# A proof of Kolyvagin’s Conjecture via the BDP main conjecture

###### Abstract.

We adapt Wei Zhang’s proof of Kolyvagin’s conjecture for modular abelian varieties over to rely on the BDP main conjecture instead of on the cyclotomic main conjecture. The main ingredient is a reduction to a case that is tractable by the BDP main conjecture, in a similar spirit to Zhang’s reduction to the rank one case. By using the BDP main conjecture instead of the cyclotomic main conjecture, our approach is more suitable than Zhang’s to extend to modular abelian varieties over totally real fields.

## 1. Introduction

### 1.1. Main result

A lot of the following notation follows [Zha14].

Fix once and for all a prime with

(-big) |

and a quadratic imaginary field of discriminant such that

(split) |

For a newform of weight level and trivial Nebentypus, we denote its field of coefficients by with ring of integers Denote by a place of above and by the order of generated by the Fourier coefficients of Let and let

(1.a) |

Let be its associated -type abelian variety over where we choose an isomorphism class with an embedding Denoting by the ring of integers of we have a Galois representation

(1.b) |

on the Tate module which is a free -module of rank As explained in [Car94], this representation is defined over the smaller subring :

(1.c) |

such that

(1.d) |

We consider the reduction of :

(1.e) |

where is a two-dimensional -vector space. Because of (LABEL:defined-smaller), there is a two dimensional -vector space such that as Galois modules.

Write such that primes are split or ramified in and primes are inert in We consider the following assumption on

(Heegner) |

We also consider the following assumption on the pair :

###### Assumption 1.1.

Assume that is such that

(-free) |

and

(good) |

We also assume that

(res-surj) |

that

(ram) |

where is the set of places ramified in and that

(not anom) |

###### Remark 1.2.

We note that if satisfies (res-surj), then is defined uniquely up to prime-to- isogeny, and hence depends only on We then denote . In this case, we may also take to be -optimal in the sense of [Zha14, Section 3.7], and we do so. We also note that (LABEL:ram) is equivalent to [Zha14, Hypothesis ] when (LABEL:sqf) holds.

###### Remark 1.3.

We note that (LABEL:split), (LABEL:good) and (LABEL:not-anom) imply that we have^{1}^{1}1Indeed, if denotes the kernel of reduction modulo we have by (LABEL:good). Applying and using that is zero by (LABEL:not-anom), we conclude that But is free of rank over and this implies that since by (LABEL:split).

(no local tor) |

A *Kolyvagin prime* for is a prime that is inert in and satisfy

(1.f) |

Let denote the set of square-free products of Kolyvagin primes for

When satisfy (LABEL:Heegner), (LABEL:good) and (res-surj), we consider the collection of cohomology classes

(1.g) |

constructed in [Zha14, Section 3.7], which are the classes of a Kolyvagin system.

###### Theorem A.

Let be a pair satisfying Assumption 1.1 and (LABEL:Heegner). Assume that the BDP main conjecture [JSW17, Conjecture 6.1.2] is true for all pairs satisfying Assumption 1.1, (LABEL:Heegner), such that and such that Then we have

###### Remark 1.4.

In the proof of the theorem, we will use the BDP main conjecture for a single pair of level for a certain product of primes that are inert in In fact, we have some choice over : for instance, we may choose it to have an arbitrarily large number of prime factors, while also avoiding any set of primes of density

###### Remark 1.5.

Although the current results on the BDP main conjecture do not allow us to obtain new results towards Kolyvagin’s conjecture as a corollary^{2}^{2}2Although Wei Zhang works in the ordinary case, one may replace the cyclotomic main conjecture of [SU14] by the one in [CÇSS18] to extend Zhang’s proof to the supersingular case as well., our method is more suitable than Zhang’s to extend to modular abelian varieties over totally real fields: the proofs of the cyclotomic main conjectures, for instance in [SU14] and [CÇSS18], rely on Kato and Beilinson–Flach elements, which we don’t have analogues of in the totally real case. On the other hand, the proofs of the BDP main conjecture rely on Heegner points, which are available in the totally real case. We also note that a large part of the methods in [Zha14] have already been extended to the totally real case in [Wan15].

