Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

We consider the generalized Jacobian {\widetilde{J}} of the modular curve {X_{0}(N)} of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the {\mathbb{Q}} -rational torsion points on {\widetilde{J}} up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on {\widetilde{J}} and its Eisenstein property.

Good families of Drinfeld modular curves

In this paper, we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach for obtaining explicit defining equations for some of these towers and, in particular, give a new explicit example of an optimal tower over a quadratic finite field.

On the rational K2 of a curve of GL2 type over a global field of positive characteristic

If is an integral model of a smooth curve X over a global field k, there is a localization sequence comparing the K-theory of and X. We show that K1 () injects into K1(X) rationally, by showing that the previous boundary map in the localization sequence is rationally a surjection, for X of “GL2 type” and k of positive characteristic not 2. Examples are given to show that the relative G1 term can have large rank. Examples of such curves include non-isotrivial elliptic curves, Drinfeld modular curves, and the moduli of -elliptic sheaves of rank 2.

THE STABILIZERS IN A DRINFELD MODULAR GROUP OF THE VERTICES OF ITS BRUHAT–TITS TREE: AN ELEMENTARY APPROACH

Let K be an algebraic function field of one variable with constant field k and let [Formula: see text] be the Dedekind domain consisting of all those elements of K which are integral outside a fixed place ∞ of K. When k is finite the group [Formula: see text] plays a central role in the theory of Drinfeld modular curves analogous to that played by SL2(ℤ) in the classical theory of modular forms. When k is finite (respectively, infinite) we refer to a group [Formula: see text] as an arithmetic (respectively, non-arithmetic) Drinfeld modular group. Associated with [Formula: see text] is its Bruhat–Tits tree, [Formula: see text]. The structure of the group is derived from that of the quotient graph [Formula: see text]. Using an elementary approach which refers explicitly to matrices we determine the structure of all the vertex stabilizers of [Formula: see text]. This extends results of Serre, Takahashi and the authors. We also determine all possible valencies of the vertices of [Formula: see text] for the important special case where ∞ has degree 1.

K1 of products of Drinfeld modular curves and special values of L-functions

Beilinson [Higher regulators and values of L-functions, Itogi Nauki i Tekhniki Seriya Sovremennye Problemy Matematiki Noveishie Dostizheniya (Current problems in mathematics), vol. 24 (Vserossiisky Institut Nauchnoi i Tekhnicheskoi Informatsii, Moscow, 1984), 181–238] obtained a formula relating the special value of the L-function of H2 of a product of modular curves to the regulator of an element of a motivic cohomology group, thus providing evidence for his general conjectures on special values of L-functions. In this paper we prove a similar formula for the L-function of the product of two Drinfeld modular curves, providing evidence for an analogous conjecture in the case of function fields.

ON THE EISENSTEIN IDEAL OF DRINFELD MODULAR CURVES

Let 𝔈(𝔭) denote the Eisenstein ideal in the Hecke algebra 𝕋(𝔭) of the Drinfeld modular curve X0(𝔭) parameterizing Drinfeld modules of rank two over 𝔽q[T] of general characteristic with Hecke level 𝔭-structure, where 𝔭 ◃ 𝔽q[T] is a non-zero prime ideal. We prove that the characteristic p of the field 𝔽q does not divide the order of the quotient 𝕋(𝔭)/𝔈(𝔭) and the Eisenstein ideal 𝔈(𝔭) is locally principal.