p-adic Arakelov theory

Amnon Besser

Introduction

Goal: ℚ_{p}-valued analogue of Arakelov theory for arithmetic surfaces. General shape: F number field X /O_{F} an arithmetic surface

D, E ∈ div(X) ↦ D · E = _{v}^{∑}[D, E]_{v} Sum over all finite places v. For v ∤ p, [D, E]_{v} similar to usual intersection pairing above v. For v|p, “p = ∞”, [D, E]_{v} comes from p-adic analysis “a la Coleman”.

Coleman integration

This theory originally developed by Robert Coleman in the 80’s Current version: Combination of (B, Math. Ann., 2002) with work of Vologodski.

Setup:

[K : ℚ_{p}] < ∞, log - choice of a p-adic log, X/K smooth.

(M, ∇) - an integral connection on X.

Vologodski (2001) gives a canonical parallel translation v_{x,y} : M_{x} → M_{y} for x, y ∈ X(K), commuting with everything you can think of Definition: An abstract Coleman function on X with values in F (coherent O_{X}-module) is a fourtuple (M, ∇, (m_{x} ∈ M_{x})_{x∈X(K̄)}, s) s.t.

• (M, ∇) as before

• v_{x,y}(m_{x}) = m_{y}

• s ∈ Hom (M, F).

Coleman functions are connected components of the category of abstract Coleman functions.

Notations: O_{Col}(X, F), O_{Col}(X) = O_{Col}(X, O_{X}), Ω_{Col}^{i}(X) = O_{Col}(X, Ω_{X}^{i}).

Key property: The sequence

0 → K → O_{Col}(X) ⟶^{d} Ω_{Col}^{1}(X) ⟶^{d} Ω_{Col}^{2}(X) is exact. Consequence: O_{Col}(X) contains iterated integrals, e.g,

∫ (η · ∫ ω) , where ω, η · ∫ ω closed Locally every Coleman function looks like this. Base change: we can work over \overline{ℚ_{p}}.

The p-adic ¯∂ operator

Definition: O_{Col ,1}(X, F) ⊂ O_{Col}(X, F), subset of functions with E_{1} ⊂ M, E_{2} = M/E_{1}, E_{i} trivial. Local definition: functions that look like f · ∫ ω. p-adic ¯∂: ¯∂ : O_{Col ,1}(X, F) → H_{dR}^{1}(X) ⊗ F(X). Local definition: ¯∂(f · ∫ ω) = [ω] ⊗ f.

• This is well defined

• The definition globalizes.

Log functions on line bundles

X/\overline{ℚ_{p}} smooth L a line bundle over X

L^{∗} = Tot (L) − 0 ⟶^{π} X

Definition: A log function on L is log_{L} ∈ O_{Col}(L^{∗}) such that:

• On a fiber L_{x} we have log_{L}(αl) = log (α) + log_{L}(l)

• d log_{L} ∈ O_{Col ,1}(L^{∗}, Ω^{1}).

This is a p-adic analogue for the log ||s||, for a metric || · ||.

Curvature

Definition: The curvature of log_{L} is

Curve (log_{L}) ∈ H_{dR}^{1}(X) ⊗ Ω^{1}(X) such that π^{∗} Curve (log_{L}) = ¯∂d log_{L} Theorem: (X complete)

1. When ch_{1}(L) ∈ Im(∪ : H_{dR}^{1}(X) ⊗ Ω^{1}(X) → H_{dR}^{2}(X)) , Curve (log_{L}) exists and ∪ Curve (log_{L}) = ch_{1}(L).

2. If ∪(α) = ch_{1}(L) there is a log function on L with curvature α.

The (almost) canonical Green function X/\overline{ℚ_{p}} smooth complete curve, Δ ∈ X × X. We define a canonical log function on O(Δ). We fix splitting H_{dR}^{1}(X) = W ⊕ Ω^{1}(X), W ∪ W = 0. p : O(Δ)^{∗} → X × X, π_{1}, π_{2} : X × X → X {ω_{1}, …, ω_{g}} a basis of Ω^{1}(X) {ω̄_{1}, …, ω̄_{g}} ⊂ W, tr(ω̄_{i} ∪ ω_{j}) = δ_{ij}

μ = \frac{1}{g} g ; ∑ ; i=1 ω̄_{i} ⊗ ω_{i} ∈ H_{dR}^{1}(X) ⊗ Ω^{1}(X) ,
Φ = π_{1}^{∗}μ + π_{2}^{∗}μ − g ; ∑ ; i=1 (π_{1}^{∗}ω̄_{i} ⊗ π_{2}^{∗}ω_{i} + π_{2}^{∗}ω̄_{i} ⊗ π_{1}^{∗}ω_{i})

∈ H_{dR}^{1}(X × X) ⊗ Ω^{1}(X × X) .

∪Φ = ch_{1}(O(Δ) ) ⇒ O(Δ) has a log function with curvature Φ (not canonical yet)

Set G = log_{O(Δ)} (1).

G can be made canonical (up to const) by imposing:

• G(x, y) = G(y, x)

• Residue condition.

Log function on determinant of cohomology

The determinant of cohomology of a line bundle L:

λ(L) = det (H^{0}(X, L)) ⊗ det (H^{1}(X, L))^{⊗−1} .

Theorem: Exist association (L, log_{L}) ↦ log function on λ(L) satisfying certain rules:

• behavior under scaling

• behavior under adding points

(analogues of classical conditions)

Global Theory

F - number field ℓ : 𝔸_{F}^{×}/F^{×} → ℚ_{p} ⇒ ℓ_{v} : F_{v}^{×} → ℚ_{p}

• v ∤ p - ℓ_{v} ↔ ℓ_{v}(π_{v}) since ℓ_{v}(O_{F_{v}}^{×}) = 0.

• v|p

[figure: F_{v}^{×}; ℓ_{v}; _{log_{v}}; ℚ_{p}; F_{v}; t_{v}]

X /O_{F} - Arithmetic surface Definition: An Arakelov divisor is D = D_{fin} + D_{∞} D_{∞} = ∑_{v|p} λ_{v}X_{v}, λ_{v} ∈ F_{v}.

Special case: s_{1}, s_{2} : O_{F} → X, D_{i} = Im(s_{i}).

[D_{1}, D_{2}] = \begin{cases}ℓ_{v}(π_{v}) · ⟨D_{1}, D_{2}⟩_{v} & v ∤ p \\ t_{v}(G_{v}(s_{1}, s_{2})) & v|p \end{cases}

Metrized line bundle ↔ Arakelov divisors:

(L, (log_{L_{v}})) + section s ↦ (s) = (s)_{fin} + (s)_{∞}, (s_{∞}) = ∑_{v|p} ι_{log_{Lv}}(s_{v}) · X_{v} where ι_{log} (s) = log (s) − G_{(s)}.

Arithmetic degree N /O_{F} - metrized line bundle F ⟶^{θ} N ⊗ F isomorphism deg (N) := ∑_{v|p} t_{v}(log_{v}(θ_{v}(1)))−∑_{v∤p} ℓ_{v}(θ_{v}^{−1}(N_{v})) Riemann-Roch L/X metrized ⟶ λ(L) metrized χ(L) = deg (λ(L)) Riemann-Roch formula:

χ(L) − χ(O_{X}) = \frac{1}{2}[L, L − ω_{X}]