## Abstract

We present a method to control gonality of nonarchimedean curves based

on graph theory. Let k denote a complete nonarchimedean valued field.We first prove

a lower bound for the gonality of a curve over the algebraic closure of k in terms of

the minimal degree of a class of graph maps, namely: one should minimize over all

so-called finite harmonic graph morphisms to trees, that originate from any refinement

of the dual graph of the stable model of the curve. Next comes our main result: we

prove a lower bound for the degree of such a graph morphism in terms of the first

eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen

as a substitute for graphs of the Li–Yau inequality from differential geometry, although

we also prove that the strict analogue of the original inequality fails for general graphs.

Finally,we apply the results to give a lower bound for the gonality of arbitraryDrinfeld

modular curves over finite fields and for general congruence subgroups Γ of Γ (1) that

is linear in the index [Γ (1) : Γ ], with a constant that only depends on the residue field

degree and the degree of the chosen “infinite” place. This is a function field analogue

of a theorem of Abramovich for classical modular curves. We present applications

to uniform boundedness of torsion of rank two Drinfeld modules that improve upon

existing results, and to lower bounds on the modular degree of certain elliptic curves

over function fields that solve a problem of Papikian.

on graph theory. Let k denote a complete nonarchimedean valued field.We first prove

a lower bound for the gonality of a curve over the algebraic closure of k in terms of

the minimal degree of a class of graph maps, namely: one should minimize over all

so-called finite harmonic graph morphisms to trees, that originate from any refinement

of the dual graph of the stable model of the curve. Next comes our main result: we

prove a lower bound for the degree of such a graph morphism in terms of the first

eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen

as a substitute for graphs of the Li–Yau inequality from differential geometry, although

we also prove that the strict analogue of the original inequality fails for general graphs.

Finally,we apply the results to give a lower bound for the gonality of arbitraryDrinfeld

modular curves over finite fields and for general congruence subgroups Γ of Γ (1) that

is linear in the index [Γ (1) : Γ ], with a constant that only depends on the residue field

degree and the degree of the chosen “infinite” place. This is a function field analogue

of a theorem of Abramovich for classical modular curves. We present applications

to uniform boundedness of torsion of rank two Drinfeld modules that improve upon

existing results, and to lower bounds on the modular degree of certain elliptic curves

over function fields that solve a problem of Papikian.

Original language | English |
---|---|

Pages (from-to) | 211-258 |

Number of pages | 48 |

Journal | Mathematische Annalen |

Volume | 361 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2015 |

## Keywords

- 05C50
- 11G09
- 11G18
- 11G30
- 14G05
- 14G22
- 14H51