invariant of A and B, or even of X, A, B. For example, let A and B be distinct isogeny factors of X 0 (N ), and let X = X 0 (N 2 ). Let f 1 , f 2 : X → X 0 (N ) be the two natural morphisms, and let (π Ai , π Bi ) be the morphisms Jac(X) → A × B obtained by composing the surjection J 0 (N ) → A × B with f i * . Then θ X,πA,i,πB,i can be nonzero (see [DR14] for examples), however if i = j, θ X,πA,i,πB,j is identically zero, since for any choice of line bundle [L] in NS(A), the associated point D [L] (b) will lie in f * i J 0 (N ), hence the projection to f j * J 0 (N ) will be torsion.

Proof of finiteness of the Chabauty-Kim set under (C)
The strategy of proof of Proposition 1.6 is very similar to that of [BD18b, Lemma 3.2]. To explain this strategy, we need to establish some notation. X, A, B are as in the proposition. Let Let U n (b) denote the maximal n-unipotent quotient of the Q p -unipotent fundamental group of X at some basepoint b as defined in [Del89,§10]. Let U be a Galois-stable quotient of U n (b) (i.e. a quotient by a Galois-stable normal subgroup of U n (b)). Let T 0 be the set of primes of bad reduction for X, and let T = T 0 ∪ {p}. Denote the maximal quotient of Gal(Q/Q) unramified outside T by G Q,T , and for v ∈ T denote Gal(Q v /Q v ) by G Qv . Then by [Kim05], [Kim09], we have a commutative diagram with the following properties 1. For G = G Q,T or G Qv , and all i < k, the sets H 1 (G, U (i) /U (k) ) have the structure of Q p points of an algebraic variety, so that the algebraic structure on H 1 (G, gr i U ) is just the usual scheme structure on a vector space, and the maps come from morphisms of algebraic varieties. The maps loc v are then algebraic for these structures.
2. For v ∈ T 0 , the map j v has finite image.
3. The image of the map j p is contained inside the subvariety H 1 f (G Qp , U ) of crystalline torsors.
The following Lemma is proved in [BD18b, Lemma 3.1] (although the result is stated only in the case A = J, the proof generalises to the case where A is an arbitrary quotient of J).
To prove Proposition 1.6, we construct a quotient U of U 2 (b) as in Lemma 3.1, with n = rk(Ker θ X,πA,πB ). We again take X to be a smooth projective geometrically irreducible curve over a field K of characteristic zero.
The group U 2 (b) is an extension Hence for any ξ ∈ Ker(NS(J) AJ * → NS(X)), we may quotient by the kernel of the dual of the Chern class cé t to obtain a quotient U Z of U 2 (b) which is an extension of V by Q p (1). Similarly, for any nice correspondence on X × X, we obtain a quotient of U 2 (b) which is an extension of V by Q p (1).
coming from a correspondence Z ⊂ X × X as above. Then the associated ex- Proof. Let E(Lie(U )) be the universal enveloping algebra of Lie(U ), and let I(Lie(U )) be the kernel of the co-unit morphism E(Lie(U )) → Q p . In [BD18b,§6], a Galois representation E Z is constructed as a quotient of E(Lie(U )). The is an isomorphism, and hence the extension class of Lie(U ) is isomorphic to D Z (b).
As explained in appendix 6, Lemma 3.2 is really a consequence of Hain and Matsumoto's computation of the extension class of Lie(U 2 ) in terms of the Ceresa cycle. Hence to complete the proof of Proposition 1.6, it will be enough to prove the following Lemma.
Proof. It will be enough to prove the corresponding statement for the Lie algebra is the composite of the commutator on U 2 , given by with the surjection Since the latter map factors through projection onto ∧ 2 V A /Q p (1), the composite map factors through projection onto V A × V A . Hence for any quotient Q of Ker(d πA ) * ⊗ Q p (1), we can construct a Lie algebra quotient of L ′ which is an extension of V A by Q. It remains to show that, when Q = Ker(θ X,πA,πB ), we can make this quotient Galois stable. That is, we first quotient out by (Ker(d πA )/ Ker(θ X,πA,πB )) * ⊗ Q p (1), to form an extension The surjection L ′′ → V B induces a Galois equivariant short exact sequence of Lie algebras and to construct the quotient U → U ′ , it is enough to show that this short exact sequence admits a Galois equivariant section. Here L ′ sits in a short exact sequence is zero.
Equivalently, we want to show that Ker(θ X,πA,πB ) is contained in the kernel of the homomorphism By Lemma 3.2, this extension class is equal to the étale Abel-Jacobi class of D cé t p (ξ) (b), and hence its V B component is equal to the étale Abel-Jacobi class of θ X,πA,πB (cé t p (ξ)).
This completes the proof of Proposition 1.6.

Bounding the number of rational points on curves satisfying (C)
Following [BD19a], the proof of finiteness of X(Q p ) 2 may be used to prove an explicit upper bound on #X(Q p ) 2 . To explain this, we introduce some notation. By [KT08, Corollary 1], for all v = p, the size of the image of X(Q v ) in H 1 (G Qv , U 2 ) is finite, and is equal to one for all primes of good reduction for X. Let T 0 denote the set of primes of bad reduction for X, and for v ∈ T 0 let n v denote the size of the image of X(Q v ) in H 1 (G Qv , U 2 ).
Corollary 3.4. Suppose X satisfies the hypotheses of Proposition 1.6, and furthermore that the rank of A(Q) is equal to its dimension, and the p-adic closure of A has finite index in A(Q p ). Let n := v∈T0 n v . Let D be an effective divisor on X, let Y ⊂ X Zp be the complement of the support of a normal crossings divisor on Y with generic fibre D, and let {ω 0 , . . . , ω 2g−1 } be a set of differentials in H 0 (X, Ω(D)) forming a basis of H 1 dR (X). Then there are a ij , a i ∈ Q p , η ∈ H 0 (X, Ω(D)) and g ∈ H 0 (X, Ω(2D)), and α 1 , . . . , α n in Q p , such that Proof. The argument is identical to the proof of [BD18a, Proposition 6.4], however as the hypotheses are different we explain the steps. There Hence we deduce (18) from the formula for h p (A Z (x)) given in [BD18a, Lemma 6.7], and the formula Corollary 3.5. Suppose X satisfies the hypotheses of Proposition, and furthermore that the rank of A(Q) is equal to its dimension. Then Remark 3.6. In [BD19b], it is proved that the size of j 2,v (X(Q v )) can be bounded by the number of irreducible components of a regular semistable model of X over a finite extension of Q v . Hence using forthcoming work of Edixhoven and Parent on stable models of X + ns (N ), one can use the above corollary, together with Theorem 1, to give explicit bounds on the size of X + ns (N ) and X + 0 (N ).

