Kolyvagin’s Conjecture, bipartite Euler systems, and higher congruences of modular forms

In the indefinite case, the first results towards Kolyvagin’s conjecture were obtained by Zhang [53], under a number of additional assumptions: that $p\geq 5,$ that the Galois representation associated to $\overline{T}_{f}$ is surjective, and additional hypotheses on the residual ramification. In particular, under the hypotheses of [53], there exists a class $c(m)$ whose reduction in $H^{1}(K,\overline{T}_{f})$ is nonzero; this is not the case in general. In the definite case, the classes $\lambda(m)$ are a novel feature of this work and were not considered in [53].

The converse theorem we obtain (Corollary C) is new in several cases, most notably when the image of the Galois action on $\overline{T}_{f}$ is dihedral, or when $p=3.$ Previous results, under various additional hypotheses, were obtained by Zhang as a corollary of his work on Kolyvagin’s conjecture, and by Skinner [46] by a purely Iwasawa-theoretic method. For converse theorems in other settings, see Burungale [7] for the CM case, Castella-Grossi-Lee-Skinner [8] for the residually reducible case, Castella-Wan [9] for the supersingular case, and Skinner-Zhang [48] for the case of multiplicative reduction.

Now suppose again that $\nu(N^{-})$ is even. While the Kolyvagin classes are constructed by varying the conductor of CM points on $X_{N^{+},N^{-}}$ over squarefree integers, one may instead $p$-adically interpolate CM points of $p$-power conductor to obtain a class:

(5)

$$\bm{\kappa}_{\infty}\in H^{1}(K,T_{f}\otimes\Lambda(\Psi)),$$

where $\Lambda=\mathcal{O}\llbracket\operatorname{Gal}(K_{\infty}/K)\rrbracket$ is the anticyclotomic Iwasawa algebra, given $G_{K}$-action by the tautological character $\Psi$. (Note that the specialization of $\bm{\kappa}_{\infty}$ at the trivial character is a multiple of $c(1)$.) The methods used to prove Theorem A also yield the following result towards Perrin-Riou’s Heegner point main conjecture.

For precise definitions of these Selmer groups and of $\bm{\kappa}_{\infty}$, which is denoted $\bm{\kappa}(1)$ in the text, see §6.1.

Finally, we have the following result on the anticyclotomic main conjecture for $f$ when $\nu(N^{-})$ is odd. Evaluating the quaternionic modular form $\phi_{f}$ on CM points of $p$-power conductor on the Shimura set $X_{N^{+},N^{-}}$, one constructs the algebraic $p$-adic $L$-function

(6)

$$\bm{\lambda}_{\infty}\in\Lambda,$$

denoted $\bm{\lambda}(1)$ in the text. The square of $\bm{\lambda}_{\infty}$ has an interpolation property for twisted $L$-values of $f$.

The opposite inclusion of ideals may be deduced directly from Skinner-Urban’s proof of one divisibility in the three-variable main conjecture [47]; indeed, this is an essential ingredient in all of our results, as explained below.

The technical hypotheses in Zhang’s proof of Kolyvagin’s conjecture were carried over to Burungale, Castella, and Kim’s proof [6] of the lower bound on the Selmer group in the Heegner point main conjecture, where it is also assumed that $p$ is not anomalous. While the methods used in this paper build on those of [6], Castella and Wan [10] have given a completely independent proof of a three-variable main conjecture when $\nu(N^{-})$ is even. Their result also requires some hypotheses on residual ramification avoided here, and that $N$ be squarefree.

For upper bounds on the Selmer group in Theorem E and Theorem F, various technical assumptions on the residual representation and on the image of the Galois action were used in prior works by Howard [24, 25] and Chida-Hsieh [11].

