Benois's Main Conjecture: the Iwasawa main conjecture for Selmer complexes attached to a regular submodule

For a pure motive $M/\mathbb{Q}$ of weight $\leqslant -1$ whose $p$ -adic realization $V$ is semistable at $p$ and satisfies standard regularity hypotheses, and for a regular $(\varphi,N)$ -submodule $D$ of the semistable module $D_{\rm st}(V)$ , Benois formulated (arXiv:1404.7386, published in LMS Lecture Note Series 420) a version of the Iwasawa main conjecture for the Selmer complex $\mathrm{SC}_{\rm Iw}(V,D)$ : one expects a $p$ -adic $L$ -function $L_p(M,D,s)=f_D(\chi(\gamma_0)^s-1)$ with $f_D$ in the distribution algebra $\mathscr{H}$ , interpolating algebraic parts of the complex $L$ -function (with an Euler-type factor involving the $\ell$ -invariant in the semistable non-crystalline case), and conjectures that $H^1_{\rm Iw}(V,D)=0$ and that $H^2_{\rm Iw}(V,D)$ is $\mathscr{H}$ -torsion with $\mathrm{char}_{\mathscr{H}}(H^2_{\rm Iw}(V,D))=(f_D)$ . This motive-level conjecture uniformly generalizes Greenberg's ordinary main conjecture and Perrin-Riou's crystalline formulation. As of 2026 it is proved only in fragments: one divisibility for modular forms (Kato), the ordinary $\mathrm{GL}_2$ case under technical conditions (Skinner–Urban, completed for good-ordinary elliptic curves by Burungale–Castella–Skinner in 2025), and conditional extra-zero results for specific families. For general motives — in particular semistable non-crystalline $V$ , where even the analytic object and its interpolation are conjectural — the problem is open.

Background

Let $p$ be a prime and let $M/\mathbb{Q}$ be a pure motive of weight $w\leqslant -1$ (for example $M=h^i(X)(m)$ , the $i$ -th cohomology of a smooth projective variety $X$ twisted by the Tate motive), with $p$ -adic realization $V$ : a finite-dimensional $\mathbb{Q}_p$ -vector space with a continuous action of the absolute Galois group $G_{\mathbb{Q},S}$ ramified only outside a finite set $S$ containing $p$ . Assume $V$ is semistable at $p$ ; then $p$ -adic Hodge theory attaches to it the semistable module $D_{\rm st}(V)$ , a filtered $\mathbb{Q}_p$ -vector space carrying a Frobenius $\varphi$ and a nilpotent monodromy operator $N$ satisfying $N\varphi=p\varphi N$ ; $V$ is crystalline exactly when $N=0$ . Write $D_c(V)=D_{\rm st}(V)^{N=0}$ .

Iwasawa theory studies $V$ over the cyclotomic $\mathbb{Z}_p$ -extension $\mathbb{Q}_\infty/\mathbb{Q}$ (the unique $\mathbb{Z}_p$ -extension inside the field of all $p$ -power roots of unity). Fix a topological generator $\gamma_0$ of $\Gamma_1=\mathrm{Gal}(\mathbb{Q}_\infty/\mathbb{Q}(\mu_p))\cong 1+p\mathbb{Z}_p$ and let $\Lambda=\mathbb{Z}_p[[\Gamma_1]]$ be the Iwasawa algebra; $\mathscr{H}$ denotes the algebra of $p$ -adic distributions on $\Gamma_1$ , the natural completion of $\Lambda[1/p]$ for which Lazard's theory assigns to every coadmissible torsion $\mathscr{H}$ -module (coadmissible meaning, roughly, a module obtained from finitely generated $\Lambda$ -modules by inverting $p$ and completing — the class on which $\mathscr{H}$ behaves like a principal ideal domain) a well-defined characteristic ideal $\mathrm{char}_{\mathscr{H}}$ . For a $G_{\mathbb{Q}}$ -stable lattice $T\subset V$ , the Iwasawa cohomology $H^i_{\rm Iw}(V)=\varprojlim_n H^i(G_{\mathbb{Q},S},\,T\otimes\mathbb{Z}_p[\mu_{p^n}]^{\vee})$ is a finitely generated $\mathscr{H}$ -module.

