Seminar des SFB/TRR 326 GAUS  Antisymmetry in the Theory of Rigid Meromorphic Cocycles

Around six years ago Darmon and Vonk initiated the theory of p-adic singular moduli for real quadratic fields by defining "rigid meromorphic cocycles". These are elements of the first cohomology group of Ihara's group SL2(Z[1/p]) with values in the group of rigid meromorphic functions on Drinfeld's upper half-plane. Using rigid meromorphic cocycles, Darmon and Vonk assign to each pair of real quadratic irrationalities a p-adic number. The two irrationalities play a vastly different role in the construction of this assignment. However, it is expected to behave like the difference of two classical singular moduli - in particular, it should be anti-symmetric in the argument. We will use the recent work of Darmon, Gehrmann and Lipnowski on rigid meromorphic cocycles for higher dimensional orthogonal groups to give a new, symmetric construction of this function.