This will be an expository lecture intended to illustrate through examples the theme of p-adic variation in the classical theory of modular forms. Classically, modular forms are complex analytic objects, but because their fourier coefficients are typically integral, it is possible to do elementary arithmetic with them. Early examples arose already in the work of Ramanujan. Today one knows that modular forms encode deep arithmetic information about elliptic curves and galois representations. A central goal of the lecture will be to motivate a beautiful theorem of Robert Coleman and Barry Mazur, who constructed the so-called Eigenvariety, which leads to a geometric approach to varying modular forms, their associated Galois representations, as well as their L-functions, in p-adic analytic families. Finally, I will define a certain modular symbol over GL(2,Q) taking values in a module of distributions on Q2 into K_2 of the tower of modular curves and will offer some speculative ideas about its “universality” with applications to Iwasawa theory.
Glenn Stevens was born in Tacoma, Washington, USA, in 1953 and obtained his PhD in Mathematics from Harvard University in 1980 under the supervision of Barry Mazur. He is, since 1984 a member of the faculty of Boston University where he was promoted to Full Professor in 1993.
What is interesting about Glenn Stevens' career is its perfect balance between research in Pure Mathematics, in Arithmetic Geometry to be more precise, and his activities in Mathematics Education. This second component of Glenn's activity started in 1989 when he founded, together with David Fried, PROMYS (the Program in Mathematics for Young Scientists) at Boston University, which is a six weeks program for high school students with exceptionally strong mathematical interests, and their teachers.
Glenn Stevens' groundbreaking achievements in Arithmetic Geometry start with the proof, with Ralph Greenberg of the celebrated Mazur-Tate-Teitelbaum conjecture and continued with his joint work with A. Ash on overconvergent p-adic modular symbols for the groups GLn. He also worked with Andreatta and Iovita on constructing overconvergent modular sheaves associated to modular curves and Hilbert modular varieties whose global sections are the overconvergent modular forms of Coleman and Coleman-Mazur. More recently with the same co-authors Glenn was able to construct overconvergent Eichler-Shimura isomorphisms which are maps connecting overconvergent modular symbols to overconvergent modular forms, interpolating p-adically the classical ones defined by G. Faltings.
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