Functoriality properties of (C)
The heart of the proof of Proposition 4.1 is an interpretation of diagonal cycles on X 0 (N ) and X ns (N ) in terms of Heegner points. The following Lemma allows us to use this to deduce something about diagonal cycles on X + 0 (N ) and X ns (N ). This lemma is a special case of a theorem of Daub [Dau13, Proposition 3.3.5].
and the result follows for D Z (b). The second item follows from the first, by (16).
Note that while the behaviour of diagonal cycles under pull-backs is tautological, their behaviour under pushforwards seems to be more complicated. For these reason it seems difficult to deduce statements about diagonal cycles on X ns (N ) from results on X s (N ), in spite of the explicit isogeny relating their Jacobians (see next section).
4 Proof of (C) for X + 0 (N ) and X + ns (N ) Given Proposition 1.6, it will be enough to prove Theorem 1.3, and the following.
Proposition 4.1. Assume Theorem 1.3. Then, for X = X + 0 (N ) or X + ns (N ) of genus at least 2, there exists an isogeny where rk(A) = dim(A) = ρ(A) ≥ 2 and such that, for all L in Ker(d πA ), θ X,πA,πB (L) = 0 is torsion (see Definition 4.3 for the choices of A and B).
We recall the definitions of some of the modular curves which appear, for example, in [Che00]. Define C + ns (N ), C + s (N ) to be choices of normalisers in GL 2 (Z/NZ) of non-split Cartan C ns (N ) and split Cartan subgroups C s (N ) of GL 2 (Z/N Z). The (normaliser of) split and nonsplit Cartan modular curves are defined by . Similarly we define X ns (N ) and X s (N ) to be the quotients of X(N ) by C ns (N ) and C s (N ) respectively. Since C ns (N ) and C s (N ) contain the centre of GL 2 (Z/N Z) and their determinant goes through all (Z/N Z) * , all X ns (N ), X s (N ) and their Atkin-Lehner quotients are geometrically connected and defined over Q.
Non-cuspidal K-points of X s (N ) (for K a field of characteristic zero) correspond to elliptic curves E together with a pair C 1 , C 2 of cyclic subgroups of E of order N generating E[N ]. We have an isomorphism which sends a point (f : The curve X s (N ) is naturally a degree two cover of X + s (N ), and there is an

Jacobians of modular curves and the asymptotics of the quadratic Chabauty condition
We recall a formula for the Picard numbers and ranks of modular Jacobians and their quotients, due to Siksek [Sik17]. Let B N k denote a normalised eigenbasis for the space of newforms in where A f denotes by the theory of Eichler-Shimura the Q-simple abelian variety associated to f (independent of the choice of representative of the orbit). Because X + s (N ) is isomorphic to X + 0 (N 2 ) as we have seen above, and by a theorem of Chen [Che00, Theorem 1], we also have a Q-isogeny Remark 4.2. As a result of (20), it is not expected that the classical Chabauty condition (1) (even for a quotient) can be applied for X + 0 (N ) and X + ns (N ). Indeed, for f in B N k , the abelian variety A f is geometrically simple and of . An eigenform f ∈ B N k has even analytic rank if and only if it satisfies w N k (f ) = −1. Hence, the Birch-Swinnerton-Dyer conjecture for modular abelian varieties implies that J + 0 (N ) has no rank zero quotient, and if one further assumes Waldschmidt's conjecture, then X + 0 (N )(Q p ) 1 = X + 0 (N )(Q p ) for all p, and similarly for X + ns (N ). Let us get back to the quadratic Chabauty condition. First, two abelian varieties A f , A g for f, g ∈ B N k are non-isogenous unless f and g are conjugate by Gal(Q/Q), and End † (A f ) is always totally real of rank dim(A f ), which proves that for each of the Jacobians (for a more general such condition for modular curves, see the main result of [Sik17]). Now, assuming the conjecture of Birch and Swinnerton-Dyer and using the previous isogenies There is a whole literature on analytic estimates for these types of analytic ranks. In particular, some tinkering with [KMV00, Theorem 1.4] to isolate the contribution from the plus part of the Jacobian gives (recall that we assume Birch and Swinnerton-Dyer) and in particular asymptotically (2) is always satisfied. It is likely that the same result can be obtained for J + ns (N ), but the square level (we are looking at J + 0 (N 2 ) new ) raises serious technical difficulties for analytic estimates of second moments used there.
On the other hand, by Corollary 7.2, Theorem 1.3 implies that we have an isogeny factor A of J satisfying ρ(A) > 1 and rk(A) = dim(A), hence to prove Proposition 4.1 it suffices to construct a nonzero [L] ∈ Ker(NS(A) → NS(X)) satisfying θ X,πA,πB ([L]) = 0, where B is the isogeny factor consisting of modular abelian varieties associated to modular forms whose analytic rank of L-functions is greater than 1. It will be shown that for any L, its image θ X,πA,πB (L) is supported (as a degree 0 divisor on X) by images of cusps or Heegner points of J in B, and hence is torsion by the generalised Gross-Zagier formula ([Zha04, Theorem 6.1]) This motivates the following definition.
A f (so that their product is isogenous to J + 0 (M ) new , not the full J 0 (M ) new ). In particular, Corollary 7.2 implies that rk(A) = dim(A) (assuming the Birch and Swinnerton-Dyer conjecture, it is the largest factor of J + 0 (M ) with this property) and the generalised Gross-Zagier formula (see §4.2 and §4.3) implies that all images of traces of Heegner points on X 0 (N ) in B are torsion. In the case of X ns (N ), there is also a notion of Heegner point due to Kohen and Pacetti, inspired by the points used in Zhang's Gross-Zagier formula for X ns (N ) (and more general Shimura curves).
The main result of the next section is the following lemma, which refers to X 0 (N ) and X ns (N ) rather than their Atkin-Lehner quotients. However, by Lemma 3.7 it implies Proposition 4.1.
Lemma 4.4. Let X = X 0 (N ) or X ns (N ), and A, B the Heegner quotient and its complement as defined above, endowed with the natural projections (π A , π B ) : Jac(X) → A × B. Then for all [L] in Ker(d πA ), θ X,πA,πB ([L]) is torsion. In particular the rank of the kernel of θ X,πA,πB is maximal (in particular at least 1 if dim A ≥ 2). 4.2 How to prove (C) using Heegner points under the analytic hypothesis: X = X 0 (N) In this section we prove Lemma 4.4. We will deduce it from the Gross-Zagier-Zhang theorem. In the case of X 0 (N ), as explained in [Dau13] or [DRS12], we could also deduce it from the Yuan-Zhang-Zhang formula for the height of diagonal cycles (see §4.4). By a Heegner point on X 0 (N ) we will mean a point such that E and E ′ have CM by the same order of an imaginary quadratic field K, not necessarily maximal but assumed to be with conductor prime to N (see [Gro84] for a review of their properties, in particular N has to be split or ramified in K). An eigenform f ∈ S 2 (Γ 0 (N )) +,new defines by Eichler-Shimura theory a Qsimple quotient π : J 0 (N ) → A f of J 0 (N ) (in fact of J 0 (N ) + ) and the Heegner points behave on A f in the following way.
is generated by the projection of a trace of a good choice of Heegner point).
2. If L ′ (f, 1) = 0, then for any P in Div 0 (X 0 (N ))(Q) Gal(Q|Q) supported on the set of Heegner points, the image π(P ) is torsion in A f (Q).
Remark 4.6. The original Gross-Zagier formula [GZ86, Theorem I.6.3] is not sufficient for the second part of the Lemma, as it only deals with Heegner points for which the discriminant of the order is squarefree (in particular, the order is maximal) and prime to N , which we cannot afford to assume here. This is why we need Zhang's formula and the ensuing technical interpretation.
Proof. The first part is given by Proposition 7.1. The second part is a consequence of the generalised Gross-Zagier formula of Zhang [Zha04, Theorem 6.1] which for this case is made completely explicit in [CST14, Theorem 1.1], which the authors also advise as a further reference for translation between Zhang's vocabulary and the classical context of modular curves [CST14, Example after Theorem 1.5]. We use the following notation: f ∈ S 2 (Γ 0 (N )) is a normalised eigenform, K an imaginary quadratic field number field in which N is not inert, c prime to N , O c = Z + cO K , and 1 c the trivial ring class character on Pic(O c ). We denote by H c the ray class field of K with conductor c. If P is a Heegner point on X 0 (N ) with CM by O c , it belongs by Heegner points theory to X 0 (N )(H c ), and we define On the other hand, considering the extension of scalars J(H c ) ⊗ C endowed with the extended Néron-Tate height, we have the decomposition in isotypical components where g goes through all eigenforms of weight 2 of J 0 (N ), so that J 0 (N ) g is exactly the isotypical part where T n acts by multiplication by a n (g). We denote by P f 1c the projection of P 1c on the f -isotypical component. The statement of [CST14, Theorem 1.1] then tells (which is sufficient for us) that L ′ (f, 1 c , 1) as defined there is proportional to the extended Néron-Tate height of P f 1c . Now, definitions of Rankin-Selberg convolutions ([Zha04, §5] but centered at 1) give the equality with 1 K the trivial class character on Pic(O K ) and χ K the Dirichlet character associated to K. In particular (and given the signs of functional equations on the right), our hypothesis L ′ (f, 1) = 0 guarantees that L(f, 1 K , s) vanishes with order at least 2 at 1, so the left-hand side of [CST14, Theorem 1.1] is zero for c = 1. This also holds for anyc prime to N , because by construction L(f, 1 c , s) is a multiple of L(f, 1 K , s) around 1 (given the definition again). We have thus proved that P g 1c is zero in J 0 (N )(H c ) ⊗ C. Now, because we have the unique decomposition P 1c = g P g 1c where g goes through all eigenforms, and this sum is also invariant by Gal(H c /K), for every 1c , which implies that P g 1c = 0 for all conjugates g of f . Now, as we also have the decomposition in stable representations of the Hecke algebra, the sum of all P g 1c for g conjugate to f is proportional to the projection π of the trace of P − (∞) (belonging to J 0 (N )(K)) in A f (K) ⊗ C, so we have proven that this projection in A f (K) is torsion.
We now explain how to deduce Lemma 4.4 from this result. Let m be an integer coprime to N . Define the Hecke correspondence C m to be the image of X 0 (mN ) in X 0 (N ) × X 0 (N ) under the product of the two natural maps X 0 (mN ) → X 0 (N ). We define to be the projection of C m onto the End(J 0 (N )) component of Pic(X 0 (N ) × X 0 (N )) (see (8)). Then C m lands in the subspace NS(J 0 (N )) ⊂ End(J 0 (N )) of endomorphisms symmetric with respect to the Rosati involution. When m is square-free, C m is the Hecke operator T m . In general, C m is a linear combination of T m/d for d divisors of m.
Recall that i 1,2 : X 0 (N ) ֒→ X 0 (N ) × X 0 (N ) denotes the diagonal morphism. A non-cuspidal point in the support of i * 1,2 ( C m ) is a cyclic N -isogeny f : E 1 → E 2 , together with cyclic subgroups G i of E i of order m such that f (G 1 ) = G 2 , and isomorphisms E i which commute with f and the induced isogeny E 1 /G 1 → E 2 /G 2 . In particular, the ring of endomorphisms of each E i , of discriminant denoted by D i , thus contains an element of norm m so there exist A i , B i in Z for which The isogeny being cyclic, A i and B i must be coprime here. The point E 1 → E 2 is a Heegner point of Y 0 (N ) if and only if D 1 = D 2 .
Lemma 4.7. Let X = X 0 (N ), let m be prime to N , and let C m be the Hecke correspondence defined above. Then the divisor i * 1,2 C m is supported on the set of Heegner points whenever m is less than N/4.

