The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains

Abstract

Abstract. Let n be a square-free ideal of F q [T ]. We study the rational torsion subgroup of the Jacobian variety J 0 (n) of the Drinfeld modular curve X 0 (n). We prove that for any prime number ℓ not dividing q(q − 1), the ℓ-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the ℓ-primary part of the cuspidal divisor class group for any prime ℓ not dividing q − 1.

Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

“…For each m ∈ A + with m|N, we let W m : X → X be the Atkin-Lehner involution associated to m (cf. [11, (1.1.5)] in the case (NF), [14,Definition 2.11] in the case (FF)). To ease the notation, we write W w = W m(w) for w ∈ W. Recall that W m restricts to an F -automorphism of C.…”

“…cit. ), and the second follows from [14,Proposition 4.2]. To show (2), we consider a commutative diagram with exact rows…”

“…Note that for e ∈ E we have (3, d(e)) = 3 if and only if e = e H in the case (NF) (see (1.4) for the definition of d(e)). In the case (FF), we have (q + 1, d(e)) = q + 1 for any e = e H , and (q + 1, d(e H )) is a power of two [14,Remark 3.6]. We also put…”

“…Our main result, explained in the next subsection, pinpoints where i fails to be an isomorphism (see Remark 1.2.4). Our proof relies on the study of classical Jacobians due to Ohta [11] and to Papikian-Wei [14]. We also discuss the action of Hecke operators on J(F ) Tor and study its Eisenstein property, see §1.3.…”

“…The referee pointed out that, in view of [14,Remark 4.4], M e admits an interpretation as the kernel of the specialization map M e → s i=1 Φ p i , where Φ p i is the group of components of the reduction of J 0 (N) at p i . A direct proof of this statement would possibly lead to a more geometric proof of Theorem 1.2.3.…”