After a brief introduction to p-adic measures, we shall explain how modular symbols may be used to construct p-adic L-functions attached to certain modular forms.

       We will describe the notion of a Λ-adic modular form and explain Hida's theorem on the existence of a universal, ordinary Λ-adic representation.

       This will be split into two talks explaining the connection between between the L-invariant of an elliptic curve E over Q with split multiplicative reduction at p, and the p-th Fourier coefficient of a certain Λ-adic cusp form associated to E.

       This talk is a continuation of two from last quarter about the SIC-POVM conjecture from quantum information theory, which postulates the existence of lines in each C^d whose orbits under a finite Heisenberg group are equiangular. In this talk, I will show how a dual characterization from design theory in terms of harmonic invariants implies the set of all such lines is a projective algebraic set. I will also discuss, for certain dimensions congruent to 7 mod 12, the class field theory underlying the structure of a real number field over which it may be easier to actually prove the existence of such lines.