Definition 64.31.1 (03VS)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 64: The Trace Formula
Section 64.31: Automorphic forms and sheaves
Definition 64.31.1 (cite)

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Definition 64.31.1. An unramified cusp form on $\text{GL}_2(\mathbf{A})$ with values in $\Lambda$ ¹ is a function

$f : \text{GL}_2(\mathbf{A}) \to \Lambda$

such that

$f(x\gamma ) = f(x)$ for all $x\in \text{GL}_2(\mathbf{A})$ and all $\gamma \in \text{GL}_2(K)$
$f(ux) = f(x)$ for all $x\in \text{GL}_2(\mathbf{A})$ and all $u\in \text{GL}_2(O)$
for all $x\in \text{GL}_2(\mathbf{A})$ ,

$\int _{\mathbf{A} \mod K} f \left(x \left( \begin{matrix} 1 & z \\ 0 & 1 \end{matrix} \right) \right) dz = 0$

see for an explanation of how to make sense out of this for a general ring $\Lambda$ in which $p$ is invertible.

[1] This is likely nonstandard notation.

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