### 1.2. Proof outline and organization of the paper

Our proof follows [Zha14] very closely. There, Zhang performs an induction on the dimension of the -Selmer group, using the level raising results of [DT94a, DT94b]. He reduces the problem to the cases of dimension and and then uses the results on the cyclotomic main conjecture of [SU14] to show that: (i) the dimension case cannot occur and (ii) the class is nonzero in the dimension case.

As Zhang already noticed, we can rule out the dimension case by using the results on the parity conjecture in [Nek13]. In the setting we are considering, we may also give a simple proof of the parity conjecture by relying on Howard’s formalism of Kolyvagin systems. This is done in Section 2.

The novelty of our paper is how we deal with the dimension case. We first perform a level raising argument to reduce the problem further to the case where the BDP Selmer group^{3}^{3}3The BDP Selmer group is defined when (LABEL:split) holds: it has the usual Bloch–Kato local condition for places the strict condition at one prime above and the relaxed condition at the other prime above is trivial. Such reduction relies on an extension of the parity lemma of Gross–Parson [GP12, Lemma 9], which we establish in Section 3. In the case of dimension the logarithm of the Heegner point can be related to the size of the BDP Selmer group, and the triviality of the latter will imply the -indivisibility of Since is the image of under the Kummer map, the -indivisibility of amounts to Such relation arises from specializing the BDP main conjecture at the trivial character: using the BDP formula of [HB15] on the analytic side and the anticyclotomic control theorem of [JSW17] on the algebraic side. We carry out such argument and conclude the proof of creftype Theorem A in Section 4.

### 1.3. A note on the hypothesis

The hypothesis (LABEL:sqf) is only needed in order to use the BDP formula of [HB15] (see [JSW17, Proposition 5.1.7]). As mentioned in [JSW17, Section 7.4.4], this can likely be dropped.

The condition (LABEL:no-local-tor) seems to be essential to our arguments: it plays an important role in the reduction to the case of trivial BDP Selmer group: it is necessary, for instance, for (LABEL:f-is-one-dim) to be true. Moreover, (LABEL:not-anom) and (LABEL:no-local-tor) are also used to deduce from the formula (LABEL:bdp-form) obtained from the anticyclotomic control theorem.

### Acknowledgments

I want to thank Francesc Castella for advising me throughout this project, and for all the encouragement and advice. I am also grateful for him for noting that the parity conjecture in the non-ordinary case could be deduced from the results in [CÇSS18]. I would also like to thank Daniel Kriz for his willingness to answer many of the questions I had when preparing this paper.

## 2. Parity conjecture

In this section, let satisfy (LABEL:Heegner), (LABEL:good) and (res-surj). Our goal is to prove that

(2.a) |

where denotes the usual -adic Selmer group.

As mentioned in the introduction, this is already covered by the result [Nek13, Theorem B]. However, we will give a simple proof of the parity conjecture in our setting by essentially following [Nek01]: combining a Kolyvagin system argument with an anticyclotomic control theorem.

In the ordinary case, the necessary ingredients are essentially already in [How04]. For the non-ordinary case, the control theorm will be a simple consequence of the work of [CÇSS18] on -Selmer groups.

For this section, denotes the -adic Tate module of Let be the anticyclotomic Galois group, with a topological generator Let be the Iwasawa algebra with Galois action given by and denote where is the unramified -extension of Denote and with diagonal Galois action, where acts on by the natural projection and acts on by

We recall some objects from [CÇSS18] in the non-ordinary case. We have Coleman maps for and as in [CÇSS18, Section 4.2], obtained by restricting the two-variable Coleman maps, first defined in [BL19], to the anticyclotomic line. We denote by the kernel of such map, and define to be its orthogonal complement under local duality. Finally, we denote by the Selmer group with the conditions for and the unramified condition outside of

###### Theorem 2.1.

Let Then the map has finite kernel and cokernel.

