Linear equations on Drinfeld modules

Introduction to Drinfeld Modules

(1) Explicit class field theory for global function fields (just as torsion of Gm gives abelian extensions of Q, and torsion of CM elliptic curves gives abelian extension of imaginary quadratic fields). Here global function field means Fp(T ) or a finite extension. (2) Langlands conjectures for GLn over function fields (Drinfeld modular varieties play the role of Shimura varieties). (3) Modular...

Rank-one Drinfeld Modules on Elliptic Curves

The sgn-normalized rank-one Drinfeld modules 4> associated with all elliptic curves E over ¥q for 4 < q < 13 are computed in explicit form. (Such 4> for q < 4 were computed previously.) These computations verify a conjecture of Dormán on the norm of j{) = aq+l and also suggest some interesting new properties of . We prove Dorman's conjecture in the ramified case. We also prove the formula...

Drinfeld Modules and Elliptic Sheaves

Siegel’s Theorem for Drinfeld Modules

We prove a Siegel type statement for finitely generated φsubmodules of Ga under the action of a Drinfeld module φ. This provides a positive answer to a question we asked in a previous paper. We also prove an analog for Drinfeld modules of a theorem of Silverman for nonconstant rational maps of P over a number field.

Integral Points for Drinfeld Modules

We prove that in the backward orbit of a nonpreperiodic (nontorsion) point under the action of a Drinfeld module of generic characteristic there exist at most finitely many points S-integral with respect to another nonpreperiodic point. This provides the answer (in positive characteristic) to a question raised by Sookdeo in [26]. We also prove that for each nontorsion point z there exist at mos...