Mock plectic points

A p-arithmetic subgroup of SL2ℚ like the Ihara group Γ:= SL2(ℤ}[1/p])acts by Möbius transformations on the Poincare upper half plane H and on Drinfeld's p-adic upper half plane Hp:= P1(Cp)-P1(ℚp).. The diagonal action of Γ on the product is discrete, and the quotient Γ(Hp×H) can be envisaged as a 'mock Hilbert modular surface'. According to a striking prediction of Nekovar and Scholl, the CM points on genuine Hilbert modular surfaces should give rise to 'plectic Heegner points' that encode nontrivial regulators attached, notably, to elliptic curves of rank two over real quadratic fields. This article develops the analogy between Hilbert modular surfaces and their mock counterparts, with the aim of transposing the plectic philosophy to the mock Hilbert setting, where the analogous plectic invariants are expected to lie in the alternating square of the Mordell-Weil group of certain elliptic curves of rank two over ℚ.