On a Generalization of the Conjecture of Mazur–Tate–Teitelbaum

We propose a generalization of the conjecture of Mazur–Tate–Teitelbaum predicting an exact shape of the p-adic ℒ-invariant of rational Tate curves (which is now a theorem of Greenberg-Stevens) to the symmetric powers of motivic two dimensional odd Galois representations over totally real fields. At p-adic places where the motive is multiplicative, the ℒ-invariant is conjectured to have the same shape as predicted by them. Then we prove our conjecture assuming a precise ring theoretic structure of the universal infinitesimal Galois deformation ring of the symmetric power.