Euler Systems and Selmer Bounds for GU(2,1)

ABSTRACT

We investigate properties of the Euler system associated with certain automorphic representations of the unitary similitude group GU(2,1) with respect to an imaginary quadratic field E, constructed by Loeffler–Skinner–Zerbes. By adapting Mazur and Rubin’s Euler system machinery we prove one divisibility of the ‘rank 1’ Iwasawa main conjecture under some mild hypotheses. When p is split in E we also prove a ‘rank 0’ statement of the main conjecture, bounding a particular Selmer group in terms of a p-adic distribution conjecturally interpolating complex L-values. We then prove descended versions of these results, at the integral level, where we bound certain Bloch–Kato Selmer groups. We will also discuss the case where p is inert, which is a work in progress.

1. INTRODUCTION

In the past decade there have been a series of developments in the theory of Euler systems, in particular, the construction of new Euler systems for Shimura varieties attached to linear algebraic groups such as |$\mathrm{GL}_2 \times_{\mathrm{GL}_1} \mathrm{GL}_2$|⁠, |$\mathrm{GSp}_4$|⁠, more recently the construction for GU(2,1) of [9] which has initiated the work of this paper. In the late 20th and early 21st century, a general machinery has been developed by Kolyvagin, Mazur, Rubin, Kato and others to use Euler systems to bound Selmer groups in terms of special values of p-adic L-functions. One main aim is to solve cases of the Bloch–Kato conjecture. With an increasing array of examples, new techniques are being developed to extend existing results.

The aim of this article is to look at a certain part of the machinery, the Iwasawa main conjectures, and formulate results for a wide class of ordinary automorphic GU(2,1) representations. One divisibility of a main conjecture is obtained below for p split in the imaginary quadratic reflex field, subject to an explicit reciprocity law. This work is a precursor to the author’s locally analytic Iwasawa theory for GU(2,1) at inert primes, and together these constitute new progress towards the Bloch–Kato conjecture for GU(2,1).

In Section 2 we discuss automorphic representations for GU(2,1) and the existence of associated Galois representations coming from the Langlands program. We also discuss ramification, Hodge–Tate weights and ordinarity conditions which will be important for later sections.

We build on this in Section 3 by looking at certain subrepresentations of our Galois representation V after twisting by Hecke characters. We compute all such subrepresentations and present the information in a diagram, which conjecturally acts as a roadmap telling us where we can find Euler systems of different ranks associated with V. Moreover, we present the existing construction of a rank 1 Euler system for V from [9], which matches the conjectural picture. We make a clear distinction when we assume that p is split in E.

In Section 4 we adapt versions of the Euler system machinery of Mazur–Rubin [14] and Kato [7] in order to prove one divisibility of the rank 1 Iwasawa main conjecture for such V (that is, a statement in terms of cohomology of a Selmer complex of Euler characteristic 1 but without p-adic L-functions) and reduce the proof to a lemma about vanishing of some local H⁰ groups. We will also use the language of Selmer complexes and derived base change to establish a version of the main conjecture over the 2-variable Iwasawa algebra Λ and also at an integral level after specializing by a Hecke character of E.

In Section 5 we will deal only with the case where p splits in E and use a 2-variable Coleman map to identify the Euler system with a distribution in the Iwasawa algebra which we call a ‘motivic p-adic L-function’. This is a p-adic distribution lying in Λ which conjecturally interpolates complex L-values. Using the results from Section 4, we will bound certain Selmer groups coming from a complex of Euler characteristic 0 using this distribution and use this result to gain information about the Bloch–Kato Selmer group. In order to tie the distribution to an analytic p-adic L-function and prove the full statements of the Iwasawa main conjecture we would need an explicit reciprocity law, and the progress of other authors towards this is briefly discussed at the end of the section. In the case where p is inert new methods are needed, in Section 6 we outline a strategy which will appear in forthcoming work of the author.

2. SETUP

This section follows notation of [9], which is the precursor of this body of work. Suppose E is an imaginary quadratic field of discriminant −D, with non-trivial automorphism |$c: x \mapsto \bar{x}$|⁠. Identify |$E \otimes \mathbb{R} \cong \mathbb{C}$| such that |$\delta= \sqrt{-D}$| has positive imaginary part. Let |$J \in \mathrm{GL}_3$| be the hermitian matrix

Let G be the group scheme over |$\mathbb{Z}$| such that for a |$\mathbb{Z}$|-algebra R,

Then |$G(\mathbb{R})$| is the unitary similitude group GU(2,1), which is reductive over |$\mathbb{Z}_l$| for all |$l \nmid D$|⁠. We are interested in cuspidal automorphic representations of G; ignoring the similitude character momentarily, we can relate (L-packets of) these representations to (certain L-packets of) cuspidal automorphic representations of |$\mathrm{Res}^{E}_{\mathbb{Q}}(\mathrm{GL}_3)$| via base change, following the notation of [9].

