The unipotent Albanese map and Selmer varieties for curves

By Minhyong Kim. 2005

119 citations recorded by the source corpus.

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, while standard conjectures about the structure of mixed motives provide control over the image of the map. As a consequence, conjectures of ‘Birch and Swinnerton-Dyer type’ are connected to finiteness theorems of Faltings-Siegel type.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9be136165de111b4a37cfdda3feaee74b819335d

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Massey products for elliptic curves of rank 1 (2009, methodology): “Since there is a Coleman map j 2,loc : X (Zp)→H(1) f (Gp, U2) with Zariski dense image for each residue disk [17], Theorem 0.”
Explicit Chabauty—Kim for the split Cartan modular curve of level 13 (2017, methodology): “The analogue of the analytic properties of AJb is the theorem that jdRn has Zariski dense image [Kim09, Theorem 1] and is given by a power series on each residue disk.”
A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves (2012, methodology): “Now, as soon as L(n) 6= 0, X (Zp)n must be finite ([14], section 3), so we are in any case investigating a finite discrepancy”