Proof.
Let (E 1 → E 2 ) be a non-cuspidal point in the support of i * 1,2 C m as above. Suppose the point is not Heegner. Since E 1 and E 2 are N -isogenous, D 2 = λ 2 D 1 for some rational λ > 0 a power of N . Since λ = 1, we must have D i divisible by N 2 for some i, and hence m > N 2 /4, by (22). Finally, if the conductor of the order was not prime to N , we would also have N 2 |D i which leads to the same inequality.
By the following Lemma (essentially just the Sturm bound) we have enough Hecke operators C m for which i * 1,2 C m is supported on cusps and Heegner points to complete the proof of the first part of Lemma 4.4. This completes the proof of case (1) of Proposition 4.1. Lemma 4.8 implies that any nice correspondence Z on X 0 (N ) can be written as a linear combination of the C m for m < N 2 /4 prime to N . By Lemma 4.7, for any such Z, D Z (b) is supported on Heegner points and cusps, so by Lemma 4.5 (part 2), its image by π B is torsion.
4.3 How to prove (C) using Heegner points under the analytic hypothesis: The second case is similar to the first, but we must replace the classical notion of Heegner point with Heegner points on non-split Cartan modular curves in the sense of Zhang/Kohen-Pacetti, and replace Gross-Zagier-Zhang on X 0 (N ) with Zhang's Gross-Zagier theorem on X ns (N ).
To make results easier to state, we use the moduli interpretation of X ns (N ) and X + ns (N ) given in [KP16] and its consequences. To do so, one fixes an ε ∈ F N which is not a square. A pair (E, φ ε ) is then an elliptic curve E together with an endomorphism φ ε of E[N ] whose square is multiplication by ε. Such an endomorphism has eigenvalues in F N 2 \F N , and two pairs (E, φ ε ) and We thus know that X ns (N ) is the moduli space of such pairs up to isomorphism [KP16, §1.2]. Furthermore, the natural involution on this modular curve is given by First, we define Hecke correspondences C m ⊂ X ns (N )×X ns (N ) (for m prime to N ) as follows. We have a curve X ns (N, m) = X ns (N ) × X(1) X 0 (m) given by adding an auxiliary Γ 0 (m) structure. We have two maps X ns (N, m) → X ns (N ), the forgetful one, and the one sending We will again use the generalised Gross-Zagier formula from Zhang from [Zha04], in a slightly different context here. We follow the notation of [Zha04,§6]. Let K/Q be an imaginary quadratic field inert at N (instead of split or ramified in the previous case), and let K ֒→ M 2 (Q) be an embedding associated to an integral basis of O K . For a choice of order O c of K of conductor c prime to N , define where U c can be defined as GL 2 (Z v ) for places v not dividing N , and (R c ⊗ Z N ) * ⊂ GL 2 (Z N ) at N (seen in GL 2 (Z N )). The important point here is that On the other hand, we say that (E, φ ε ) ∈ Y ns (N ) is a Heegner point (in the sense of Kohen-Pacetti) with multiplication by O c if End(E) ∼ = O c (with c prime to N ) and φ ε comes from an endomorphism β of E. Notice that it automatically implies that N is inert in O c , because the minimal polynomial of β modulo N , being X 2 − ε, must be irreducible which is only possible when This discussion thus implies the following equivalence of definitions. Let f be an eigenform in S 2 (Γ 0 (N 2 )) +,new . It can be seen as an automorphic form on an M Uc as above, using the isomorphism of Hecke modules S 2 (Γ 0 (N 2 )) +,new ∼ = S 2 (Γ + ns (N )) and the isomorphism M Uc (C) ∼ = Y ns (N ) C and we again have by Eichler-Shimura theory a Q-simple quotient A f of J + ns (N ). The consequence of Zhang's result that we will use is the following. If L ′ (f, 1) = 0, then P f 1c = 0 and π f (P 1c ) is torsion in A f (H c ). Proof. Using the previous lemmas and discussion, we can translate everything in terms of the Shimura curve M Uc : the Heegner point P becomes a CM point in the sense of Zhang and f becomes an automorphic representation φ. These changes are compatible with Hecke operators and Galois actions, so they preserve the decompositions into isotypical components above. We can then proceed along the same lines as the proof of Lemma 4.5 part 2 to deduce the conclusion from Zhang's theorem.
We are now ready to prove the analogue of Lemma 4.7 with X 0 (N ) replaced by X + ns (N ).
Lemma 4.11. Let X = X ns (N ), let m be prime to N , and let C m be the Hecke correspondence defined above. Then the divisor i * 1,2 C m is supported on Heegner points in the sense of Kohen-Pacetti and cusps whenever m is less than N 2 /4.