The Selmer complex formalism of Nekovář refines Selmer groups into objects of the derived category: given cohomological local conditions at the places of $S$ , the Selmer complex is the mapping cone of the sum of the localization maps to those conditions, shifted by one. Benois imposes, at places $v\neq p$ , the Bloch–Kato condition $H^1_f$ (the subgroup of local cohomology classified by the $p$ -adic exponential map of de Rham cohomology), and at $p$ the condition defined by the Iwasawa cohomology $R\Gamma_{\rm Iw}(\mathbf{D})$ of a $(\varphi,\Gamma)$ -module $\mathbf{D}$ — the Robba-ring avatar of a $p$ -adic representation of $G_{\mathbb{Q}_p}$ (an étale module over the ring of Laurent series convergent on an annulus, with commuting Frobenius and $\Gamma$ -actions, a category equivalent to $p$ -adic representations of $G_{\mathbb{Q}_p}$ ).

The local condition at $p$ is determined by a regular submodule. Following Perrin-Riou, a $(\varphi,N)$ -submodule $D\subset D_{\rm st}(V)$ is regular if $D\cap\mathrm{Fil}^0 D_{\rm st}(V)=0$ and the regulator map

$r_{V,D}:\,H^1_f(\mathbb{Q},V)\longrightarrow D_{\rm st}(V)/(\mathrm{Fil}^0D_{\rm st}(V)+D)$

is an isomorphism (weight $\leqslant -2$ ), or if the projection $D\to t_V=D_{\rm dR}(V)/\mathrm{Fil}^0$ is an isomorphism (weight $-1$ ). Regular submodules generalize Greenberg's ordinarity filtration — every semistable representation is ordinary in Greenberg's sense — and interpolate between Bloch–Kato and Greenberg local conditions; to $D$ one attaches canonically a $(\varphi,\Gamma)$ -submodule $\mathbf{D}$ of $D^{\dagger}_{\rm rig}(V)$ , the $(\varphi,\Gamma)$ -module of $V$ .

On the analytic side, the expectation is that each regular $D$ determines a $p$ -adic $L$ -function $L_p(M,D,s)=f_D(\chi(\gamma_0)^s-1)$ , $f_D\in\mathscr{H}$ , interpolating algebraic parts of the special values of the complex $L$ -function $L(M,s)$ : at $s=0$ ,

$\frac{L^*_p(M,D,0)}{\Omega_p(M,D)}=\mathscr{E}(V,D)\,\frac{L^*(M,0)}{\Omega_\infty(M)},$

where $\Omega_\infty(M)$ is the transcendental period predicted by the Beilinson–Deligne conjecture on special values, $\Omega_p(M,D)$ is the determinant of the regulator $r_{V,D}$ (weight $\leqslant -2$ ) or of the $p$ -adic height pairing $\langle\,,\,\rangle_{V,D}$ (weight $-1$ ), and $\mathscr{E}(V,D)$ is an Euler-like factor at $p$ , given in the crystalline case by Perrin-Riou's conjectural formula. Extra (trivial) zeros are compatible with the vanishing hypothesis C3) $D_c(V)^{\varphi=1}=0$ imposed below: they are detected not on $D_c(V)$ itself, but on subquotients of $D^{\dagger}_{\rm rig}(V)$ . To a regular $D$ one attaches an increasing filtration $(D_i)_{i=-2}^{2}$ of $D_{\rm st}(V)$ with $D_0=D$ , which induces a filtration of $D^{\dagger}_{\rm rig}(V)$ ; write $W=D_1/D_{-1}$ and $W_0$ for the graded piece with $D_{\rm st}(W_0)^{\varphi=p^{-1}}=D_{\rm st}(W_0)$ . In case a) $D_{\rm st}(V)^{\varphi=1}=0$ , so that $W=W_0$ ; in case b) $H^1_f(\mathbb{Q},V)=H^1_f(\mathbb{Q},V^*(1))=0$ and, under a structural condition M) of the source, $W$ contains a subquotient $M^{\flat}$ fitting into an exact sequence $0\to M_0\to M^{\flat}\to M_1\to 0$ with $\mathrm{rank}(M_0)=\mathrm{rank}(M_1)$ (the typical example being $V=W_f(k)$ , where $W_f$ is the $p$ -adic representation attached to a split multiplicative newform $f$ of weight $2k$ on $\Gamma_0(N)$ with $p\mid N$ ). In both cases $L_p$ is expected to acquire a zero of order $e=\mathrm{rank}(W_0)$ at $s=0$ , governed by the $\ell$ -invariant $\ell(V,D)$ — a $p$ -adic analytic invariant of the local representation at $p$ , defined as a determinant on $D_c(W_0)$ in case a) and on $D_c(M_1)$ in case b). The precise leading-term formula is the Extra Zero Conjecture:

$\lim_{s\to 0}\frac{L_p(M,D,s)}{s^{e}}=\ell(V,D)\,\mathscr{E}^{+}(V,D)\,\frac{\Omega_p(M,D)\,L(M,0)}{\Omega_\infty(M)},$

where $\mathscr{E}^{+}(V,D)$ is obtained from $\mathscr{E}(V,D)$ by excluding the zero factors. Following the source, this formula is written with $L(M,0)$ , whereas the interpolation formula above is written with the leading term $L^*(M,0)$ ; the numerator carries the regulator factor $\Omega_p(M,D)$ , which equals $1$ in case b), where $H^1_f(\mathbb{Q},V)=H^1_f(\mathbb{Q},V^*(1))=0$ , but is the regulator determinant and generally nontrivial in case a).

