On component groups of Jacobians of Drinfeld modular curves

Mihran Papikian[1]

  • [1] Stanford University, Department of Mathematics, Stanford, CA 94305 (USA)

Annales de l'Institut Fourier (2004)

  • Volume: 54, Issue: 7, page 2163-2199
  • ISSN: 0373-0956

Abstract

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Let J 0 ( 𝔫 ) be the Jacobian variety of the Drinfeld modular curve X 0 ( 𝔫 ) over 𝔽 q ( t ) , where 𝔫 is an ideal in 𝔽 q [ t ] . Let 0 → B → J 0 ( 𝔫 ) → A → 0 be an exact sequence of abelian varieties. Assume B , as a subvariety of J 0 ( 𝔫 ) , is stable under the action of the Hecke algebra 𝕋 ⊂ End ( J 0 ( 𝔫 ) ) . We give a criterion which is sufficient for the exactness of the induced sequence of component groups 0 → Φ B , ∞ → Φ J , ∞ → Φ A , ∞ → 0 of the Néron models of these abelian varieties over 𝔽 q [ [ 1 t ] ] . This criterion is always satisfied when either A or B is one-dimensional. Moreover, we prove that the sequence of component groups is always exact on ℓ -power torsion for any prime ℓ not dividing ( q - 1 ) . In particular, the sequence is always exact when q = 2 .

How to cite

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Papikian, Mihran. "On component groups of Jacobians of Drinfeld modular curves." Annales de l'Institut Fourier 54.7 (2004): 2163-2199. <http://eudml.org/doc/116171>.

@article{Papikian2004,
abstract = {Let $J_0(\{\mathfrak \{n\}\})$ be the Jacobian variety of the Drinfeld modular curve $X_0(\{\mathfrak \{n\}\})$ over $\{\mathbb \{F\}\}_q(t)$, where $\{\mathfrak \{n\}\}$ is an ideal in $\{\mathbb \{F\}\}_q[t]$. Let $0\rightarrow B \rightarrow J_0(\{\mathfrak \{n\}\}) \rightarrow A \rightarrow 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0(\{\mathfrak \{n\}\})$ , is stable under the action of the Hecke algebra $\{\mathbb \{T\}\} \subset $ End $ (J_0(\{\mathfrak \{n\}\}))$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0 \rightarrow \Phi _\{B, \infty \} \rightarrow \Phi _\{J, \infty \} \rightarrow \Phi _\{A, \infty \} \rightarrow 0$ of the Néron models of these abelian varieties over $\{\mathbb \{F\}\}_q [\![\{1 \over t\}]\!]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that the sequence of component groups is always exact on $\ell $-power torsion for any prime $\ell $ not dividing $(q-1)$. In particular, the sequence is always exact when $q=2$.},
affiliation = {Stanford University, Department of Mathematics, Stanford, CA 94305 (USA)},
author = {Papikian, Mihran},
journal = {Annales de l'Institut Fourier},
keywords = {Component groups; Drinfeld modular curves; monodromy pairing; component groups},
language = {eng},
number = {7},
pages = {2163-2199},
publisher = {Association des Annales de l'Institut Fourier},
title = {On component groups of Jacobians of Drinfeld modular curves},
url = {http://eudml.org/doc/116171},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Papikian, Mihran
TI - On component groups of Jacobians of Drinfeld modular curves
JO - Annales de l'Institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 7
SP - 2163
EP - 2199
AB - Let $J_0({\mathfrak {n}})$ be the Jacobian variety of the Drinfeld modular curve $X_0({\mathfrak {n}})$ over ${\mathbb {F}}_q(t)$, where ${\mathfrak {n}}$ is an ideal in ${\mathbb {F}}_q[t]$. Let $0\rightarrow B \rightarrow J_0({\mathfrak {n}}) \rightarrow A \rightarrow 0$ be an exact sequence of abelian varieties. Assume $B$, as a subvariety of $J_0({\mathfrak {n}})$ , is stable under the action of the Hecke algebra ${\mathbb {T}} \subset $ End $ (J_0({\mathfrak {n}}))$. We give a criterion which is sufficient for the exactness of the induced sequence of component groups $0 \rightarrow \Phi _{B, \infty } \rightarrow \Phi _{J, \infty } \rightarrow \Phi _{A, \infty } \rightarrow 0$ of the Néron models of these abelian varieties over ${\mathbb {F}}_q [\![{1 \over t}]\!]$. This criterion is always satisfied when either $A$ or $B$ is one-dimensional. Moreover, we prove that the sequence of component groups is always exact on $\ell $-power torsion for any prime $\ell $ not dividing $(q-1)$. In particular, the sequence is always exact when $q=2$.
LA - eng
KW - Component groups; Drinfeld modular curves; monodromy pairing; component groups
UR - http://eudml.org/doc/116171
ER -

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