Arithmetic statistics of noncommutative modular symbols
Final Report Abstract
Noncommutative modular symbols are a natural generalization of modular symbols that appear in central problems of number theory concerning L-functions of elliptic curves. While these objects are difficult to study one by one, patterns do emerge when one studies how the family of noncommutative modular symbols is distributed among the complex numbers. The present work presents a first step in a general study of the arithmetic statistics of noncommutative modular symbols. In particular, while classical/commutative modular symbols was previously known to be normally distributed (which is, in a sense, a maximally "random” distribution), already the length two case features much more exotic distributions which are not well known outside of the statistics literature. While both the length one and the length two case can be tackled using a combination of classical tools from statistics (method of moments) combined with some more recent analytic objects (twists of real analytic Eisenstein series), it came as a surprise came that in higher length these tools are seemingly insufficient to determine the precise distribution of noncommutative modular symbols.
Publications
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The distribution of Manin’s iterated integrals of modular forms. Journal für die reine und angewandte Mathematik (Crelles Journal), 0(0).
Matthes, Nils & Risager, Morten S.
