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This is the note for a series of lectures that the author gave at the Centre de Recerca Matem´atica (CRM), Bellaterra, Barcelona, Spain on October 19–24, 2009. The aim is to give a comprehensive description of some recent work of the author and his students on generalisations of the Gross-Zagier formula, Euler systems on Shimura curves, and rational points on elliptic curves.

Based on a global estimate of the heat kernel, some important inequalities such as Poincaré inequality and log-Sobolev inequality, furthermore a tight logarithm Sobolev inequality are presented on graphs, just under a positive curvature condition CDE'(n,K) with some K > 0. As consequences, we obtain exponential integrability of integrable Lipschitz functions and moment bounds at the same assumption on graphs.

No abstract uploaded!

No abstract uploaded!