Proof.
By the moduli interpretation of X ns (N ) and the Hecke correspondences, a noncuspidal point of i * 1,2 C m is a pair (E, φ ε ) such that there exists an endomorphism α of E of norm m with cyclic kernel (of order m) such that if α is the induced endomorphism of E[N ], α • φ ε • α −1 = φ ε . By elementary arguments on GL 2 (Z/N Z), this implies that α belongs to the nonsplit Cartan subgroup associated to φ ε (which is also the group of invertible elements of Z[φ ε ]). Now, α is not a scalar element: indeed, otherwise we would have α = k + N β with k ∈ Z and β ∈ End(E) and looking at the norm in the underlying quadratic imaginary ring O K , we would have N/2 dividing the double of the second coordinate of α in (1, √ D K ) (nonzero because α is not scalar itself) therefore N 2 /4 is less than the norm of α which is m, but we have m < N 2 /4. From this, we deduce that Z[α] = Z[φ ε ], as both are Z/N Z-vector spaces of dimension 2 and the former is included in the latter. This implies that φ ε is induced by the action of an element of Z[α] ⊂ End(E) on E[N ], and the ring of endomorphisms has conductor prime to N for the same reasons as in X 0 (N ), and its discriminant is automatically prime to N as discussed after defining Heegner points in the sense of Kohen-Pacetti.
By the compatibility with Hecke correspondences on X 0 (N 2 ) (which is a consequence of Chen's theorem without quotient by Atkin-Lehner involutions, e.g. [dSE00, Théorème 2]), Lemma 4.8 implies that any nice correspondence Z on X + ns (N ) can be written as a linear combination of C m for m < N 2 /4 prime to

An alternative approach
In this subsection, we sketch an alternative and less ad hoc approach for proving Proposition 4.1 in the case X = X + 0 (N ), using the Theorem of Yuan-Zhang-Zhang on the heights of diagonal cycles. . Let X = X 0 (N ), and let f, g be non-conjugate eigenforms in S 2 (Γ 0 (N )). Let Z ∈ NS(J 0 (N )) lie in the image of NS(A g ). Suppose ǫ(f ) = −1 and ǫ(Sym 2 (g) ⊗ f ) = 1. If the projection of D Z (b) to A f is non-torsion, then L ′ (f, 1) = 0.
The result above holds for arbitrary N , but is most useful when N is prime, since in this case we have ǫ(f ⊗ g ⊗ g) = −a N (f )a N (g) 2 = −a N (f ) (see e.g. [GK92]). Hence in this case Theorem 4.12 implies that the image of D Z (b) in A f is torsion for all eigenforms f in S + 2 (Γ 0 (N ))., which implies that we get an alternative proof for X + 0 (N ). One way to view Proposition 4.1 is that it shows that it is easier to prove diagonal cycles are torsion than it is to prove they are non-torsion. On the other hand, one can show directly that the image of D Z (b) in A f is torsion for all eigenforms f satisfying w N (f ) = −f , as explained in [Dau13, Theorem 3.3.8]: by Lemma 3.7, we have w * N (D Z (b)) = D w * N (Z) (b). Since w * N (Z) = Z, and w * N acts as (-1) on A f , we deduce π f * (D Z (b)) is torsion.

Proof of the analytic part
In this section, we prove Theorem 1.3 using analytic weighted averages techniques, following guiding principles e.g. from [IS00] and [Ell04]. For convenience and consistency, the notation below is as close as possible to those from [LF17]. Notation • N is a prime number and M = N or N 2 in all of the following.
• If f, g ∈ S 2 (Γ 0 (M )), we denote their Petersson scalar product by where D is a fundamental domain of Γ 0 (M ), and the associated Petersson norm by · M .
• For ε = ±1, the space S 2 (Γ 0 (M )) ε refers to the subspace of modular forms f of S 2 (Γ 0 (M )) such that f |wM = ε · f , where w M is the Fricke involution of S 2 (Γ 0 (M )). Note that in weight 2, this is the space of modular forms f such that L(f, s) has root number −ε.
• For A, B linear forms on S 2 (Γ 0 (M )) (resp. on a subspace indicated by superscripts), we write where f goes through an orthogonal basis of S 2 (Γ 0 (M )) (it is readily checked not to depend on this choice of basis), resp. of the prescribed subspace. We will add superscripts +, −, new to refer to the sum restricted to an orthogonal basis of the corresponding subspaces of S 2 (Γ 0 (M )).
• We denote by a m (for m ∈ N ≥1 ) and L ′ the linear forms on S 2 (Γ 0 (M )) which to f associate respectively the m-th coefficient of the q-expansion of f , and L ′ (f, 1) (defined properly in the next paragraph).
• Unless stated otherwise, for any integers a, b, c, (a, b) and (a, b, c) refer to the greatest common divisor respectively of a and b and of a, b, c respectively.
• For any positive number B, O 1 (B) refers to a complex number of absolute value ≤ B.
The proof of Theorem 1.3 relies on the following lemma. 3), this implies that L ′ (g, 1) = 0 for all normalised newforms g which are conjugates of f by Gal(Q/Q), thus Theorem 1.3 holds for M unless the field of coefficients of f is Q and this f is unique, which we assume now. As f is normalised, those coefficients are algebraic integers hence belong to Z. Now, one has by hypothesis, so a 2 (f ) / ∈ Z which leads to a contradiction and Theorem 1.3 holds.
Remark 5.2. The statement of this lemma appears quite ad hoc so let us explain the main motivations behind it.
• As we will see later, as long as m is small compared to √ M , one has with explicit implied constants. This proves that the hypotheses of the lemma are indeed satisfied for large M .
• The error terms of the estimate above are smaller when the m's are smaller, hence the choices of m = 1 and 2 for the ratio.
• There are far better asymptotic estimates on the number of newforms f in S 2 (Γ 0 (M )) +,new such that L ′ (f, 1) = 0, e.g. : by [KMV00] (at least for M = N prime), the proportion of such forms is asymptotically at least 7/8, in particular there are far more than just 2 for M large). These techniques, using also estimates of second moments and of the norms f M , are harder to make explicit, and we suspect the effective bounds obtained by following step-by-step the arguments would be huge. Lemma 5.1, while very crude (and giving a weaker result) is tailor-made to be efficient enough for precise estimates and approachable bounds.

Splitting of the terms to estimate the first moments
The starting point to estimate the weighted averages a m , L ′ new N is the following trace formula of Petersson adapted by Akabary (and proven in greater generality in [LF17]).