At the time of the source paper (2014) the conjecture was known only in fragments: Greenberg's ordinary formulation; Kato's Euler system giving one divisibility for modular forms; Skinner–Urban's proof of both divisibilities for ordinary $\mathrm{GL}_2$ under technical hypotheses; and compatibility with Perrin-Riou's crystalline theory via work of Pottharst. The nonvanishing of $\ell(V,D)$ was known only for Dirichlet motives (Ferrero–Greenberg) and for elliptic curves over $\mathbb{Q}$ (Barré-Sirieix–Diaz–Gramain–Philibert).

Problem Statement

Let $M/\mathbb{Q}$ be a pure motive of weight $\leqslant -1$ containing no submotive isomorphic to $\mathbb{Q}(m)$ , with $p$ -adic realization $V$ semistable at $p$ and satisfying:

C1) $H^0(\mathbb{Q},V)=H^0(\mathbb{Q},V^*(1))=0$ ;
C2) the Frobenius $\varphi$ on $D_{\rm st}(V)$ is semisimple at the eigenvalues $1$ and $p^{-1}$ ;
C3) $D_c(V)^{\varphi=1}=0$ , where $D_c(V)=D_{\rm st}(V)^{N=0}$ ;
C4) either $H^1_f(\mathbb{Q},V^*(1))=0$ (weight $\leqslant -2$ regime) or $V$ is self-dual (weight $-1$ regime).

Let $D\subset D_{\rm st}(V)$ be a regular $(\varphi,N)$ -submodule, $\mathbf{D}$ the attached $(\varphi,\Gamma)$ -submodule of $D^{\dagger}_{\rm rig}(V)$ , and $H^i_{\rm Iw}(V,D)$ the cohomology of the associated Selmer complex $\mathrm{SC}_{\rm Iw}(V,D)$ over $\mathscr{H}$ . Prove, or refute, Benois's Main Conjecture:

(i) $H^1_{\rm Iw}(V,D)=0$ ;

(ii) $H^2_{\rm Iw}(V,D)$ is a coadmissible torsion $\mathscr{H}$ -module and

$\mathrm{char}_{\mathscr{H}}\left(H^2_{\rm Iw}(V,D)\right)=(f_D),$

where $f_D\in\mathscr{H}$ is defined by the $p$ -adic $L$ -function $L_p(M,D,s)=f_D(\chi(\gamma_0)^s-1)$ , whose existence and interpolation property are part of the problem: at $s=0$ it must satisfy

$\frac{L^*_p(M,D,0)}{\Omega_p(M,D)}=\mathscr{E}(V,D)\,\frac{L^*(M,0)}{\Omega_\infty(M)},$

with $\Omega_p(M,D)$ the determinant of the regulator $r_{V,D}$ (weight $\leqslant -2$ ) or of the $p$ -adic height pairing $\langle\,,\,\rangle_{V,D}$ (weight $-1$ ), and $\mathscr{E}(V,D)$ the Euler-type factor at $p$ : Perrin-Riou's conjectural factor when $V$ is crystalline, and in the semistable non-crystalline case with extra zeros the factor given by the Extra Zero Conjecture described in the Background — $L_p(M,D,s)$ vanishes to order $e=\mathrm{rank}(W_0)$ at $s=0$ and its leading term is $\ell(V,D)\,\mathscr{E}^{+}(V,D)\,\Omega_p(M,D)\,L(M,0)/\Omega_\infty(M)$ , with $\mathscr{E}^{+}(V,D)$ obtained from $\mathscr{E}(V,D)$ by excluding the zero factors, $\Omega_p(M,D)$ the regulator factor defined above (equal to $1$ in case b), generally nontrivial in case a)), and $\ell(V,D)$ the $\ell$ -invariant attached to the subquotients $W_0$ , respectively $M_1$ , of the filtration induced by $D$ .

The common objective is the identification of the analytic $p$ -adic $L$ -function with the algebraic characteristic ideal for every regular submodule in the stated class; the construction of the analytic object, its interpolation property, and the ideal equality are jointly the content of the conjecture.

References

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