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NCTS Seminar on Number Theory at Hsinchu
13:30 - 14:30, November 22, 2017 (Wednesday)
Lecture Room B, 4th Floor, The 3rd General Building, NTHU
(清華大學綜合三館 4樓B演講室)
Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms
Shin Hattori (Kyushu University)

Let be a rational prime, a -power and $P$ a non-constant irreducible polynomial in . The notion of Drinfeld modular form is an analogue over of that of elliptic modular form. On the other hand, following the analogy with -adic elliptic modular forms, Vincent defined -adic Drinfeld modular forms as the -adic limits of Fourier expansions of Drinfeld modular forms. Numerical computations suggest that Drinfeld modular forms should enjoy deep -adic structures comparable to the elliptic analogue, while at present their -adic properties are far less well understood than the -adic elliptic case.
In this series of talks, I will explain how basic properties of -adic Drinfeld modular forms are obtained in a geometric way, using the duality theories of Taguchi for Drinfeld modules and finite -modules. Key ingredients are the theory of canonical subgroups of Drinfeld modules with ordinary reduction and Hodge-Tate-Taguchi maps, which give torsion comparison isomorphisms between the etale and de Rham sides.


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