 

Here we have used |$\bullet^c$| to denote |$\text{Gal}(E/\mathbb{Q})$|-conjugation and |$\bullet^\vee$| to denote contragredient dual. The following theorem justifies why we take this definition over |$\mathrm{GL}_3$| rather than working directly over GU(2,1), by proving an instance of Langland’s functoriality for the inclusion of linear algebraic groups GU(2,1) |$\hookrightarrow \mathrm{GL}_3 \times \mathrm{GL}_1$|⁠. The result is stated as in op. cit., but it is derived from work of [13] amongst others.

([9], Theorem 2.6)

  

Given such an automorphic representation, it is natural to ask whether we can associate with it a Galois representation in line with the Langlands program. The next theorem shows that the answer is yes, and after this we can exposit the constructions of associated Euler systems.

Let |$(\Pi, \omega)$| be a RAECSDC representation of |$\mathrm{GL}_3/E$|⁠, and let w be a prime for which |$\Pi_w$| is unramified. Let |$q=\mathrm{Nm}(w)$|⁠, and define |$P_w(\Pi,X)$| to be the Hecke polynomial, that is, satisfying

([2], Theorem 1.2)

 

where |$\mathrm{Frob}_w$| is the arithmetic Frobenius.

The rest of Section 2 will establish a few properties of this representation which we will need later.

([20], Theorem 2)

 

This justifies an assumption we use from now on that |$V_{\mathfrak{P}}(\Pi)$| is irreducible.

Recall Π is regular algebraic, so we can define its weight at each embedding |$\tau \hookrightarrow F_{\Pi}$| which is a triple of integers |$a_{\tau,1} \geq a_{\tau,2} \geq a_{\tau,3}$|⁠. Twisting by a Hecke character if necessary, we can assume that Π has weight |$(a+b,b,0)$| at the identity embedding and |$(a+b,a,0)$| at the conjugate embedding for some |$a,b \geq 0$|⁠. When discussing Galois representations, for convenience we will use the convention of Hodge numbers (negative of Hodge–Tate weights), so the cyclotomic character will have Hodge number −1.  

From here on we will assume that |$\Pi_v$| is unramified for each place |$v \mid p$|⁠, so the Hecke polynomials are defined. We want to make further assumptions about the local behaviour of Π, which we will do in the next few sections.

 

Following arguments in [2, section 2], using p-adic Hodge theory one can show that Π is ordinary at v if and only if the dual representation |$V_{\mathfrak{P}}(\Pi)^*$| has a codimension 1 Galois-invariant subspace |$\mathcal{F}^1_v V_{\mathfrak{P}}(\Pi)^*$| at v such that |$V_{\mathfrak{P}}(\Pi)^*/\mathcal{F}^1_v$| is unramified and |$\mathrm{Frob}_v$| acts on the quotient by α_v. Since Π is conjugate self-dual (up to a twist), we can see that such a subspace exists if and only if |$V_{\mathfrak{P}}(\Pi)$| has a codimension 2 invariant subspace at v with compatible action of |$\mathrm{Frob}_{\bar{v}}$|⁠. If Π is ordinary at all primes above p then |$V_{\mathfrak{P}}(\Pi)$| and its dual preserve a full flag of invariant subspaces at each prime above p, and we will use the notation |$\mathcal{F}^i_v$| for both:

We will henceforth assume ordinarity at all primes above p as we will need the full flag. Relaxed ordinarity conditions may be considered in future work.

 

The crystalline characters are graded pieces of |$V|_{\mathrm{Gal}(\overline{E_v}/E_v)}$| with Hodge–Tate weights in decreasing order as i increases, and this will help us understand how the crystalline Frobenius map ϕ acts on the Dieudonné module |$\mathbb{D}_{\mathrm{cris}}(V)$|⁠. Later we will use known classifications of local crystalline characters due to Conrad to obtain information about cohomology groups of V. This will differ in the cases of p split and inert.

3. LOCAL CONDITIONS

Let V be a p-adic Galois representation over a number field K which is de Rham at the primes above p.

 

• |$V^+_v$| is stable under |$G_{K_v}$|⁠,

• |$V^+_v$| has all Hodge–Tate weights > 0,

• |$V/V^+_v$| has all Hodge–Tate weights |$\leq 0$|⁠.

Such |$V_v^+$| is unique if it exists and gives us a local condition on V which relates to the Bloch–Kato Selmer group we will see in later sections.

Define the rank of V as |$ r(V)= \max(0,r_0(V))$| combined with the following; fix considering infinite places v of K, let σ_v denote complex conjugation in |$\overline{K_v}$| for a real place v and let L be the field of definition of V. Then

We will call V r-critical if |$r(V)=r$| and |$r(V^*(1))=0$|⁠. If moreover there exist Panchishkin subrepresentations |$V^+_v$| for all |$v \mid p$|⁠, say V satisfies the rank r Panchishkin condition. These subrepresentations are important because of the following prediction, originally from Perrin–Riou but is stated clearly from newer work.

