Formal Drinfeld modules. (Q1421292)

The author develops the theory of formal Drinfeld modules. The definition is similar to the definition of a Drinfeld module; the main feature is that the valuation ring \(R\) of a local field of positive characteristic acts by power series in the Frobenius on some \(R\)-algebra. The most interesting instance of such an object arises naturally as follows. Let \(\rho\) be a Drinfeld \(A\)-module on a finite extension \(E\) of the quotient field of \(A\). Let \(P\) be a prime ideal of \(A\), and let \(\widehat{A_P}\) be the completion of \(A\) in the \(P\)-adic topology. If \(\rho\) has stable reduction at some place \(w\) of \(E\) above \(P\), one can extend \(\rho\) to a formal Drinfeld \(\widehat{A_P}\)-module \(\widehat{\rho}\) on the completion \(E_w\). This formal completion \(\widehat{\rho}\) contains information on the torsion of \(\rho\) (in a similar way as the formal group of an elliptic curve at a finite prime contains information on the torsion of the elliptic curve). As an application the author proves a partial result on the (still open) uniform boundedness conjecture for Drinfeld \({\mathbb F}_q[T]\)-modules \(\rho\) of rank \(2\) on a fixed finite extension \(E\) of \({\mathbb F}_q(T)\), namely: Fix a finite set \(S\) of primes of \({\mathbb F}_q[T]\). Then the torsion of \(\rho\) can be bounded by a constant that depends only on \(E\), \(S\) and the minimal pole order of the \(j\)-invariant of \(\rho\) at the places of \(E\) above \(S\).