Proposition 5.3.
Let m, n, M be three positive integers, and ε = ±1. Then, we have where S is the notation for Kloosterman sums S(m, n; c) = k∈(Z/cZ) * e 2iπ(mk+nk −1 )/c (except for c = 1 where its value is 1 by convention), Q −1 means the inverse of Q modulo d in the Kloosterman sums and J 1 is the Bessel function of first type and order 1.
The sums on the right-hand side are absolutely convergent thanks to the following well-known uniform bounds: |J 1 (x)| ≤ |x|/2 for all x, and the Weil bounds Lemma 5.4. For any M ≥ 1 and any f ∈ S 2 (Γ 0 (M )) + , one has where E 1 is the exponential integral function, defined on ]0, +∞[ by Proof. We define the completed L-function Λ associated to L by By usual results (e.g. [Bum96], section 1.5), this function extends to an holomorphic function on C and satisfies the functional equation The expression of L ′ (f, 1) is then deduced from the functional equation of Λ by integration of residues on vertical axes and Mellin transform (see e.g. [IK04] (26.10) where the definition of L is translated by 1/2).
With this formula and by uniform convergence of the terms involved, we obtain: where and The main term in (27) will be E 1 (2πm/ √ M ) as long as m ≪ √ M . The trace formula does not separate between the old and new spaces, which we need for M = N 2 . This is taken care of in the following lemma. Proof. By orthogonality of the new and old subspaces, To prove the formula on the oldpart, we need to be a bit careful with the definitions of completed L-functions: although the definition of L(f, s) does not depend on the ambient space of modular forms, the definition of completed Lfunction Λ(f, s) in (25) does! The degeneracy operators are denoted by A n as in the original article [AL70]. Let hence also Consequently, an orthogonal (see the computations of section 4 of [LF17] for example) basis of S 2 (Γ 0 (N 2 )) +,old is given by the f |A1 + (f |A1 ) |W N 2 , where f goes through an eigenbasis of S 2 (Γ 0 (N )). The mentioned computations also If N does not divide m (so that a m (f |AN ) = 0), this implies that a m , L ′ +,old where f goes through an orthonormal basis of S 2 (Γ 0 (N )). Now, by the functional equation of Λ(f, s) in (26), , 1)).
The first equality is a direct application of the definition of Λ, the second one uses that L(f |AN , 1) = L(f, 1) (easy to show by the integral formula of L(f, 1)) and the results above. Thus, to compute L ′ (f |A1 + (f |A1 ) |W N 2 , 1), it is enough to know the sum of the two right-hand terms which is the sum of the two left-hand terms, mutually equal. Now, if ε f = 1 then L(f, 1) = 0 by sign of the functional equation of Λ(f, s) (in level N here !), and if ε f = −1, Λ ′ (f, 1) = 0. We thus obtain in this case and get the lemma by summation on those forms f 's gathered by sign of ε f .

First estimates
We recall that M = N or N 2 with N > 2 prime for the following.
Lemma 5.6. Using the Weil bounds, we get for every c multiple of M and d prime to M : For m = 2 and c, d even, these estimates are improved to Proof. In the definitions of S(c) (and similarly for T (d)), we separate the terms in n depending on the values of (m, n, c) = m ′ which is a divisor of (m, c). Then, using |J 1 (x)| ≤ |x|/2, it only remains to control the sum of the E 1 (2πm ′ n/ √ M ) for n from 1 to +∞, which after sum-integral comparison and variable change is smaller than √ M /(2πm ′ ). In the specific case where m = 2 and c or d even, the cases are made from the beginning on the values of (m, n, c) 1/2 instead of bounding by (m, c) 1/2 , and a careful computation gives those bounds. hence Actually, the exact same bounds are found for as the integral of e −t on [0, +∞[ is equal to 1 like the one of E 1 . Thus, by similar computations, Gathering those bounds, we get for all m a m , L ′ +,new and slightly better ones for m = 2 coming from refinements above (it suffices to replace 86mg(m) by 213 and 43mg(m) by 97 above). By computations on Sage, we deduce the following first estimates. hence Lemma 5.1 applies and Theorem 2 is true for N ≥ 45341 for X 0 (N ) + and for N ≥ 269 for X + ns (N ). For M = N , the estimates of a m , L ′ N are readily obtained, but the slowness of convergence is much more visible. This is mainly due to the fact that the error term is in m/ √ N instead of m/N .

Improving the estimates for prime level
To attain from N ≥ 45341 a range where all remaining primes can be checked by a different method, one needs to improve upon the worst error term appearing in a m , L ′ + N , which is in m/ √ N and comes from the estimates of T (d) after looking at (31).
The following arguments rely on cancellations of Kloosterman sums not exploited by the Weil bounds. For d = 1, the Kloosterman sum is always 1 (see the convention) so this case has to be dealt with separately. A careful analysis proves that which will slightly improve the bounds later. Assume now that d ≥ 2.
The main term contributing to the bound is E 1 (2πn/ √ N ), hence we write where T M (d) is the sum of terms for which n ≤ 3 √ N /π and T R (d) is the remainder.
By Weil bounds, using the fact that the integral of E 1 on [5, +∞[ is less than 10 −4 , we obtain where λ m = 43 for m = 1 and 97 for m = 2 as before, so this contribution will be very small. For T M (d), we will exploit Polyà-Vinogradov-type estimates ([LF16], Lemma 5.9).
Proposition 5.8. For every d > 1, every k invertible modulo d and every m, K, K ′ ∈ N, Now, assume N ≥ 1000, so that for m = 1 or 2 and n ≤ 5 √ N /(2π), 4π √ mn/(d √ N ) ≤ 1.5. This implies that in the considered range for n, the function t → J 1 (4π is decreasing and positive (as the product of two such functions). Its total variation on [1, 5 √ N /2π] is then bounded by its first value (itself controlled by E 1 (2π/ √ N )/2). By Abel transform and the previous proposition, we thus obtain Compared to Weil bounds in Lemma 5.6, the new bound is approximately the best for d ≤ f (N ) = ⌊N/(2.5 2 E 1 (2π/ √ N ) 2 )⌋. We then obtain with lemma 5.11 of [LF16]. By Weil bounds and the same lemma, for m = 1, and for m = 2, (log(f (N )) + 4).
Combining these arguments, we get, for N ≥ 1000, We now discuss how to deal with the remaining cases, namely those for which N ≤ 8641 and g(X + 0 (N )) ≥ 2, and those for which N ≤ 151 and g(X + ns (N )) ≥ 2. The most natural approach is the following: for any small N , compute a basis of eigenforms for S 2 (Γ 0 (M )) +,new , and for every f (normalised) in this basis, compute L ′ (f, 1) up to sufficient precision to ensure that L ′ (f, 1) = 0.
Recall that by ( [GZ86], Corollary V.1.3), if L ′ (f, 1) = 0 under the same assumptions, the same is true for the Galois conjugate eigenforms, so only one check needs to be performed for the Galois orbit. Theorem 1.3 requires exactly that the sum of sizes of those Galois orbits is at least 2, so we only need to check that for two Galois orbits of size 1 (or one of size at least 2), one has L ′ (f, 1) = 0.
We have performed these verifications in MAGMA, and obtained that : • For any prime N ≤ 2000 such that X 0 (N ) + is of genus at least two, there are at least two distincts normalised newforms such that L ′ (f, 1) = 0, hence Theorem 2 holds. Actually, we have also checked that for all such N , L ′ (f, 1) = 0 for all the eigenforms in S 2 (Γ 0 (N )) + , therefore by Proposition 7.1, rank J + 0 (N )(Q) = dim J + 0 (N ) unconditionally for all those small primes. • Similarly, for any prime N ≤ 53 such that X + ns (N ) is of genus at least two, L ′ (f, 1) = 0 for all the eigenforms in S 2 (Γ 0 (N 2 )) +,new , therefore by the same arguments, rank Jac(X + ns (N ))(Q) = dim Jac(X + ns (N )) for all those small primes. Unfortunately, these algorithms require explicit embeddings of the fields of coefficients K f of f into C, which makes them very slow when N becomes larger than 2000 (then, the degree of K f can be larger than 100). We thus could not complete the argument by using only this method, let us explain how to deal with the intermediary range N ∈ [2000, 9000] for X + 0 (N ) and N ∈ [59, 151] for X + ns (N ).
The idea is to look at the simple quotients of the two relevant Jacobians which are elliptic curves. If there are none, in this range, we have proved that a 1 , L ′ +,new M = 0 so we must have f such that L ′ (f, 1) = 0, and it generates a simple quotient of dimension at least 2 by hypothesis, so we are done. Now, if there are elliptic curves in there, it is sufficient to find two of them of rank 1 for the same reasons. Quotients of J 0 (M ) +,new of dimension 1 are in one-to-one correspondence with isogeny classes of elliptic curves of conductor N and root number −1 (the fact that this correspondence is surjective is a consequence of Cremona's tables in this range but also a particular case of modularity theorems).
One can thus eliminate all levels N except the ones for which there exists exactly one (up to isogeny) elliptic curve E of analytic rank 1 and conductor N . Using Cremona's tables, we obtain a list of respectively 70 (M = N ) and 7 (M = N 2 ) possible exceptions, namely N in {61, 67, 73, 101, 109, 113} for the latter.
Now, we use a last argument: if the modular form f E associated to E is really the only one such that L ′ (f, 1) = 0 in the space, one should have (the fact that this equality holds without a normalisation factor comes from the Manin constant being equal to 1 here, which is true in this range by results of Cremona). Now, the left-hand side is larger than 4/5 for M = N , N ≥ 2000 and than 1/2 for M = N 2 , N ≥ 53 by the (optimised) lower bounds given above, and the right-hand side is computable in terms of periods of E. Using this idea turns out to eliminate all remaining possible exceptions in both cases of M , which concludes the proof. Remark 5.9. In some sense, this heuristic is natural: all terms in the sum defined by a 1 , L ′ +,new M are positive (another consequence of Gross-Zagier formula), hence there is no cancellation among those, and the idea is that one of them alone cannot be enough to approach the estimates given for the sum.