([11], Conjecture 6.5)

 

satisfying the Euler system compatibility relation, where F varies over finite abelian extensions of K. Moreover, c_K is non-zero if and only if the associated complex L-function of V satisfies |$L^{(r)}(V^*(1),0) \neq 0$|⁠.

This tells us that we expect to find a (rank 1) Euler system attached to any such V which is 1-critical and a rank 0 Euler system (which is a p-adic L-function) when V is 0-critical. For |$r \geq 2$|⁠, little is known about the existence of rank r Euler systems so we will only consider |$r=0,1$|⁠. This p-adic L-function will conjecturally interpolate values of the complex L-function of V, although this is beyond the scope of this paper and will follow in future work.

Back to our specific case; first let us assume that p splits and fix a choice of prime |$\wp \mid p$|⁠. Let η be an algebraic Hecke character of conductor dividing |$\mathfrak{m}p^\infty$| for some ideal |$\mathfrak{m} \subset \mathcal{O}_E$| (which we can often take to be the unit ideal) and infinity type (s, r), and let |$V = V_{\mathfrak{P}}(\Pi)^*$|⁠. Then the Hodge numbers of |$V(\eta^{-1})$| are

We can see that |$V(\eta^{-1})$| can satisfy different rank Panchishkin conditions depending on the choice of |$\infty$|-type (s, r) of η, and we summarize this below—plotting the regions where different rank r Panchishkin conditions are met in Figure 1, tabulating the ranks and subrepresentations in Figure 2. Any unlabelled regions are not t-critical for any t. Note that we are still assuming ordinarity at all primes above p and therefore have the full flag of subrepresentations of V locally at any choice of |$v \mid p$|⁠. Later we will also denote |$\mathcal{F}^i_vT = \mathcal{F}^i_vV \cap T$| and will use shorthand of |$\mathcal{F}^i$| in the tabulation of our data below.

The important boxes we focus on are the two conjugate rank 0 boxes |$\Sigma^{(0)}=\{a+1 \leq r \leq a+b+1, 0 \leq s \leq b \}$| and |$\Sigma^{(0^{\prime})}=\{0 \leq r \leq a, b+1 \leq s \leq a+b+1 \}$| and the rank 1 box |$\Sigma^{(1)}=\{0 \leq r \leq a, 0 \leq s \leq b \}$|⁠.

When p is inert in E, we have the same Hodge–Tate weights at embeddings of E_p into |$\overline{F_{\Pi,\mathfrak{P}}}$| (rather than at the primes dividing p). Instead of two flags we obtain one flag with a self-duality relation. We will have the same conditions on r and s for each region, but despite p-ordinarity we no longer have Panchishkin subrepresentations for every region. The rank 1 Panchishkin subrepresentation for |$\Sigma^{(2)}$| still exists and is defined in the same way (which is crucial in the next subsection), but the rank 0 ones no longer exist—when p is inert the conjugate self-duality condition is only satisfied at boxes on the ‘diagonal’ line swapping the pairs (r, a) and (s, b). As a result the case of p inert requires new ideas and will be considered separately in upcoming work.

3.1. Rank r Euler systems

From the data and the preceding conjecture, we echo a prediction of Panchishkin, Perrin–Riou and others that the existence of a rank r Panchishkin subrepresentation attached to a box in the diagram above should be equipped with a rank r Euler system which relates to L-values of twists with |$\infty$|-type in the box. In particular this predicts a rank 1 Euler system to exist for |$V(\eta^{-1})$| when the infinity type lies in the region |$\Sigma^{(2)}$|—this is exactly the construction [9, section 12] (presented in Theorem 3.3). In the split prime case we expect two conjugate rank 1 Euler systems to exist in boxes |$\Sigma^{(3)}$| and |$\Sigma^{(3^{\prime})}$|⁠, but these are yet constructed. We can also predict two conjugate pairs of p-adic L-functions to exist when |$V(\eta^{-1})$| lies in the 0-critical regions. For every pair of adjacent boxes, these predictions suggest a bridge connecting the corresponding rank r + 1 and rank r Euler systems, which we call a rank lowering operator. As we only have knowledge about one Euler system, we can only look for p-adic L-functions interpolating twisted L-values related to |$\Sigma^{(0)}$| (or a conjugate construction). When p is inert, since we do not have a Panchishkin subrepresentation attached to rank 0 boxes the resulting p-adic L-function is more mysterious.

