# The motivic anabelian geometry of local heights on abelian varieties

@article{Betts2017TheMA, title={The motivic anabelian geometry of local heights on abelian varieties}, author={L. Alexander Betts}, journal={arXiv: Number Theory}, year={2017} }

We study the problem of describing local components of height functions on abelian varieties over characteristic $0$ local fields as functions on spaces of torsors under various realisations of a $2$-step unipotent motivic fundamental group naturally associated to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the $\mathbb Q_\ell$- and $\mathbb Q_p$-pro-unipotent etale realisations when the base field is $p$-adic, and in terms of the…

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## References

SHOWING 1-10 OF 63 REFERENCES

### Galois-Teichmüller theory and arithmetic geometry

- Mathematics
- 2012

Remarks on the Milnor conjecture over schemes by A. Auel On the decomposition of motivic multiple zeta values by F. C. S. Brown Combinatorics of the double shuffle Lie algebra by S. Carr and L.…

### A non-abelian conjecture of Birch and Swinnerton-Dyer type for hyperbolic curves

- Mathematics
- 2012

We state a conjectural criterion for identifying global integral points on a hyperbolic curve over $\mathbb{Z}$ in terms of Selmer schemes inside non-abelian cohomology functors with coefficients in…

### Selmer varieties for curves with CM Jacobians

- Mathematics
- 2010

We study the Selmer variety associated to a canonical quotient of the $\Q_p$-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over $\Q$ whose Jacobian…

### On p-adic height pairings

- Mathematics
- 2002

The aim of the present work is to construct p-adic height pairings in a sufficiently general setting, namely for Selmer groups of reasonably behaved p-adic Galois representations over number fields.…

### Canonical Height Pairings via Biextensions

- Mathematics
- 1983

The object of this paper is to present the foundations of a theory of p-adic-valued height pairings
$$A\left( K \right) \times A'\left( K \right) \to {Q_p}$$
(*)
, where Λ is a abelian…

### The l-component of the unipotent Albanese map

- Mathematics
- 2006

We prove a finiteness theorem for the local l ≠ p-component of the $${\mathbb{Q}}_p$$ -unipotent Albanese map for curves. As an application, we refine the non-abelian Selmer varieties arising in the…

### Schematic homotopy types and non-abelian Hodge theory

- MathematicsCompositio Mathematica
- 2008

Abstract We use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the schematization functor$X \mapsto…

### Cohomology

- Lectures on the Geometry of Manifolds
- 2018

. Let B denote the upper triangular subgroup of SL 2 ( C ), T its diagonal torus and U its unipotent radical. A complex projective variety Y endowed with an algebraic action of B such that the ﬁxed…

### The unipotent Albanese map and Selmer varieties for curves

- Mathematics
- 2005

We study the unipotent Albanese map that associates the torsor of paths for p-adic fundamental groups to a point on a hyperbolic curve. It is shown that the map is very transcendental in nature,…