Andreas Schweizer

THE MINIMUM INDEX OF A NON-CONGRUENCE SUBGROUP OF SL2 OVER AN ARITHMETIC DOMAIN

AbstractLetA be an arithmetic Dedekind ring with only finitely many units. It is known that (i)A = ℤ, the ring of rational integers, (ii)A =Od, the ring of integers of the imaginary quadratic field

On the ^{}-torsion of elliptic curves and elliptic surfaces in characteristic

\mathbb{Q}\left( {\sqrt { - d} } \right)

Value-sharing of meromorphic functions on a Riemann surface

, whered is a square-free positive integer, or (iii)A=C=C(C,P,k), the coordinate ring of the affine curve obtained by removing a closed pointP from a smooth projective curveC over afinite fieldk. serre has shown that, in comparison with other low rank arithmetic groups, the groups SL2(A) have “many” non-congruence subgroups.Let ncs (A) denote the smallest index of a non-congruence subgroup of SL2(A). It is well-known that ncs(ℤ) = 7. Grunewald and Schwermer have proved that, with 4 exceptions, ncs(Od) = 2. In this paper we prove that ncs(C)=2, for “most”, but not all,C.

On the torsion of elliptic curves over cubic number fields

Let be a function field of genus with a finite constant field . Choose a place of of degree and let be the arithmetic Dedekind domain consisting of all elements of that are integral outside . An explicit formula is given (in terms of , and ) for the minimum index of a non-congruence subgroup in SL. It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL. The minimum index of a normal non-congruence subgroup is also determined.

On the uniform boundedness conjecture for Drinfeld modules

We study the extension generated by the x-coordinates of the pe-torsion points of an elliptic curve over a function field of characteristic p. If S → C is a non-isotrivial elliptic surface in characteristic p with a p e -torsion section, then for p e >11 our results imply restrictions on the genus, the gonality, and the p-rank of the base curve C, whereas for p e < 11 such a surface can be constructed over any base curve C. We also describe explicitly all occurring p e in the cases where the surface S is rational or K3 or the base curve C is rational, elliptic or hyperelliptic.

Non-standard automorphisms and non-congruence subgroups of SL2 over Dedekind domains contained in function fields

Let K be a function field with constant field k, and let ∞ be a fixed place of K. Let 𝒞 be the Dedekind domain consisting of all those elements of K which are integral outside ∞. The group G = GL 2(𝒞) is important for a number of reasons. For example, when k is finite, it plays a central role in the theory of Drinfeld modular curves. Many properties follow from the action of G on its associated Bruhat–Tits tree, 𝒯. Classical Bass–Serre theory shows how a presentation for G can be derived from the structure of the quotient graph (or fundamental domain) G \ 𝒯. The shape of this quotient graph (for any G) is described in a fundamental result of Serre. However, there are very few known examples for which a detailed description of G \ 𝒯 is known. (One such is the rational case, 𝒞 =k[t], i.e., when K has genus zero, and ∞ has degree one.) In this article, we give a precise description of G \ 𝒯 for the case where the genus of K is zero, K has no places of degree one, and ∞ has degree two. Among the known examples a new feature here is the appearance of vertex stabilizer subgroups (of G) which are of quaternionic type.

Shared values of meromorphic functions on compact Riemann surfaces

We present some results on two meromorphic functions from S to Cˆ sharing a number of values where S is a Riemann surface of one of the following types: compact, compact minus finitely many points, the unit disk, a torus, the complex plane.