Appendix: Chow-Heegner points and Ceresa cycles
In this appendix we explain how Lemma 3.2 is a consequence of Hain and Matsumoto's work relating the extension [Lie(U 2 )] to the Ceresa cycle.

Ceresa cycles and Gross-Kudla-Schoen cycles
We recall some properties of modified diagonal cycles studied in [GS95], [CvG93] and [DRS12]. As our discussion applies in fairly broad generality, we take X to be a smooth geometrically irreducible projective curve over a field K of characteristic zero. Let π S denote the projection X n → X #S defined by projecting onto the coordinates in S as in (7). The Gross-Kudla-Schoen cycle is defined to be It defines an element of the group CH 2 (X 3 ) of codimension two cycles in the triple product X × X × X. By [GS95, Proposition 3.1], the class of ∆ GKS lies in the subspace CH 2 0 (X 3 ) of homologically trivial cycles. Now let Z ⊂ X × X be a correspondence, and let where the second map is the intersection product with Z × X 2 ⊂ X 4 .

The Gross-Kudla-Schoen cycle and the Ceresa cycle
Since [∆ GKS ] is homologically trivial, it has ( §2.1) an étale Abel-Jacobi class By [GS95, Corollary 2.6], the cycle class AJé t ([∆ GKS ]) lies in the image of the Kunneth projector and hence may be thought of as an element of H 1 (G K , V ⊗3 (−1)) (here V := H 1 et (X K , Q p (1))). The action of S 3 on X 3 induces an action on V ⊗3 (−1), which is given by ǫ ⊗ σ, where ǫ is the sign of a permutation and σ is the natural action of S 3 on V ⊗3 . Since ∆ GKS is invariant under the S 3 action, it lies in the image of H 1 (G K , ∧ 3 V (−1)) under the map induced by the inclusion For the relations to fundamental groups, it will be helpful to recall the relation between ∆ GKS and the Ceresa cycle. By [GS95,Proposition 5.3], the image of ∆ GKS in CH g−1 (J) under the map is rationally equivalent to The Ceresa cycle C b is defined to be Proposition 6.2 (Colombo-van Geemen, [CvG93], Proposition 2.9). We have We first recall Hain and Matsumoto's description of the Galois action on U 2 . We again take X to be a smooth projective geometrically irreducible curve over a field K of characteristic zero. The group U 2 is an extension with V = T p J ⊗ Q p again. We define and write the image of v 1 ∧ v 2 in ∧ 2 V as v 1 ∧ v 2 . Taking the Lie algebra L 2 of U 2 , we obtain an element [L 2 ] ∈ Ext 1 GK (V, ∧ 2 V ), or equivalently an element of H 1 (G K , V (−1) ⊗ ∧ 2 V ). The following theorem of Hain and Matsumoto characterises this extension class in terms of the Gross-Kudla-Schoen cycle. Theorem 6.3 (Hain-Matsumoto [HM05], Theorem 3). Let α : is the class of the Ceresa cycle in CH g−1 (J), and AJé t ( Via the relation between the Ceresa cycle and the Gross-Kudla-Schoen cycle, this has the following corollary. Proof. Let ι : ∧ 3 V → V ⊗3 be the inclusion (39), and τ ′ : Hence we deduce from Theorem 6.3 that We now return to the case where K = Q. Via the commutative diagram Lemma 6.5. Let Z ⊂ X × X be a codimension 1 cycle. Let i 1 , i 2 , i 3 : X ֒→ X × X be the closed immersions defined by the subschemes {b} × X, X × {b} and the diagonal ∆ X of X × X respectively. For j = 1, 2, {1, 2}, let i * j denote the pull-back morphism Then the extension class in H 1 (G K , V ) associated to the Lie algebra L Z is given by AJé t (D Z (b)), with D Z (b) as in (15).
Proof. The class [L Z ] is the image of [L 2 ] under the morphism induced by π Z : ∧ 2 V → Q p (1). We have a commutative diagram By Theorem 6.3, the extension class [L 2 ] is given by AJé t (∆ GKS ), hence by Lemma 6.1.

Appendix: Proof of the Kolyvagin-Logachev type result
In this appendix, we fix the following notation: • M is a fixed odd level (which for our applications will be N or N 2 ) • f ∈ S 2 (Γ 0 (M )) +,new is a normalised eigenform.
• A = A f is its associated quotient of J 0 (M ), together with the canonical projection π : J 0 (M ) → A independent of the choice of f in its Galois orbit.
We explain here the following result, attributed to Kolyvagin and Logachev.
If L ′ (f, 1) = 0, the rank of A(Q) is exactly g := dim A. Proof of the Corollary. By proposition 7.1 the rank of A is equal to its dimension as it is true for each of its factors A f . Now, we recall that all endomorphisms of an A f are symmetric and the latter is of GL 2 -type, in particular End † (A f ) is of rank dim A f (see §4.1) . Finally, for f, g non Galois conjugates, there is no morphism between A f and A g (by multiplicity one in the newpart) so the endomorphism ring splits and we get the last equality.
Remark 7.3. This result is well-known if dim A = 1 ( [Kol90] for the original reference, [Gro91] for a survey), and proven in much greater generality in [Nek07], all these along the lines of a stronger result in the rank zero case proved in [KL90]. It is also (a slightly weaker version of) the main result in Tian's thesis [Tia03] and of a paper of Tian and Zhang in preparation [TZ] for which we could not find quotable material. In any case, we felt it sufficiently different from the former references (to which we borrow constantly) to deserve a proof for the nonexperts. For the same reasons, we will simply refer to those papers for parts of the proofs which generalise seamlessly and focus on the more technical points.
Convention We use a well-chosen prime number p to obtain Proposition 7.1. As we only need one such p, in all this Appendix, when a property holds when p is large enough, we then automatically assume it is without further mention.
We will prove Proposition 7.1 by reducing it successively to other statements which will be emphasized.
Notation Throughout this text, τ denotes the usual complex conjugation and when it acts on an Z-module M, M + and M − denote the spaces of m ∈ M respectively fixed and reversed by τ . If M is finite of odd order, M = M + ⊕ M − , which we will frequently use implicitly.
Given an Galois extension L/K of number fields and P a prime ideal of L unramified over p, (P, L/K) denotes the Frobenius of P for this extension, and (p, L/K) the conjugacy class of such Frobenius's in Gal(L/K).

