# Algebra/Number Theory Seminar

The structure of Drinfeld modular forms of level \(\Gamma_0(T)\) and applications.

Tarun Dalal, Indian Institute of Technology, Hyderabad, India

11:00 AM - 12:00 PM

Zoom id 566 385 6457 (no password)

Let \(q\) be a power of an odd prime \(p\). Let \(A:=\mathbb{F}_q[T]\) and \(C\) denote the completion of an algebraic closure of \(\mathbb{F}_q((\frac{1}{T}))\). For any ring \(R\) with \(A \subseteq R \subseteq C\), we let \(M(\Gamma_0(\mathfrak{n}))_R\) denote the ring of Drinfeld modular forms of level \(\Gamma_0(\mathfrak{n})\) with coefficients in \(R\). In 1988, Gekeler showed that the \(C\)-algebra \(M(\mathrm{GL}_2(A))_C\) is isomorphic to \(C[X,Y]\). As a result, the properties of the weight filtration for Drinfeld modular forms for \(\mathrm{GL}_2(A)\) are studied by Gekeler in 1988 and by Vincent in 2010.

In this talk, we discuss about the structure of the \(R\)-algebra \(M(\Gamma_0(T))_R\) and study the properties of the weight filtration for Drinfeld modular forms of level \(\Gamma_0(T)\). As an application, we prove a result on mod-\(\mathfrak{p}\) congruences for Drinfeld modular forms of level \(\Gamma_0(\mathfrak{p} T)\) for \(\mathfrak{p} \neq (T)\). This is a joint work with Narasimha Kumar.

Event contact: dinesh dot thakur at rochester dot edu

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