Mock Plectic Points in Hilbert Modular Surfaces and Elliptic Curves

Explore a lecture on mock plectic points and their connections to Hilbert modular surfaces. Delve into the Ihara group's actions on the Poincaré upper half plane and Drinfeld's p-adic counterpart, examining the resulting quotient as a "mock Hilbert modular surface." Investigate the correspondence between Oda's conjecture on periods of Hilbert modular surfaces and the exceptional zero conjecture of Mazur-Tate Teitelbaum. Compare complex ATR points with Stark-Heegner points over ring class fields of real quadratic fields. Analyze Nekovar and Scholl's prediction regarding CM points on genuine Hilbert modular surfaces and their relation to "plectic Heegner points." Examine the analogy between Hilbert modular surfaces and their mock counterparts, focusing on transposing the plectic philosophy to the mock Hilbert setting. Consider the consistency of these predictions with the "anti-cyclotomic exceptional zero conjecture" and its implications for the Nekovar-Scholl philosophy. This joint work by Michele Fornea and Henri Darmon offers insights into advanced topics in algebraic number theory and arithmetic geometry.