[Paper Review] On higher special elements of $p$-adic representations

David Burns, Takamichi Sano+1 more|Research Portal (King's College London)|Sep 11, 2018

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper introduces a novel algebraic framework for higher special elements in the exterior power biduals of $p$-adic Galois cohomology, generalizing Euler systems to higher rank. It establishes integrality properties, annihilator formulas, and congruence relations via canonical height pairings, and applies the theory to refine conjectures on derivatives of Dirichlet $L$-series and the Quillen–Lichtenbaum conjecture, proving a refined class number formula for $K_{2m-1}$-groups.

ABSTRACT

As a natural generalization of the notion of `higher rank Euler system', we develop a theory of `higher special elements' in the exterior power biduals of the Galois cohomology of $p$-adic representations. We show, in particular, that such elements encode detailed information about the structure of Galois cohomology groups and are related by families of congruences involving natural height pairings on cohomology. As a first concrete application of the approach, we use it to refine, and extend, a variety of existing results and conjectures concerning the values of derivatives of Dirichlet $L$-series.

Motivation & Objective

• To develop a general algebraic theory of higher special elements in the exterior power biduals of $p$-adic Galois cohomology groups.
• To establish integrality and annihilator properties of these elements without restrictive assumptions on input data.
• To formulate and prove congruence relations between higher special elements using canonical height pairings.
• To apply the theory to refine and extend conjectures on derivatives of Dirichlet $L$-series and the Quillen–Lichtenbaum conjecture.
• To provide a refined class number formula for $K_{2m-1}(\mathcal{O}_F)$ via higher special elements in the context of cyclotomic elements.

Proposed method

• Define higher special elements $\eta_{(C,\lambda,\mathcal{L},x_\bullet)}$ in the $a$-th exterior power of $H^1(C)$ using a complex $C$ of $\mathfrak{A}$-modules, an isomorphism $\lambda$, and a generator $\mathcal{L}$ of a determinant module.
• Prove that these elements compute higher Fitting ideals of $H^2(C)$ and generate annihilators of subquotients, even without restrictions on the tuple $x_\bullet$.
• Construct a canonical perfect $\mathfrak{A}$-bilinear pairing between quotients of exterior powers and quotients of $\mathfrak{A}$, using the Gorenstein condition.
• Establish congruence relations between higher special elements via a canonical algebraic height pairing on cohomology groups.
• Apply the theory to compactly supported $p$-adic cohomology and Nekovář–Selmer complexes in arithmetic settings.
• Verify that the higher special element associated with cyclotomic data coincides with Deligne–Soulé’s cyclotomic element, linking to $K$-theory.

Experimental results

Research questions

• RQ1 How can higher special elements be systematically defined in the exterior power biduals of $p$-adic Galois cohomology without restrictive assumptions on the input data?
• RQ2 What integrality and annihilator properties do higher special elements possess, and how do they relate to the structure of Galois cohomology groups?
• RQ3 How are higher special elements related across different data sets, and can these relations be captured by canonical height pairings?
• RQ4 Can the theory refine known conjectures on derivatives of Dirichlet $L$-series and the Quillen–Lichtenbaum conjecture?
• RQ5 Does the higher special element construction recover or refine the classical class number formula for $K_{2m-1}(\mathcal{O}_F)$?

Key findings

• The higher special element $\eta$ computes the higher Fitting ideals of $H^2(C)$ and generates annihilators of subquotients of $H^2(C)$, even without restrictions on the tuple $x_\bullet$, as shown in Theorem 3.10.
• Under the Gorenstein condition, a canonical perfect $\mathfrak{A}$-bilinear pairing exists between the quotient of $\bigwedge^a_{\mathfrak{A}}H^1(C)$ by the submodule generated by $\eta$ and the quotient of $\mathfrak{A}$ by an ideal determined by $\eta$, as established in Theorem 3.27.
• Congruence relations between higher special elements are governed by a canonical algebraic height pairing on cohomology, as proven in Theorem 3.36.
• The higher special element associated with the cyclotomic data coincides with Deligne–Soulé’s cyclotomic element, confirming its role in $K$-theory.
• The theory proves a refined class number formula for $K_{2m-1}(\mathcal{O}_F)$, showing that $\left(\varepsilon_m^- \cdot K_{2m-1}(\mathcal{O}_F)^\prime / \mathbb{Z}'[G] \cdot c_F(m)\right)^\vee \cong \mathbb{Z}'[G]\varepsilon_m^- / \mathrm{Fit}^0_{\mathbb{Z}'[G]}(K_{2m-2}(\mathcal{O}_{F,\Sigma})^\prime)$, as stated in Theorem 5.15.
• The result corrects an error in El Boukhari’s [15, Th. 6.5] by providing a properly defined factor $\prod_{\ell \in \Sigma \setminus \{\infty\}} (1 - \mathrm{Fr}_\ell^{-1} \cdot \ell^{m-1}) e_{I_\ell}$, resolving an incorrectly defined 'Q(0)' factor.