For now we will consider both cases where p is split or inert. Fix |$c \in \mathbb{Z}_{\geq1}$| coprime to 6pS for a finite set S of primes (containing the primes at which E and Π are ramified), and let |$\mathrm{Spl}_{E/\mathbb{Q}}$| be the set of rational primes splitting in E. Define

([9], Theorem 12.3)

 

for all |$\mathfrak{m} \in \mathcal{R}$| such that

• For |$\mathfrak{m} \mid \mathfrak{n}$|⁠, we have

  $$\mathrm{norm}^{\mathfrak{n}}_{\mathfrak{m}}(\textbf{c}_{\mathfrak{n}}^\Pi) = \left( \prod_{w \mid \frac{\mathfrak{n}}{\mathfrak{m}}} P_w(\Pi,\mathrm{Frob}_w^{-1}) \right)\textbf{c}_{\mathfrak{m}}^\Pi.$$

• For an algebraic Hecke character η of conductor dividing |$\mathfrak{m}p^\infty$| and infinity type (s, r), with |$0 \leq r \leq a$| and |$0 \leq s \leq b$|⁠, the image of |$\textbf{c}^\Pi_{\mathfrak{n}}$| in |$H^1_{\mathrm{Iw}}(E[\mathfrak{m}p^\infty], V(\eta^{-1}))$| is the étale realization of a motivic cohomology class.

• For all |$v \mid p$|⁠, the projection of |$\mathrm{loc}_v(\textbf{c}_{\mathfrak{m}}^\Pi)$| to the group |$H^1_{\mathrm{Iw}}(E_v \otimes_E E[\mathfrak{m}p^\infty], T/\mathcal{F}^1_vT)$| is zero.

 

We will use this result without further reference.

 

where |$\Gamma=\mathrm{Gal}(E[p^\infty]/E)$|⁠, |$\Lambda=\mathbb{Z}_p[[\Gamma]]$| is the Iwasawa algebra and |$\mathbf{j}: \Gamma \rightarrow \Lambda^*$| is the canonical character associated with Λ. We call |$\mathbb{T}=T \otimes \Lambda(-\mathbf{j})$| the universal twist of T. We can lift our Panchishkin subrepresentations to the universal twist, using the notation |$\mathcal{F}_v^i\mathbb{T}$| for the image of |$\mathcal{F}_v^iT$| in |$\mathbb{T}$|⁠.

|$\mathbb{T}$| considers all twists of T by Hecke characters η of p-power conductor by evaluating |$\mathbf{j}$| at the corresponding Galois character |$\tilde{\eta}$| arising from η due to global class field theory. This gives a correspondence between such Hecke characters and characters of Λ; for an idele x, we define this by |$\eta([x])=\tilde{\eta} (\mathbf{j}$|(⁠|$\phi_E([x]))$|⁠, where ϕ_E is the global Artin map defined by sending the image of a prime v in the ideles to the arithmetic Frobenius at v. Every such |$\tilde{\eta}$| gives rise to such a Hecke character η by the universal property of |$\mathbf{j}$| and by surjectivity of the Artin map. We will often switch between thinking of η as a Hecke character and a Λ-character.

4. BOUNDING SELMER GROUPS

Our main objective in constructing this Euler system is to use it to find a relationship between p-adic L-functions and certain Selmer groups. In particular we want to compute the dimension of |$H^1_f(E,V)$|⁠, the Bloch–Kato Selmer group defined by the unramified local condition at primes |$v \nmid p$| and the crystalline local condition when |$v \mid p$|⁠. We want to bound this by the critical nonzero values of a complex L-function as predicted by the Bloch–Kato conjecture, but in order to do so we must first develop some more general machinery to deal with Selmer groups.

The first intermediate goal is to prove a Kato style main conjecture for our |$\mathrm{GU}(2,1)$|-representation V. We do so by using results from [3], which generalize the standard Euler system machinery from [14, section 5] to work over any base number field K and to allow more general Panchishkin local conditions. In order to apply these results, we need to place some conditions on V in addition to ordinarity at all primes above p.

4.1. Galois torsion elements

Let |$\Lambda = \mathbb{Z}_p[[\Gamma]]$|⁠, where |$\Gamma=\mathrm{Gal}(E[p^\infty]/E)$|⁠, where |$E[p^\infty]$| is the maximal abelian extension of E unramified outside p. Then |$\mathrm{Gal}(E[p^\infty]/E) \cong \mathbb{Z}_p^2 \times \Delta$|⁠, where Δ is a finite abelian torsion group. We assume E is such that Δ has order coprime to p. It is sufficient to assume |$p \geq 5$| and |$p \nmid h_E = |\text{Cl}(E)|$| the ideal class number. We demonstrate by class field theory,

and we have the exact sequence

Since p does not divide h_E, any p-torsion element in Δ will come from the image of |$\prod_{v \mid p} \mathcal{O}_v^\times$|⁠. Then p times this element will lie in |$\mathcal{O}_E^\times$| by exactness, which is finite as E is imaginary quadratic. Since |$p \geq 5$|⁠, E_v does not contain pth roots of unity (p ≠ 2 is sufficient when p splits, and we also take p ≠ 3 for inert p), so this is a contradiction. This gives us an explicit size |$|\Delta|=\frac{(p-1)^2|\mathrm{Cl}(E)|}{|\mathcal{O}_E^\times|}$| when p splits and |$\frac{(p^2-1)|\mathrm{Cl}(E)|}{|\mathcal{O}_E^\times|}$| when p is inert. Thus as p gets large, the density of ‘good’ characters converges to 1.

