Elliptic points of the Drinfeld modular groups

Abstract

Let K be an algebraic function field with constant field F q . Fix a place ∞ of K of degree δ and let A be the ring of elements of K that are integral outside ∞. 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 Ω and on the Drinfeld modular curve G\Ω. 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 δ = 1 we obtain from this a surprising free product decomposition for P GL 2 (A).

Quasi-inner automorphisms of Drinfeld modular groups

“…Suppose now that δ is even and that C is a cyclic subgroup of order q 2 − 1 (maximally finite or not). Then from the proof of [MS4,Proposition 2.3] it follows that C fixes µ ∈ K.F q 2 \K. In this case however µ ∈ K ∞ as δ is even.…”

“…If H is a finite index subgroup of G, the quotient space H\Ω will, after adding Cusp(H), be the C ∞ -analog of a compact Riemann surface, which is called the Drinfeld modular curve associated with H. Moreover, in the covering of Drinfeld modular curves induced by the natural map H\Ω → G\Ω ramification can only occur above the cusps and elliptic points of G. Also, for (classical and Drinfeld) modular forms, analyticity at the cusps and elliptic points requires special care. This paper is a continuation and extension of [MS4] which is concerned with the elliptic points of G. There the starting point [Ge,p.51] is the existence of a bijection between Ell(G) and ker N, where N : Cl( A) → Cl(A) is the norm map and A = A.F q 2 . It can be shown [MS4] that Cl( A) 2 ∩ ker N, the 2-torsion subgroup of ker N, is in bijection with Ell(G) = = {Gω : ω ∈ E(G), Gω = Gω}, where ω, the conjugate of ω, is the image of ω under the Galois automorphism of K.F q 2 /K.…”

“…This paper is a continuation and extension of [MS4] which is concerned with the elliptic points of G. There the starting point [Ge,p.51] is the existence of a bijection between Ell(G) and ker N, where N : Cl( A) → Cl(A) is the norm map and A = A.F q 2 . It can be shown [MS4] that Cl( A) 2 ∩ ker N, the 2-torsion subgroup of ker N, is in bijection with Ell(G) = = {Gω : ω ∈ E(G), Gω = Gω}, where ω, the conjugate of ω, is the image of ω under the Galois automorphism of K.F q 2 /K. (In [MS4] Ell(G) = is denoted by Ell(G) 2 .)…”

“…It can be shown [MS4] that Cl( A) 2 ∩ ker N, the 2-torsion subgroup of ker N, is in bijection with Ell(G) = = {Gω : ω ∈ E(G), Gω = Gω}, where ω, the conjugate of ω, is the image of ω under the Galois automorphism of K.F q 2 /K. (In [MS4] Ell(G) = is denoted by Ell(G) 2 .) Here we show that, when δ is odd, Cl(A) 2 and the 2-torsion in ker N are isomorphic.…”

From the Carlitz Exponential to Drinfeld Modular Forms

“…The interesting point in this discussion is that if we set L = K ∞ = F((π)), A = H 0 (C\{∞}, O C ) ⊂ K ∞ , F = C ∞ etc. the group PGL 2 (A) acts on Ω = P 1,an F \P 1,an K∞ but the action is in general not free; there usually are elliptic points (this happens, for instance, when [F : F q ] is odd, see [44]). Even more seriously, the group itself is not finitely generated (see Serre's book [56] for more details), so that PGL 2 (A) is not a Schottky group.…”