Mock plectic points

Abstract

A $p$-arithmetic subgroup of $\mathrm{SL}_2(\mathbb{Q})$ like the Ihara group $\Gamma := \mathrm{SL}_2(\mathbb{Z}[1/p])$ acts by M\"obius transformations on the Poincar\'e upper half plane $\mathcal{H}$ and on Drinfeld's $p$-adic upper half plane $\mathcal{H}_p := \mathbb{P}_1(\mathbb{C}_p)\setminus\mathbb{P}_1(\mathbb{Q}_p)$. The diagonal action of $\Gamma$ on the product is discrete, and the quotient $\Gamma\backslash(\mathcal{H}_p\times \mathcal{H})$ can be envisaged as a "mock Hilbert modular surface". According to a striking prediction of Nekov\'a$\check{\text{r}}$ and Scholl, the CM points on genuine Hilbert modular surfaces should give rise to "plectic Heegner points" that encode non-trivial 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 $\mathbb{Q}$.