When |$p \nmid |\Delta|$|⁠, we have a decomposition of the Iwasawa algebra into local rings

where δ ranges over the characters of Δ into |$\mathbb{C}^{\times}$|⁠. This also gives us a decomposition of Λ-modules into a direct sum of |$\Lambda_\delta$|-modules. For |$\mathbb{T}$| we will denote this by |$\mathbb{T}=\bigoplus_{\delta} \mathbb{T}_{\delta}$|⁠. For the maximal ideals |$\mathfrak{m}_{\delta}$| of Λ, and for each δ we set |$\overline{\mathbb{T}}_{\delta}=\mathbb{T}/\mathfrak{m}_{\delta}$|⁠. We will denote the projection maps onto each component by e_δ.

We can ask what the right choice of δ is—in this work we will vary our Galois representation V by twisting by a Hecke character η of E, and conditions we need to make will be placed on η rather than on the starting representation. Given such η with conductor prime to p, we can think of it as a character on |$\text{Gal}(E[p^\infty]/E)$| by global class field theory. Thus it reduces to the canonical choice |$\overline{\eta}:=\eta\mid_{\Delta}$|⁠. We use this from now on; fixing a character η satisfying the necessary conditions (to be established in the next few sections) we can fix a choice of |$\mathbb{Z}_p^2$|-extension |$\Gamma_\delta$| determined by |$\delta=\overline{\eta}$|⁠. When we have not yet specialized by such a character, we will use general δ and apply the idempotents e_δ.

4.2. Some assumptions

For a prime |$v \mid p$| such that |$\Pi_v$| is unramified, |$V_{\mathfrak{P}}(\Pi) \mid_{\mathrm{Gal}(\overline{E_v}/E_v)}$| is crystalline and the eigenvalues of the power of crystalline Frobenius |$\phi^{[E_v:\mathbb{Q}_v]}$| on |$\mathbb{D}_{\mathrm{cris}}(V_{\mathfrak{P}}(\Pi) \mid_{\mathrm{Gal}(\overline{E_v}/E_v)})$| are the reciprocal roots of |$P_v(\Pi,qX)$| by [2, Theorem 1.2]. Using this, we can get conditions for the vanishing of certain local cohomology groups when p is split (p = q) and inert (⁠|$q=p^2$|⁠).

Consider η as a character of |$\mathrm{Gal}(E[p^\infty]/E)$| by Remark 2.5, and denote its restrictions at each prime |$v \mid p$| as |$\eta_v:G_{\mathbb{Q}_p}^{\text{ab}} \rightarrow \mathcal{O}_L^\times$| where we assume L contains |$F_{\mathfrak{P}}$|⁠. By local class field theory,

is the decomposition into the unramified part and inertia, such that the p-adic cyclotomic character is trivial on |$\hat{\mathbb{Z}}$|⁠.

By considering prime-to-p torsion elements in the exact sequence before the lemma, the mod p character on Δ can be pulled back to a character |$\overline{\eta}_v$| of |$\mathbb{F}_p^\times$| for each |$v \mid p$|⁠. When p is split and |$\mathrm{Hom}(\mathbb{F}_p^\times, \bar{\mathbb{Q}}_p^\times)$| is canonically isomorphic to |$\mathbb{Z}/(p-1)\mathbb{Z}$| and generated by a fixed Teichmüller character τ, we can write |$\overline{\eta_v}=\tau^{s_v}$| for |$s_v \in \mathbb{Z}/(p-1)\mathbb{Z}$|⁠. When p is inert we similarly can define |$s_p \in \mathbb{Z}/(p^2-1)\mathbb{Z}$| depending on a choice of identity embedding and a choice of Teichmüller character. We can use these invariants to computationally determine whether the cohomology of our mod p representation twisted by the reduction of η vanishes; they can be thought of as a ‘mod p analogue of Hodge–Tate weights’ of η although this is technically inaccurate.  

