Legendre Drinfeld modules and universal supersingular polynomials

Drinfeld Modules with No Supersingular Primes

We give examples of Drinfeld modules φ of rank 2 and higher over Fq(T ) that have no primes of supersingular reduction. The idea is to construct φ so that the associated mod ` representations are incompatible with the existence of supersingular primes. We also answer a question of Elkies by proving that such obstructions cannot exist for elliptic curves over number fields. Elkies [El1] proved t...

Addendum to "Factoring polynomials over finite fields with Drinfeld modules"

After my paper [2] was electronically published by Mathematics of Computation, I came across the PhD thesis of professor I. Y. Potemine [6]. In Section 4.3 of his thesis, an algorithm for factoring polynomials is proposed which is equivalent to the algorithm discussed in my paper. Potemine’s algorithm is acknowledged in my PhD thesis [1]. Our algorithms were found independently, both as analogu...

Factoring polynomials over finite fields with Drinfeld modules

In the following, we describe a way of factoring polynomials in Fq[X] with Drinfeld modules. We furthermore analyse the complexity of the algorithm and compare it to the well-known Cantor-Zassenhaus algorithm. 1. Defining Fq[X ]-module structures with Drinfeld modules Throughout this paper we will denote A = Fq[X ], where q is a power of some prime p, and N ∈ A for the polynomial which is to be...

Hecke Modules and Supersingular

Let F be a nonarchimedean local field of odd residual characteristic p. We classify finite-dimensional simple right modules for the pro-pIwahori-Hecke algebra HC(G, I(1)), where G is the unramified unitary group U(2, 1)(E/F ) in three variables. Using this description when C = Fp, we define supersingular Hecke modules and show that the functor of I(1)-invariants induces a bijection between irre...

Drinfeld Modules and Elliptic Sheaves