To prove Theorems A and D, we extend Kolyvagin’s construction to a larger system of classes

(7)

$$c(m,Q_{1})\in H^{1}(K,T_{f}/\wp^{M}),\;\;\lambda(m,Q_{2})\in\mathcal{O}/\wp^{M},$$

where $M$ is a fixed integer, and $m,Q_{1},Q_{2}$ are squarefree product of auxiliary primes satisfying certain congruence conditions, such that $\nu(N^{-}Q_{1})$ is even and $\nu(N^{-}Q_{2})$ is odd. The classes (7) form a bipartite Euler system in the sense of Howard [25] for each fixed $m$ and a Kolyvagin system for each fixed $Q_{1}.$ If $\nu(N^{-})$ itself is even, then the classes $c(m,1)$ agree with Kolyvagin’s original construction. The Euler system relations are of the form:

(8)

$$\operatorname{loc}_{q}c(m,Q_{1})\sim\lambda(m,Q_{1}q)\sim\partial_{q^{\prime}}c(m,Q_{1}qq^{\prime}),$$

where $q,q^{\prime}$ are two additional auxiliary primes not dividing $Q_{1}$; and

(9)

$$\operatorname{loc}_{\ell}^{\pm}c(m,Q_{1})\sim\partial_{\ell}^{\mp}c(m\ell,Q_{1}),$$

where $\ell$ is an additional auxiliary prime not dividing $m$. (Here $\operatorname{loc}_{q}$, $\partial_{q^{\prime}}$, $\operatorname{loc}_{\ell}^{\pm}$, $\partial^{\pm}_{\ell}$ are certain localization maps landing in subspaces of the local cohomology free of rank one over $\mathcal{O}/\wp^{M}.$) The classes $c(m,Q_{1})$ were introduced by Zhang, although the $\lambda(m,Q_{2})$ are only implicit in [53].

If $c(m,Q_{1})\neq 0$, then one can use the Kolyvagin system relation to find an auxiliary $\ell$ — either prime or equal to $1$— such that $\partial_{q}c(m\ell,Q_{1})\neq 0$. By the bipartite Euler system relation, this implies $\lambda(m\ell,Q_{1}/q)\neq 0.$ On the other hand if $\lambda(m,Q_{2})\neq 0$ and $q|Q_{2},$ then $c(m,Q_{2}/q)\neq 0.$ Combining these two observations, we reduce the non-vanishing of some class $c(m,1)$ or $\lambda(m,1)$ — depending on the parity of $\nu(N^{-})$ — to exhibiting a single $Q_{2}$ such that $\lambda(1,Q_{2})\neq 0.$

Now, if there exists a newform $g$ of level $NQ_{2}$ with a congruence to $f$ modulo $\wp^{M}$, then $\lambda(1,Q_{2})$ is essentially the reduction of the algebraic $L$-value $L^{\text{alg }}(g/K,1)$ modulo $\wp^{M}$, which is related to the length of the Selmer group of $g$ by the Iwasawa main conjecture [47, 51]. To complete the proof, it therefore suffices to choose a suitable $Q_{2}$ and construct such a $g$ with a small Selmer group. We remark that our results can only be obtained by working modulo $\wp^{M}$ for a large $M$, since in general it will not be possible to choose $g$ such that $L^{\text{alg}}(g/K,1)$ is a $\wp$-adic unit; in [53], $M=1$ is fixed throughout, and the need to show that the $L$-value is a unit is responsible for most of the additional hypotheses.

To construct $g$, we use the deformation-theoretic techniques developed by Ramakrishna [41]. Standard level-raising methods work by producing a modulo $\wp$ eigenform of the desired level, and then using that all modulo $\wp$ eigenforms lift to characteristic zero, but this is not the case modulo $\wp^{M}$. Instead, we deform the representation $T_{f}/\wp^{M}$ to a $\wp$-adic Galois representation of a suitable auxiliary level, and then apply modularity lifting to ensure the resulting representation is modular. The auxiliary level $Q_{2}$ must be chosen to control two Selmer groups: the adjoint Selmer group governing the deformation problem, and the Selmer group $\operatorname{Sel}(K,A_{g}[\wp^{\infty}])$ that is related to the $L$-value.