  

Henceforth we will assume that η is such that the hypothesis of Lemma 4.1 holds, and so the corresponding degree 0 and 2 cohomology groups vanish. In fact it follows that |$H^j(E_v, \overline{\mathbb{T}}_{\delta})=H^j(E_v, \mathbb{T}) = 0$| for each δ, each |$v \mid p$| and |$j=0,2$|⁠. We further need to assume some hypotheses known collectively as ‘big image’ criteria. These assume that for our choice of η:

• either |$p \geq 5$| or |$\mathrm{Hom}_{\mathbb{F}_p[[G_E]]}(\overline{\mathbb{T}}_{\bar{\eta}},\overline{\mathbb{T}}_{\bar{\eta}}^*(1)) = 0$|⁠;

• |$\overline{\mathbb{T}}_{\bar{\eta}}$| is absolutely irreducible as a G_E-module;

• there is an element |$\tau \in G_E$| such that |$\mathbb{T}/(\tau-1)\mathbb{T}$| is free of rank 1 and τ acts trivially on p-power roots of unity.

These are expected to hold for all but finitely many p for a fixed non-endoscopic Π (similar results have already been proven for |$\mathrm{GL_2}$| and |$\mathrm{GSp}_4$|⁠, for example, see [5]), so these are fairly safe assumptions.

We conclude this subsection by summarizing e. We give a version for the universal twist |$V \otimes \Lambda(-\mathbf{j}) $| and a second ‘descended’ version for V twisted by a Galois character η. We let δ be a character of |$\Delta=\Gamma_{\text{tors}}$|⁠. its restriction onto Δ.  

A comment on the last assumption is that this is a nontrivial claim that should follow for certain twists of V from the construction of the Loeffler–Skinner–Zerbes Euler System and indeed has been shown for their analogous construction for |$\mathrm{GSp}_4$|-representations. From Theorem 3.3 (ii) we can see that the Euler system comes from a motivic cohomology class after applying twists in region |$\Sigma^{(2)}$| of Figure 1; this motivic cohomology class is highly expected to be non-zero (in which case the whole Euler system is not identically zero), but it has not yet been proven to be due to the difficult nature of these sorts of problems.  

4.3. Selmer complexes

Let M be a finitely generated |$\mathbb{Z}_p$|-module with a continuous action of |$G_{E,S}$|⁠.  

Following work of [15], summarized in [8, section 11.2] we can define the associated Selmer complex |$\widetilde{\mathrm{R}\Gamma}(E,M;\Delta)$|⁠. We can compute the cohomology of this Selmer complex, which we will denote by |$\widetilde{H}^i(E,M;\Delta)$|⁠. We will mainly use this for |$i=1,2$| when M is a twist of T, the |$\mathbb{Z}_p$|-lattice inside |$V=V_{\mathfrak{P}}(\Pi)^*$| or a subquotient of T.

A Selmer structure is called simple if for all v the map

is an isomorphism for i = 0, injection for i = 1 and zero for i = 2. In this case, the local condition is determined by the subspace |$i(H^1(U_v^+)) \subseteq H^1(E_v,M)$| just as a usual Selmer group. This language of Selmer complexes still has advantages even in this simple case.

We define three Selmer structures which we will use, which are all simple; first define the Bloch–Kato structure Δ^BK by the saturation of the unramified condition at each |$v \nmid p$| and by the crystalline local condition for |$v \mid p$|⁠. We also define Δ¹ by the unramified local condition for |$v \nmid p$| and above p by the inclusions

where |$T^+$| is a |$\mathbb{Z}_p$|-lattice in the Panchishkin subrepresentation of V in the region |$\Sigma^{(2)}$|⁠. We similarly define Δ⁰ for the region |$\Sigma^{(0)}$|⁠. Note that both of these are simple local conditions by the vanishing of local H⁰ groups in Assumption (⁠|$\mathbf{\ast \ast})$|⁠; as the result shows every |$G_{E_v}$|-subquotient of |$\overline{\mathbb{T}}$| has no non-trivial |$G_{E_v}$| invariant vectors and the required results for global cohomology groups follow. Using [8, Prop 11.2], we see that for each of these simple structures,

where |$H^i(E,-)$| is cohomology of |$G_{E,S}$| modules and |$? \in \{0,1,\mathrm{BK} \}$|⁠, and we use |$H^1_\Delta$| to denote the usual Selmer group defined by inclusion of classes satisfying the simple condition Δ into |$H^1(E,M)$|⁠. Here we have used |$\bullet^\vee$| to denote Pontryagin duality. By simplicity and local Tate duality, when M is a p-adic Galois representation, |$\widetilde{H}^i_{\Delta_{\mathrm{BK}}}(E,M)$| computes the global Bloch–Kato Selmer group |$H^1_f(E,M)$| for i = 1 and |$H^1_f(E,M^*(1))$| for i = 2.

4.4. Kato’s main conjecture for GU(2,1)

We can now formulate one divisibility in the rank 1 Iwasawa main conjecture, which we will prove below under the assumptions we have listed throughout the paper, in particular Lemma 3.1. In the case of inert p we still assume that η is such that the relevant H⁰ and H² groups vanish, but we no longer have a tidy condition on η.

