Abstract
Let \(K\) be an algebraic function field with constant field \({\mathbb {F}}_q\). Fix a place \(\infty \) of \(K\) of degree \(\delta \) and let \(A\) be the ring of elements of \(K\) that are integral outside \(\infty \). We give an explicit description of the elliptic points for the action of the Drinfeld modular group \(G=GL_2(A)\) on the Drinfeld’s upper half-plane \(\Omega \) and on the Drinfeld modular curve \(G{\setminus }\Omega \). It is known that under the building map elliptic points are mapped onto vertices of the Bruhat–Tits tree of \(G\). We show how such vertices can be determined by a simple condition on their stabilizers. Finally for the special case \(\delta =1\) we obtain from this a surprising free product decomposition for \(PGL_2(A)\).
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Abbreviations
- \({\mathbb {F}}_q\) :
-
The finite field of order \(q\)
- \(K\) :
-
An algebraic function field of one variable with constant field \({\mathbb {F}}_q\)
- \(g(K)\) :
-
The genus of \(K\)
- \(L_K(u)\) :
-
The \(L\)-polynomial of \(K\)
- \(\infty \) :
-
A chosen place of \(K\)
- \(\delta \) :
-
The degree of the place \(\infty \)
- \(A\) :
-
The ring of all elements of \(K\) that are integral outside \(\infty \)
- \(\widetilde{K}\) :
-
The quadratic constant field extension \({\mathbb {F}}_{q^2}K\) of \(K\)
- \(\widetilde{A}\) :
-
\({\mathbb {F}}_{q^2}A\), the integral closure \(A\) in \(\widetilde{K}\)
- \(\nu \) :
-
The additive, discrete valuation of \(K\) defined by \(\infty \)
- \(\pi \) :
-
A local parameter at \(\infty \) in \(K\)
- \(K_{\infty }\) :
-
\(\cong {\mathbb {F}}_{q^{\delta }}((\pi ))\), the completion of \(K\) with respect to \(\infty \)
- \({\mathcal {O}}_{\infty }\) :
-
\(\cong {\mathbb {F}}_{q^{\delta }}[[\pi ]]\), the valuation ring of \(K_\infty \)
- \(C_\infty \) :
-
The completion of an algebraic closure of \(K_\infty \)
- \(\Omega \) :
-
\(=C_\infty -K_\infty \), Drinfeld’s upper half-plane
- \({\mathcal {T}}\) :
-
The Bruhat–Tits tree of \(GL_2(K_\infty )\)
- \(G\) :
-
The group \(GL_2(A)\)
- \(G_w\) :
-
The stabilizer in \(G\) of \(w \in \mathop {{\mathrm {vert}}}({\mathcal {T}}) \cup \mathop {{\mathrm {edge}}}({\mathcal {T}})\)
- \(G_\omega \) :
-
The stabilizer in \(G\) of \(\omega \in \Omega \)
- \(Z\) :
-
The centre of \(G\)
- \({\mathrm {Cl}}(R)\) :
-
The ideal class group of the Dedekind ring \(R\)
- \({\mathrm {Cl}}^0(F)\) :
-
The divisor class group of degree \(0\) of the function field \(F\)
- \(E(G)\) :
-
The elliptic elements of \(G\) on \(\Omega \)
- \(\mathop {{\mathrm {Ell}}}(G)\) :
-
The elliptic points of \(G\) on \(G{\setminus }\Omega \)
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Acknowledgments
This paper is part of a project supported by grant 99-2115-M-001-011-MY2 from the National Science Council (NSC) of Taiwan. The biggest part was written while the second author was working at the Institute of Mathematics at Academia Sinica in Taipei. He wants to thank Julie Tzu-Yueh Wang, Wen-Ching Winnie Li, Jing Yu, Liang-Chung Hsia and Chieh-Yu Chang for help in general as well as for help with the application for that grant. During the final stage the second author was supported by ASARC in South Korea. Several ideas in the paper were developed during research visits of the second author at Glasgow University. The hospitality of their Mathematics Department is gratefully acknowledged. Finally, the second author thanks Ernst-Ulrich Gekeler for helpful discussions during a research visit to Saarbrücken.
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Mason, A.W., Schweizer, A. Elliptic points of the Drinfeld modular groups. Math. Z. 279, 1007–1028 (2015). https://doi.org/10.1007/s00209-014-1400-9
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DOI: https://doi.org/10.1007/s00209-014-1400-9
Keywords
- Drinfeld modular group
- Drinfeld modular curve
- Elliptic point
- Bruhat–Tits tree
- Vertex stabilizer
- Free product