We now make some remarks on the construction of the Euler system. The elements $c(m,Q_{1})$ (resp. $\lambda(m,Q_{2})$) are constructed from CM points of conductor $m$ on the Shimura curve $X_{N^{+},N^{-}Q_{1}}$ (resp. Shimura set $X_{N^{+},N^{-}Q_{2}}$). Similar Euler system constructions have been made by many authors, e.g. in [11, 3] as well as in [53], but all have relied on certain hypotheses ensuring an integral multiplicity one property for the space of algebraic modular forms on $X_{N^{+},N^{-}Q_{i}}$, which we do not impose here. Instead, we obtain a control on the failure of multiplicity one, using the work of Helm [22] on maps between Jacobians of modular curves and Shimura curves. The construction of the Euler system is intimately related to level-raising, and so our method can also be viewed as improving results on level-raising of $f$ to algebraic eigenforms modulo $\wp^{M}$ new at multiple auxiliary primes, which had previously been restricted to the multiplicity one case.

The proof of Theorem E is similar to that of Theorem A: the $p$-adically interpolated Heegner class $\bm{\kappa}_{\infty}$ is viewed as the bottom layer of an Euler system $\left\{\bm{\kappa}(Q_{1}),\bm{\lambda}(Q_{2})\right\}$. (The squarefree conductor $m$ no longer plays a role.) If $g$, as above, is a newform of level $NQ_{2}$ with a congruence to $f$, then $\bm{\lambda}(Q_{2})$ is congruent to Bertolini and Darmon’s anticyclotomic $p$-adic $L$-function of $g$ [3]. Using this and an Euler system argument, we reduce the lower bound on the Selmer group in the Heegner point main conjecture to the anticyclotomic main conjecture for $g$, which was proven in [47]. Finally, the upper bound on the Selmer group in the Heegner point main conjecture, as well as Theorem F, follow by standard arguments from the construction of the Euler system.

In the text, the arguments described above are phrased in the language of ultrapatching, which amounts to a formalism for letting $M$ tend to infinity; this also forces each prime factor of $m$, $Q_{1}$, $Q_{2}$ to tend to infinity in order to satisfy the congruence conditions. (The number of prime factors of $m$, $Q_{1},$ and $Q_{2}$ remain bounded.) This method was inspired by [45], where ultrapatching was applied to the Taylor-Wiles construction. Our setting is different in that we patch Galois cohomology groups and Selmer groups rather than geometric étale cohomology groups. The benefit of ultrapatching is that it allows us to consider the Euler system classes as characteristic zero objects in patched Selmer groups, significantly streamlining the Euler system arguments. For instance, with patching, we are able to make precise the heuristic that the non-vanishing of each Euler system class $c(m,Q_{1})$ or $\lambda(m,Q_{2})$ is equivalent to the $(m,Q_{i})$-transverse Selmer group being rank one or zero, respectively, cf. Lemma 8.3.4.

In §2, we review basic properties of ultrafilters and introduce patched cohomology and Selmer groups. In §3, we present a simplified version of the theory of bipartite Euler systems that appeared in [25], using patched cohomology. In §4, we recall the geometric inputs that will be used to construct bipartite Euler systems: the work of Helm on maps between modular curves and Shimura curves, and the behavior of Heegner points on Shimura curves under reduction and specialization. In §5, we prove the modulo $\wp^{M}$ level-raising result and present a general framework for constructing bipartite Euler systems out of CM points, which we then specialize in §6 for our applications. In §7, we give the deformation-theoretic input to construct the newform $g$ (in fact a sequence $g_{n}$ satisfying increasingly deep congruence conditions). Finally, we prove the main results in §8. An additional calculation in cyclotomic Iwasawa theory is required for Kolyvagin’s conjecture when $p$ is non-ordinary or inert in $K$; this is done in the appendix.

I am grateful to Mark Kisin for suggesting this problem and for his ongoing encouragement. Special thanks are additionally due to Francesc Castella, who first alerted me to the relation between Kolyvagin’s conjecture and the Heegner point main conjecture. It is also a pleasure to thank many other people with whom I had helpful conversations and correspondence over the course of this project: Robert Pollack, Xin Wan, Christopher Skinner, Alexander Petrov, Sam Marks, Aaron Landesman, and Alexander Smith. This work was supported by NSF Grant #DGE1745303.

The facts recalled in this subsection are discussed in more detail in [32].