Given a finitely generated torsion module M over a Noetherian Krull domain R, we can define its characteristic ideal as follows: the structure theorem of Iwasawa theory (for example, see [1, section 2]) tells us that there are pseudo-null R-modules P, Q (that is, the localizations of P and Q at all height 1 primes of R are zero) such that the following sequence is exact for some |$n \in \mathbb{N}$| and some height 1 primes |$\mathfrak{p}_i$| of R and integers e_i,

Then we define the characteristic ideal of M (over R) by

and we will say |$\mathrm{char}_R(M)=0$| when M is non-torsion. Using our Euler system and an adaptation of classical Euler system machinery, we develop a divisibility of characteristic ideals over the Iwasawa algebra Λ, which is part of the Iwasawa main conjecture for GU(2,1). We will denote the projection maps onto each component by e_δ. Since |$p \nmid |\Delta|$|⁠, this commutes with taking cohomology.

 

• |$e_{\delta} \widetilde{H}^2(E,\mathbb{T};\Delta^1)$| is |$\Lambda_{\delta}$|-torsion,

• |$e_{\delta}\widetilde{H}^1(E,\mathbb{T};\Delta^1)$| is torsion-free of generic |$\Lambda_{\delta}$|-rank 1,

• |$\mathrm{char}_{\Lambda_{\delta}}\left( e_{\delta} \widetilde{H}^2(E,\mathbb{T};\Delta^1) \right) \mid \mathrm{char}_{\Lambda_{\delta}}\left( \dfrac{e_{\delta} \widetilde{H}^1(E,\mathbb{T};\Delta^1)}{e_{\delta} c^\Pi} \right). $|

 

4.5. Descending to integral level

We will develop a useful base change property of Selmer complexes, then by following the structure of arguments in [8, section 11] we can apply a descent method to Theorem 4.3 to get a bound on the size of the Selmer group. First we need a technical lemma; for the following we will take |$\mathcal{O}=\mathcal{O}_{F_\Pi}$|⁠, so that everything we work with has the structure of an |$\mathcal{O}$|-module.  

           

5. MOTIVIC p-ADIC L-FUNCTIONS

Our next immediate goal is to start tying our results to some sort of p-adic L-function. To do so, we will have to assume p splits in E for this section. The standard tool for connecting an Euler system to a p-adic L-function is Perrin-Riou’s big logarithm map or in this case a 2-variable version: one variable for the cyclotomic extenstion of |$E_{\wp} = \mathbb{Q}_p$| and the other for its conjugate. This construction only works when we locally have a |$\mathbb{Z}_p$|-extension, and hence we are forced to assume p splits. This will map the first class |$c^\Pi_1$| of our Euler system to a bounded p-adic distribution in Λ, which we call a ‘motivic p-adic L-function’ and is our guess for what a p-adic L-function should be but is as yet not connected to complex L values of V. The right map has been constructed in [10, section 4], and by choosing a basis carefully we use it to define a Coleman map when p is split in E. The following statement comes from Theorem 4.7 op. cit., taking image inside bounded distributions by part (2)

where |$\mathbb{T}^i$| and Vⁱ are the images of the rank i Panchishkin local condition in |$\mathbb{T}$| and V for |$i=0,1$|⁠, respectively. Note that our choice of |$\wp \mid p$| at which we restrict is fixed as before, the unique prime of E where local rank 0 and rank 1 conditions differ, and this also determines the 1-dimensional |$G_{E_\wp}$|-subquotient |$T^1/T^0$| of T on which we apply the regulator map. The bottom Euler system class lies in this domain, so we can apply the Coleman map to it. We want to fix a |$\mathbb{Q}_p$|-basis |$\{\xi \}$| of the crystalline character |$\mathbb{D}_{\mathrm{cris}}(V^1/V^0)$| in order to renormalize the Coleman map; this normalized version is denoted |$\mathrm{Col}^{\Pi,\xi}$| and has image contained in Λ. The choice of ξ is not important now, but in later work the right ξ may need to be chosen to compare our map to existing p-adic analytic constructions, and hence we include it in notation.

Define |$L_p^*(\Pi, 1 + \mathbf{j})$| := |$ \mathrm{Col}^\Pi(c^\Pi)$| to be the ‘motivic p-adic L-function’. We denote the ξ-normalised version by:

where |$\text{loc}_\wp$| is localization to local Iwasawa cohomology at the choice of |$\wp \mid p$|⁠, for some non-zero scalar Ω which will depend on our choice of ξ. With some exact sequences, we can restate the main conjecture to include the ideal generated by this L-function.

