Néron-Tate height pairings are defined on jacobians of curves over number fields. The jacobian admits the so-called universal vector extension, the moduli of line bundles with connections. I will describe an attempt to extent the Néron-Tate pairing to the universal vector extension, and the relation to the complex valued analytic torsion introduced by Cappell-Miller. The link betweem both is a Riemann-Roch type formula in Arakelov geometry, a variant of Kronecker's second limit formula.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/246