Glenn Stevens

Boston University


Modular Symbols with Values in Beilinson-Kato Distributions


Heilbronn Number Theory Seminar


12th February 2025, 4:00 pm – 5:00 pm
Fry Building, 2.04


In recent years, the “classical” Siegel distribution µ taking values in the group of modular units
of all levels, has played a central role in a number of important number theoretic developments,
including the construction of Kato’s Euler systems with applications to p-adic analogs of the Birch
and Swinnerton-Dyer conjecture, and also in defining the p-adic measures used by Darmon and
Dasgupta in their construction of global units in ring class extensions of real quadratic fields.
In this talk I will describe a sequence, {ξn}∞
n=1, which starts with ξ1 := µ, and, for n>1, is a
modular symbol over GLn(Q) (in the sense of Avner Ash and Lee Rudolph) with coefficients in
a group of distributions valued in the Milnor Kn-group of the field of all modular functions of all
levels. As we will see, the key ingredient needed for the construction of ξn is Suslin reciprocity
applied to K3 of the field of meromorphic functions on the universal elliptic curve over the modular
curve Y(N).

This is joint work with Cecilia Busuioc, Jeehoon Park, and Owen Patashnick.

This talk was organised with the support of the London Mathematical Society, the Heilbronn Institute for Mathematical Research, King's College London and UKRI/EPSRC Additional Funding Programme for Mathematical Sciences.





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