Structure of the p-torsion and reduction to Selmer groups
Let K f be the number field of coefficients of f . By ( [KL90], section 2.1), there is an isomorphism The inverse image of End Q A is thus an order in K f denoted by O, and A is endowed with a structure of O-module. We now fix p an odd prime totally split in K f and prime to the conductor of O (there are infinitely many such primes by Cebotarev density theorem), so that pO = P 1 · · · P g as a decomposition into prime ideals. In all the following, the notation P will run through P 1 , · · · , P g . Remark 7.4. It is likely the proof still holds for any type of decomposition of p but this hypothesis makes the exposition much more symmetric (and there are infinitely many of them so we can choose it as large as necessary). In the opposite situation, if there is an inert prime in K f , the proof should be a bit simpler.
One of the key ideas to get closer to the case of elliptic curves is decomposing every structure of O/(p)-modules using those prime ideals. Our tool is the following Lemma, often used without mention.
If L is a number field, for every place v of L, the natural localisation maps loc v give rise to a commutative diagram δ locv locv locv δv (43) inherited by flatness from the commonly known analogous diagram for the ideal (p) (for references on those facts and the Selmer groups, see [HS00], Appendix C.4). Let us define the P-Selmer group as again canonically identified to Sel p (L, A)[P] hence fitting by the same arguments into the exact sequence Now, consider an imaginary quadratic field K whose discriminant D K < −4 is squarefree, prime to the level M and a square modulo M . These conditions guarantee that there is a Heegner point (we fix definitively n, [a 0 ]) in the notation of [Gro84], where H is the Hilbert class field of K. As f |wM = f , π • w M = π therefore by elementary properties of Heegner points ( [Gro84], formulas (4.1) to (5.2)), for y 1 = π((x) − (∞)) ∈ A(H), one has Now, using a theorem of Waldspurger [Vig81, Théorème 2.3], let us fix once and for all a K such that L(f ⊗ ε K , 1) = 0 where ε K is the Dirichlet character associated to K. By Gross-Zagier formula ( [GZ86], Theorem I.6.3) the point y K is then nontorsion in A(K) and has an integer multiple in A(Q) by (48). The subgroup O · y K is thus a subgroup of A(K) of rank g (as nonzero elements of O act by isogenies), which leads us to Reduction 1 'Prove that O · y K is of finite index in A(K)'. Now, for p large enough, which further leads by (45) to Reduction 2 'Prove that for all P, δ(y K ) generates Sel P (K, A)'.
Proof. If this claim holds, every Sel P (K, A) is an O/P ∼ = F p -vector space of dimension 1, so A(K)/PA(K) is of dimension at most 1 by (45), and is of dimension at most g over F p . This imposes that the Mordell-Weil rank of A(K) over Z is at most g, hence the equality using O · y K .
To conclude this paragraph, τ acts naturally on A(Q), A[P], H 1 (K, A[P]) and Sel P (K, A), and the action of O and the morphisms between those in (42) and (43)   is injective, with the action of G on Gal(L ab /L) defined by conjugation in Gal(Q/Q).
Remark 7.8. Here is an important difference with the dim A = 1 case: the Galois representation Gal(Q/Q) → GL(A[P]) ∼ = GL 2 (F p ) is not proven to be surjective ( [Rib76] does not cover the square M case), but we will manage with (a) and (b) although it introduces significant changes compared to some arguments in [Gro91].
We now choose S a finite sub-O-module of H 1 (K, A[P]), stable by τ (this will be Sel P (K, A) and then an auxiliary module for the proof). By Proposition 7.7 (c), there is a pairing which is injective on the left. We define L S the extension of L whose absolute Galois group is the orthogonal of S, and thus obtain a nondegenerate pairing between finite abelian p-torsion groups Keeping track of the actions of τ and the σ ∈ G, we have that In particular, the extension L S /Q is Galois.
Lemma 7.9. This pairing induces a perfect bilinear pairing from S ε × H + S to A[P] ε ∼ = F p , hence a duality between S ε and H + S . Proof. By (50), these two pairings (for ε = ±1) are well-defined, let us prove they are injective on the left and on the right, they will then be perfect as everything is finite (-dimensional Lemma 7.10. Fix ε = ±1 and I + S a proper subgroup of H + S . Then, s ∈ S ε is a 0 if for all ρ ∈ H + S \I + S , [s, ρ] S = 0. Proof. It is a trivial consequence of the perfect duality above, knowing that the sub-F p -vector space generated by H + 0 \I + 0 is H + 0 itself, e. g. by a counting argument. Reduction 3 'For all P, apply Lemma 7.10 to (s 0 = 0, ε = −1) (resp. δy K , ε = 1) to prove that Sel P (K, A) − = 0 (resp. Sel P (K, A) + = δy K )'.
The next subsection will show us how to compute the pairing [·, ·] S .
• A Kolyvagin prime ℓ is a prime number such that: − ℓ does not divide D K M p (or the conductor of O), so is unramified in L.
− The conjugacy class of (ℓ, L/Q) is the one of τ in Gal(L/Q). In particular, ℓO K =: λ ℓ is inert over ℓ. We will often shorten it to λ if ℓ is nonambiguous, and for any extension K ′ of K, λ K ′ will be a choice of prime ideal of O K ′ above λ (in a consistent fashion if multiple extensions are considered).
• A Kolyvagin number n is a squarefree product of Kolyvagin primes ℓ.
Proposition 7.12. For a Kolyvagin prime ℓ, λ splits completely in L. Furthermore: p|a ℓ (f ), p|ℓ + 1 in O, and all the points of A[P] are defined over K λ . Moreover, for the eigenspaces of the action of Frob(ℓ) on A(K λ ), each space (A(K λ )/PA(K λ )) ε is of dimension 1 over F p .
Proof. Up to conjugation, (λ L , L/K) = (λ L , L/Q) f (λ/ℓ) = τ 2 = Id so λ L /λ is totally split. Now, by Eichler-Shimura theory ( [KL90], formula (2.1.8)), the characteristic polynomial of the Frobenius endomorphism Frob(ℓ) on the reduction A of A modulo ℓ (as an O-linear endomorphism) is X 2 − a ℓ (f )X + ℓ and the one of the complex conjugation is X 2 − 1, and they must agree on A[p]. In particular, Frob(ℓ) 2 acts trivially on A[P] so A[P] = A[P](F λ ) and we can lift those points to K λ . By the same arguments, on also has the decomposition in two nontrivial spaces, given the characteristic polynomial of Frob(ℓ), so each of the two spaces on the right-hand side is of dimension 1 over F p . We deduce immediately by the structure of finite abelian groups that as groups, which proves that each ( A(F λ )/P A(F λ )) ε must be of dimension 1 over F p , and this also lifts to K λ (without increasing the dimension as the group of elements reducing to 0 modulo λ is p-divisible).
To state the next result, recall that for a finite place v ∤ p of good reduction of A, the image of , called the unramified part. The latter is isomorphic to A[p] when all the p-torsion is defined over K v , via the evaluation of the cocycles at Frob(v) the topological generator of Gal(K unr v /K v ). The same argument translates for A[P] by tensoring by O/P again.
Proposition 7.13. Let L be an unramified prime ideal of L S whose Frobenius in Gal(L S /Q) is τ h for h ∈ H S . It is above a Kolyvagin prime ℓ and for every s ∈ S whose localisation at λ = ℓO K is unramified, is clearly commutative, which establishes the equality by definition.
Remark 7.14. The set of all (τ h) 2 thus obtained is exactly H + S , by Cebotarev density theorem. Now, for any place v of K, we can construct ( [Tat58], section 2) a canonical bilinear pairing obtained from Tate duality ·, · Kv : The key use of Tate duality is the following Proposition, which is a slight generalisation of [Gro91, Proposition 8.2].
Proof. By its definition, (51) comes from the Weil pairing in the sense that the latter induces a cup product (·, ·) Kv : for which δ v (A(K v )/pA(K v )) is isotropic, and the resulting quotiented pairing is exactly ·, · Kv . It also implies that the P and P ′ -parts are mutually orthogonal because they are for the Weil pairing. Now, the so-called global Tate duality states that for any s ∈ Sel p (K, A), γ ∈ H 1 (K, A)[p], v∈MK inv v δ −1 v loc v s, loc v γ Kv = 0 ∈ Q/Z, where inv v : Br(K v ) → Q/Z is the Brauer invariant isomorphism for all v. Indeed, let us lift γ to γ, so that for every v ∈ M K , with the analogous definition of (·, ·) K on K, and loc v,Br : Br(K) → Br(K v ) the usual localisation. Now, by properties of Brauer groups, the sum of inv v • loc v is 0 on Br(K) hence the formula. Under our assumptions on γ and s, we thus have loc λ γ = 0 and δ −1 λ loc λ s, loc λ γ K λ = 0, let us show how this implies that loc λ s = 0.
By the original arguments of [Tat58], the pairing ·, ·, K λ is a perfect pairing. Being inherited from the Weil pairing, the P and P ′ -parts for P = P ′ are orthogonal, so it induces a duality A(K λ )/PA(K λ ) × H 1 (K λ , A)[P] → Z/pZ. Proof. (a) for [P n ] is inherited from (55) by the construction of P n (see Proposition 5.4 of [Gro91]), and deduced for c(n), d(n) by τ -equivariance of the morphisms of (57).
(b) is obtained by tensoring (57) by O/P, which preserves exactness by flatness and [P n ] seen in A(K n )/pA(K n ) ⊗ O/P is exactly the image of P n in A(K n )/PA(K n ). The proof of (c) is given by Proposition 6.2 of [Gro91].
For (d), if D is the Galois group of (K n ) λn over K λ , it is cyclic generated by some σ ℓ and as such, we have injective arrows where for a cocycle c ∈ Z 1 (D, A), red(c) = c(σ ℓ ) mod λ n , and invariant up to coboundary because K n /K m is totally ramified at λ m , so red is well-defined. As A 1 ((K n ) λn ) is a pro-ℓ-group, H 1 (D, A 1 )[p] = 0 which proves that red is injective. The map ι is the quotiented connecting homomorphism, automatically injective. As A(F λ ) is a finite abelian group, the orders of A(F λ )[p] and A(F λ )/p A(F λ ) are readily seen to be equal so ι is also an isomorphism. By ([Gro91], Proposition 6.2 (2)), the image of loc λ d(n) in A(F λ )[p] by red is where R m is any choice of p-th root of P m in A. Its image by Frob(ℓ) is then ℓ(Frob(ℓ) 2 − Id) R m = −(Frob(ℓ) 2 − Id) R m , but the injection ι from (58) is explicitly given by taking a p-th root and applying (Frob(ℓ) 2 − Id), as Frob(ℓ) 2 = Frob(λ) ([KL90], Lemma 3.4.2 for details). The image of loc λ d(n) in A(F λ )/p A(F λ ) via (58) is thus exactly − Frob(ℓ) −1 · P m , and its P-part is trivial if and only if the P-part of P m is. Finally, A 1 (K λm ) is pdivisible hence the equality of O/(p)-modules A(K λm )/pA(K λm ) ∼ = A(F λ )/p A(F λ ), so finally loc λ d(n) P is trivial if and only if [P m ] ∈ A(K λm )/pA(K λm )[P], which is equivalent to P m ∈ PA(K λm ) and the equivalence in terms of c(m) is straightforward.