###### Proof.

We first analyze the local conditions for Let and be its image under the natural map As in [HL19, Proposition 2.12], we have that if and only if Since is defined to be the kernel of this means that we have the following natural map in cohomology

(2.b) |

This also means that the map in the commutative diagram below has the same kernel as the evaluation at map.

(2.c) |

By a Snake lemma, this means that we have an exact sequence

(2.d) |

but as in [CÇSS18, Proposition 2.3], we can prove that

(2.e) |

has finite cokernel, and hence that the first module in (LABEL:eq_control) is finite. We also have that is finite, since it is dual to which is finite by the proof of [JSW17, Proposition 3.3.7: case 3(b)]. Hence we conclude from (LABEL:eq_control) that is finite.

Now the control theorem follows from standard arguments in Iwasawa theory: consider the following diagram

(2.f) |

where

(2.g) |

and

(2.h) |

As in [Gre99, Lemmas 3.1, 3.2], we have that and that is finite.

Let be the factors of the map From the proof of [JSW17, Proposition 3.3.7], is finite when and is when has good reduction at For the analysis of the local conditions above imply that is finite, since it is dual to Hence finite, and we can conclude so is

By a Snake lemma, we conclude that and are finite. ∎

###### Theorem 2.2.

Let be a pair that satisfies (LABEL:Heegner), (LABEL:good) and (res-surj). Then we have

(2.i) |

###### Proof.

For the ordinary case, we consider the ordinary Selmer group as in [How04, Definition 3.2.2]. By [How04, Theorem 3.4.2], we have a pseudo-isomorphism^{4}^{4}4As explained in [BCK19, Theorem 3.1], we may take in [How04, Theorem 3.4.2].

(2.j) |

for a -torsion module

Now (LABEL:howard-structure-ord) implies that has odd -corank. The control theorem [How04, Lemmas 3.2.11, 3.2.12] says that the natural map has finite kernel and cokernel, and from this we may conclude that is odd.

For the non-ordinary case, we repeat the argument above, but with -Selmer groups. For we have, as in the proof of [CÇSS18, Theorem 5.7]^{5}^{5}5The -Selmer groups admit Kolyvagin systems in the sense of [How04], as constructed in [CÇSS18, Proposition 5.6], and so (LABEL:howard-structure) follows from the proof of [How04, Theorem 3.4.2]., that

(2.k) |

for a -torsion module

Now (LABEL:howard-structure) implies that has odd -corank, and together with Theorem 2.1 this implies that is odd. ∎

## 3. Galois cohomology

### 3.1. Selmer structures

We recall the setup of [MR04, Chapter 2] for Selmer structures.

Let be a finite extension and its ring of integers. Let be a number field, and be an -module with a continuous -linear action of that is unramified except for finitely many primes.

###### Definition 3.1.

A Selmer structure for is a collection of -submodules indexed by the places of

(3.a) |

such that, for all but finitely many we have

(3.b) |

We consider the associated Selmer group

(3.c) |

We recall a well-known consequence of Poitou–Tate global duality:

###### Lemma 3.2 ([Ddt94, Theorem 2.19]).

Let have finite order, and be a Selmer structure for . Then

(3.d) |

If is also self-dual, we can rephrase this theorem in a way which will be more useful to us:

###### Corollary 3.3.

Let have finite order and be self-dual. Let be a Selmer structure for . Then

(3.e) |

###### Proof.

By local duality and by the self-duality of we have

(3.f) |

Let

(3.g) |

Then is a finite set, and satisfy

(3.h) |

So we only need to prove that

(3.i) |

The square of the left side of such expression is, by (LABEL:eq_loc-dual), simply