5.1. Main conjecture with p-adic distribution

 

• |$e_{\delta} \widetilde{H}^1(E,\mathbb{T};\Delta^0) = 0 $|⁠,

• |$e_{\delta} \widetilde{H}^2(E,\mathbb{T};\Delta^0)$| is torsion,

• |$ \mathrm{char}_{\Lambda_{\delta} } \left(e_{\delta} \widetilde{H}^2(E,\mathbb{T};\Delta^0)\right) \huge\vert \frac{e_{\delta} L^*_p(\Pi, 1+ \mathbf{j})}{\Omega}$|⁠.

 

First note that |$e_{\delta}\widetilde{H}^1(E, \mathbb{T}; \Delta^0)$| injects into |$e_{\delta}\widetilde{H}^1(E, \mathbb{T}; \Delta^1)$| which is torsion free of rank 1 by Theorem 4.3. However, the latter has an element |$e_{\delta}c^\Pi$| whose image in |$e_{\delta}H^1(E_\wp, \mathbb{T}^1/\mathbb{T}^0)$| is non-torsion. This gives us part (i).

Using part (i), from the exact triangle we get the following exact sequence,

The last map is actually surjective due to the vanishing assumptions of local H² groups (interpreted here as a non-exceptional zero condition). This gives us part (ii) as the first two modules are torsion. Applying the divisibility of characteristic ideals from the rank 1 main conjecture, we get

We also have an exact sequence induced by the Coleman map:

By [10, Proposition 4.10], |$\mathrm{Col}^{\Pi,\xi}$| has free image of rank 1 over |$\Lambda_{\delta}$|⁠, and therefore |$\mathrm{coker}(e_{\delta} \circ \mathrm{Col}^{\Pi,\xi})$| is a torsion Λ-module and has non-zero characteristic ideal. We can therefore divide out by it to obtain

Thus we have the required divisibility.

 

This has established a strong link between the Selmer groups we are considering and the L-function as a distribution, but we would still like to go further and gather information about special values of the L-function after specializing |$\mathbb{T}$| by a character η with all the conditions we have placed above. We will define |$L^*_p(\Pi, 1+ \eta) := \eta(L^*_p(\Pi, 1+\textbf{j})) \in \mathcal{O}$| as the L-value at η. Similar to Section 4.5, we want to prove a descended version of Theorem 4.1 at a finite level.  

  

5.2. Bounding a Bloch–Kato Selmer group

Now we have bounded a Selmer group with Δ⁰ local conditions in terms of the p-adic L-function, and we want to use this to obtain a result for the Bloch–Kato Selmer group. Thankfully, this is closely related to the Selmer groups we have already worked with, as shown by the following result.  

  

6. THE CASE OF INERT p

In Section 5 we exclusively dealt with the rank 0 conjecture when p splits in K, but as mentioned previously there are a few reasons we cannot repeat the arguments for inert p. Firstly, the correct rank 0 Panchishkin subrepresentation (and respectively the Selmer group) does not exist when p is inert and, secondly, we do not have a local |$Z_p^2$|-extension analogue of the Coleman map which maps our Euler system to a p-adic distribution. One can consider an approach using the theory of L-analytic regulator map following ideas in [19], where |$L=E_p=\mathbb{Q}_{p^2}$| is the unramified degree 2 extension of |$\mathbb{Q}_p$|⁠. This will be presented in forthcoming work. We will need some additional constraints on our twist η, so that the middle piece of our representation looks like a power of a Lubin–Tate character (or more generally an E_p-analytic character) and therefore the corresponding distribution varies in 1 |$\mathcal{O}_{E_p}$|-variable rather than 2 |$\mathbb{Z}_p$|-variables. We can then obtain similar results to Theorem 4.3 but over an E_p-analytic distribution algebra without a formal scheme model. The module theory over this algebra is more complicated, and we have to make some compromises. A full account and statement of a main conjecture are given in [12].

Moreover, we are yet to connect our motivic p-adic L-function to an automorphic one interpolating complex L-values (after all this is the purpose of an Iwasawa main conjecture), which has been constructed in the split case in the work of Oh [17] using a higher Hida theory for GU(2,1) at split primes developed by Manh-Tu Nguyen [16]. Future work will aim to reconcile these definitions via an explicit reciprocity law, subsequently doing the same in the inert case despite the lack of p-adic analytic constructions or a higher Hida theory at the time of writing.

Acknowledgements

This paper was written during the author’s time as a PhD student at the University of Warwick. The author expresses gratitude to his PhD supervisor David Loeffler for suggesting the problem and for guidance on the many paths that have emerged from it. He is also grateful for the useful and constructive conversations with Kazim Büyükboduk, Pak-Hin Lee, Rob Rockwood and many others.

Funding

This work was supported by the Warwick Mathematics Institute Centre for Doctoral Training, and the author gratefully acknowledges funding from the University of Warwick and the UK Engineering and Physical Sciences Research Council (Grant number: EP/V520226/1).

References