I will define the heat Eisenstein series and sketch some of their theory, as developed by Jorgenson and Lang in the 90s and early 00s. Then, I will describe their role in an ongoing project to generalize theta relations and spectral zeta functions to arithmetic quotients of real Lie groups in higher rank. Finally, I will state some of my results on exact fundamental domains and explain how they are used or are expected to be used in this program.

Let E be an elliptic curve over Q without complex multiplication. Mazur asked if the universal deformation ring of the p-torsion Galois representation of E is unobstructed for all but finitely many p. In this talk we discuss the relationship between this question and conjectures on existence of p-adic p-torsion points of E and prove that Mazur's question is at least true on average. This is joint work with Chantal David.

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