Rank-one Drinfel’d modules on elliptic curves

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...

Torsion bounds for elliptic curves and Drinfeld modules

We derive asymptotically optimal upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L : K]. Our main tool is the adelic openness of the image of Galois representations attached to elliptic curves and Drinfeld modules, due to Serre and Pink-Rütsche, respectively. Our approach...

Drinfeld Modules and Elliptic Sheaves

On the rank of certain parametrized elliptic curves

In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.

On the Torsion of Drinfeld Modules of Rank Two

We prove that the curve Y0(p) has no F2(T )-rational points where p ⊳ F2[T ] is a prime ideal of degree at least 3 and Y0(p) is the affine Drinfeld modular curve parameterizing Drinfeld modules of rank two over F2[T ] of general characteristic with Hecke-type level p-structure. As a consequence we derive a conjecture of Schweizer describing completely the torsion of Drinfeld modules of rank two...