(3.j) |

Using the formulas for the local Euler characteristics in [Hid00, Theorem 4.45] and [Hid00, Theorem 4.52], we have

(3.k) |

and since and this becomes

(3.l) |

### 3.2. Local conditions

We consider the *strict* local condition and the *relaxed* local condition

Given a Selmer structure for and given products of places and that do not share any place, we denote by the Selmer structure that differs by by being strict at and relaxed at that is,

(3.m) |

For the module we also consider the *finite* local condition

(3.n) |

where is the local Kummer map. These form a Selmer structure We also define the finite condition for by propagation: it is the pre-image of in

(3.o) |

We note that is self-dual, and that is its own annihilator under Tate local duality. In particular, we have Moreover, such local conditions are the unramified condition for all but finitely many primes, and hence form a Selmer structure, which we denote by

Since is self-dual, from (LABEL:split) and (LABEL:no-local-tor) we have, when that

(3.p) |

and hence

(3.q) |

We also consider the *transverse* local condition for certain primes. A prime is caled *admissible* for if it is inert in and satisfy

(3.r) |

We denote by the set of square-free products of admissible primes. We also denote by the subset of elements coprime with

The following results are from [Zha14, Lemma 4.2]. Let We have a unique direct sum decomposition

(3.s) |

as -modules, which induces

(3.t) |

Moreover, we have the identification We define the *transverse* condition to be

(3.u) |

and these satisfy

(3.v) |

We can extend the notation in (LABEL:Selmer-notation) as follows: for a Selmer structure for we let be the Selmer structure defined by

(3.w) |

where do not share any places and

We also record the following well-known application of Chebotarev regarding admissible primes.

### 3.3. The parity lemma

For this section, let with satisfying Assumption 1.1.

We record the following consequence of the parity lemma of Gross–Parson [GP12, Lemma 9].

###### Lemma 3.5 ([Zha14, Lemma 5.3]).

Let be a prime. Then we have

(3.x) |

Moreover, we have either

(3.y) |

or

(3.z) |

###### Corollary 3.6.

If then there is a positive density of admissible primes such that

(3.aa) |

Now we prove an extension of the parity lemma, and deduce as a consequence a Chebotarev-type result that will be useful to deal with the rank one case.

###### Lemma 3.7.

Let be any place of and let be a prime. Assume that

(3.ab) |

that is, that both and are strict at

Then an analogous of the parity lemma holds for the relaxed Selmer groups: we have

(3.ac) |

Moreover, we have either

(3.ad) |

or

(3.ae) |

###### Proof.

Lemma 3.5 tell us that and are and in some order. By assumption, these are strict at that is, we have

(3.ag) |

Hence we have

(3.ah) |

and thus (LABEL:eq_relating-r) becomes

(3.ai) |

which is the first part of what we want to prove.

Then (LABEL:eq_same-for-ms) and (LABEL:eq_relating-r2), together with the inclusions

(3.al) |

imply the rest of the theorem. ∎

###### Corollary 3.8.

Let be any place of with and assume that Then there is a positive density of admissible primes such that

###### Proof.

We have by Corollary 3.3, so we know that there is a positive density of admissible primes such that by Lemma 3.4.

We will prove such primes suffice. Assume by contradiction that we had This would mean that

and since we have the last statement in Lemma 3.7 would imply that

which cannot be true since we chose such that ∎

## 4. Proof of the main result

### 4.1. Level raising

We denote by the subset of elements with an even number of prime factors.

The following is a summary of some of the properties of the constructions done in [Zha14].

###### Theorem 4.1.

Let be a pair satisfying Assumption 1.1. For any there is an eigenform and a prime above such that Furthermore, the pair satisfy Assumption 1.1.

Fix an identification for all If then the Selmer structure of is given by

If satisfy (LABEL:Heegner) and then also satisfy (LABEL:Heegner), and we denote by the class Then if with primes, there is a suitable isomorphism

(4.a) |