Computation of the unipotent Albanese map on elliptic and hyperelliptic curves

Abstract

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map j dr n on elliptic and hyperelliptic curves over number fields via their universal unipotent connections U.Several algorithms forming part of the computation of finite level versions j dr n of the unipotent Albanese maps are presented. The computation of the logarithmic extension of U in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on U over affine elliptic and odd hyperelliptic curves.We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well.Here we introduce unipotent connections and background material on the unipotent Albanese map. Throughout both sections [23] is used as a primary reference for definitions and results.Let K be a field of characteristic 0, let X be a K-scheme, and suppose we have a fixed basepoint b ∈ X(K). Let V be a vector bundle on X.Here Ω 1 X/K is the sheaf of 1-forms on X/K. Remark 2.2. We will often refer to a vector bundle V with connection ∇ simply as a connection and write such objects either as (V, ∇) or simply as V.Remark 2.3. We may extend ∇ to a covariant derivative ∇ 1 :Definition 2.4. We say that V is a flat or integrable connection if the induced morphismis the zero map. Note that if X is a curve, then any connection V is automatically flat.Given a connection (V, ∇) with V of rank n, there is a matrix Ω ∈ gl n ⊗ Ω 1 X/K called the connection matrix which determines ∇: suppose that we have a local basis e i : O X ֒→ V (1 ≤ i ≤ n). Let U ⊂ X be some trivialising neighbourhood in X. Then ∇(e i ) ∈ V ⊗ Ω 1 X/K (U ), and so there are ω ij ∈ Ω 1 X/K (U ) such thatWe let Ω := (ω ij ). We may show that in matrix notation ∇(e · f ) = e · (df + Ω · f ), and so ∇ acts locally as d + Ω. Remark 2.5. A connection (V, ∇ = d + Ω) is flat if and only if dΩ + Ω ∧ Ω = 0Definition 2.6. A morphism of connections (V, ∇) → (W, ∇ ′ ) is a morphism f : V → W of sheaves preserving the connection.Definition 2.7. A connection V is unipotent with index of unipotency less than or equal to n if there is a decreasing sequence of sub-connectionssuch that the quotients V i+1 /V i are isomorphic to a direct sum of copies of (O X , d) i.e. they are trivial.We obtain the following category on X:Definition 2.8. Let Un n (X) be defined to be category whose objects are unipotent vector bundles on X with flat connection having index of unipotency less than or equal to n with morphisms being morphisms of connections. Define Un(X) to be ∪ n≥1 Un n (X)Given so...

Quadratic Chabauty and Rational Points II: Generalised Height Functions on Selmer Varieties

“…, and let T denote the map H → H α sending P to P ⊗ ([IA(b)] + τ * (P )). Then by the multilinearity of generalised heights we have (7) h(P ) = h(S • T (π * P )) for all P ∈ Sel(U ) α . To use this to write down equations for loc p (Sel(U ) α ), we introduce some notation for resultants (see e.g.…”