End of the proof
Let S = Sel P (K, A). By (49), P 1 = y K / ∈ PA(K), hence it defines a nonzero s K := c(1) ∈ S + (Proposition 7.18 (a)). Fixing s ∈ S, for every h ∈ H S , by Cebotarev density theorem, there is a prime ideal L such that (L, L S /Q) = τ h, and by Proposition 7.13, where λ is the prime ideal of K below L, and above ℓ which is a Kolyvagin prime. Outside of I + S (defined as the +-part of the orthogonal of s K ), this formula proves that loc λ s K = 0, so loc λ d(ℓ) P = 0 and all other localisations of d(ℓ) P are trivial by Proposition 7.18. By Proposition 7.15, if s ∈ S − , loc λ s = 0 so [s, (τ h) 2 ] S = 0, hence S − = 0 by Lemma 7.10. Now, consider s ∈ S + such that for some L as above (fixed, so it fixes λ and h above), loc λ s = 0. We have loc λ s K = 0 by hypothesis on h, so in turn loc λ d(ℓ) P = 0 by Proposition 7.18 (d) and c(ℓ) P does not belong to S. By the perfect pairing result of Lemma 7.9 applied to S, c(ℓ) if (τ h) 2 / ∈ I + S , the extensions L S and L c(ℓ) are linearly disjoint over L, which allows, for any h ′ ∈ H S , to choose L ′ a prime ideal of L S L c(ℓ) whose Frobenius restricted to L S is τ h ′ and whose Frobenius restricted to L c(ℓ) is of the shape τ h 0 and not orthogonal to c(ℓ) P . Denoting ℓ ′ the corresponding Kolyvagin prime and λ ′ the ideal of O K , we thus have [c(ℓ) P , (τ h 0 ) 2 ] = loc λ ′ c(ℓ) P (Frob(λ ′ )), this formula being legitimate because loc λ ′ (d(ℓ) P ) = 0 by Proposition 7.18 (c). All this proves that loc λ ′ c(ℓ) P = 0 so loc λ ′ d(ℓℓ ′ ) P = 0 by Proposition 7.18 (d), and it belongs to H 1 (K, A) + [P]. Now, for our s above, the global Tate duality between s and d(ℓℓ ′ ) in the proof of Proposition 7.15 has two possible nonzero terms (in λ and λ ′ ), but by hypothesis loc λ s = 0 so the λ ′ -term is alone, therefore 0 as well. This implies by Proposition 7.15 that loc λ ′ s = 0 for all such λ ′ , therefore s = 0 in this case by Lemma 7.10.
Finally, for s ∈ S + , as loc λ s K = 0 and the space (A(K λ )/PA(K λ )) + is one-dimensional (Proposition 7.12), there is k ∈ Z such that s − ks K satisfies the previous hypothesis and then s = ks K , so we have proved that S + = s K .