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- W392563441 abstract "Let $frak{p}$ be a prime ideal of $mathbb{F}_q[T]$. Let $J_0(frak{p})$ be the Jacobian variety of the Drinfeld modular curve $X_0(frak{p})$. Let $Phi$ be the component group of $J_0(frak{p})$ at the place $1/T$. We use graph Laplacians to estimate the order of $Phi$ as $mathrm{deg}(frak{p})$ goes to infinity. This estimate implies that $Phi$ is not annihilated by the Eisenstein ideal of the Hecke algebra $mathbb{T}(frak{p})$ acting on $J_0(frak{p})$ once the degree of $frak{p}$ is large enough. We also obtain an asymptotic formula for the size of the discriminant of $mathbb{T}(frak{p})$ by relating this discriminant to the order of $Phi$; in this problem the order of $Phi$ plays a role similar to the Faltings height of classical modular Jacobians. Finally, we bound the spectrum of the adjacency operator of a finite subgraph of an infinite diagram in terms of the spectrum of the adjacency operator of the diagram itself; this result has applications to the gonality of Drinfeld modular curves." @default.
- W392563441 created "2016-06-24" @default.
- W392563441 creator A5021892978 @default.
- W392563441 date "2015-05-26" @default.
- W392563441 modified "2023-10-01" @default.
- W392563441 title "Graph Laplacians, component groups and Drinfeld modular curves" @default.
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- W392563441 doi "https://doi.org/10.17879/35209679465" @default.
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