STNB2017(31è any)

Stark's conjectures and generalized Kato classes


Óscar Rivero


During the last years, a great progress in the study of BSD and Block-Kato conjectures has been made. Here, we are interested in two di erent aspects: on the one hand, the conjecture suggested by Darmon, Lauder and Rotger relating the value of a p-adic iterated integral (whose value may be encoded in Harris-Kudla triple p-adic L-functions) with Stark's units and the value of regulators de ned in terms of p-adic logarithms of units in number elds (this can also be formulated for points over elliptic curves and gives inter- esting results for BSD in analytic rank 2). On the other hand, several works of Bertolini, Darmon, Rotger and others allow us to construct families of co- homology classes satisfying good properties and related with special values of the p-adic L-functions. These classes are obtained via the image through etale and syntomic regulators of distinguished cycles in certain algebraic va- rieties. In our work, we use the construction of families of cohomology classes in- terpolating two cuspidal forms to formulate a conjecture about the good behavior of these Kato classes and their relation with Ohta periods, that would imply the main result suggested in the paper of Darmon, Lauder and Rotger.


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