The Drinfeld Modular Jacobian J 1 ( n ) has connected fibers

Sreekar M. Shastry[1]

  • [1] Tata Institute of Fundamental Research School of Mathematics Dr Homi Bhabha Rd Mumbai 400 005 (India)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 4, page 1217-1252
  • ISSN: 0373-0956

Abstract

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We study the integral model of the Drinfeld modular curve X 1 ( n ) for a prime n ∈ 𝔽 q [ T ] . A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod n . A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order n in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of X 1 ( n ) which, after contractions in the special fiber, gives a regular model with geometrically integral fiber over n . Thus the mod n component group of J 1 ( n ) is trivial, i.e.  J 1 ( n ) has connected fibers.

How to cite

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Shastry, Sreekar M.. "The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers." Annales de l’institut Fourier 57.4 (2007): 1217-1252. <http://eudml.org/doc/10256>.

@article{Shastry2007,
abstract = {We study the integral model of the Drinfeld modular curve $X_1(n)$ for a prime $n\in \mathbb\{F\}_q[T]$. A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod $n$. A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order $n$ in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of $X_1(n)$ which, after contractions in the special fiber, gives a regular model with geometrically integral fiber over $n$. Thus the mod $n$ component group of $J_1(n)$ is trivial, i.e. $J_1(n)$ has connected fibers.},
affiliation = {Tata Institute of Fundamental Research School of Mathematics Dr Homi Bhabha Rd Mumbai 400 005 (India)},
author = {Shastry, Sreekar M.},
journal = {Annales de l’institut Fourier},
keywords = {Component groups; Drinfeld modular curves; Igusa curves; component groups},
language = {eng},
number = {4},
pages = {1217-1252},
publisher = {Association des Annales de l’institut Fourier},
title = {The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers},
url = {http://eudml.org/doc/10256},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Shastry, Sreekar M.
TI - The Drinfeld Modular Jacobian $J_1(n)$ has connected fibers
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 4
SP - 1217
EP - 1252
AB - We study the integral model of the Drinfeld modular curve $X_1(n)$ for a prime $n\in \mathbb{F}_q[T]$. A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod $n$. A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order $n$ in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of $X_1(n)$ which, after contractions in the special fiber, gives a regular model with geometrically integral fiber over $n$. Thus the mod $n$ component group of $J_1(n)$ is trivial, i.e. $J_1(n)$ has connected fibers.
LA - eng
KW - Component groups; Drinfeld modular curves; Igusa curves; component groups
UR - http://eudml.org/doc/10256
ER -

References

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