IMSc Webinar

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

Tarun Dalal

IIT Hyderabad

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Abstract:

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 $\mathfrak{p}
eq (T)$. This is a joint work with Narasimha Kumar.