Minimal models for Drinfeld modules of rank 2 with complex multiplication

2000 ◽  
Vol 74 (3) ◽  
pp. 192-200
Author(s):  
S. Bae ◽  
D. Jeon
Keyword(s):  
Complex Multiplication ◽  
Drinfeld Modules ◽  
Minimal Models ◽  
Rank 2

Torsion Points of Drinfeld Modules

1995 ◽  
Vol 38 (1) ◽  
pp. 3-10
Author(s):  
Sunghan Bae ◽  
Jakyung Koo
Keyword(s):  
Drinfeld Module ◽  
Drinfeld Modules ◽  
Locally Compact ◽  
Torsion Points ◽  
Complete Field ◽  
Rank 2

AbstractThe finiteness of K-rational torsion points of a Drinfeld module of rank 2 over a locally compact complete field K with a discrete valuation is proved.

scholarly journals Algebraic relations among periods and logarithms of rank 2 Drinfeld modules

2011 ◽  
Vol 133 (2) ◽  
pp. 359-391 ◽  
Author(s):  
Chieh-Yu Chang ◽  
Matthew A. Papanikolas
Keyword(s):  
Drinfeld Modules ◽  
Rank 2

GAUSSIAN LAWS ON DRINFELD MODULES

2009 ◽  
Vol 05 (07) ◽  
pp. 1179-1203 ◽  
Author(s):  
WENTANG KUO ◽  
YU-RU LIU
Keyword(s):  
Finite Field ◽  
Polynomial Ring ◽  
Valuation Ring ◽  
Finite Extension ◽  
Drinfeld Modules ◽  
Good Reduction ◽  
Prime Divisors ◽  
Frobenius Automorphism ◽  
Tate Module ◽  
Rank 2

Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, write [Formula: see text], where [Formula: see text] is the valuation ring of 𝔓 and [Formula: see text] its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with [Formula: see text], we prove that there exists a normal distribution for the quantity [Formula: see text] For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ and obtain similar results.

EXPLICIT SURJECTIVITY RESULTS FOR DRINFELD MODULES OF RANK 2

2017 ◽  
Vol 234 ◽  
pp. 17-45 ◽  
Author(s):  
IMIN CHEN ◽  
YOONJIN LEE
Keyword(s):  
Elliptic Curves ◽  
Upper Bound ◽  
Riemann Hypothesis ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Drinfeld Module ◽  
Torsion Points ◽  
Rank 2 ◽  
Invent Math

Let $K=\mathbb{F}_{q}(T)$ and $A=\mathbb{F}_{q}[T]$. Suppose that $\unicode[STIX]{x1D719}$ is a Drinfeld $A$-module of rank $2$ over $K$ which does not have complex multiplication. We obtain an explicit upper bound (dependent on $\unicode[STIX]{x1D719}$) on the degree of primes ${\wp}$ of $K$ such that the image of the Galois representation on the ${\wp}$-torsion points of $\unicode[STIX]{x1D719}$ is not surjective, in the case of $q$ odd. Our results are a Drinfeld module analogue of Serre’s explicit large image results for the Galois representations on $p$-torsion points of elliptic curves (Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331; Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323–401.) and are unconditional because the generalized Riemann hypothesis for function fields holds. An explicit isogeny theorem for Drinfeld $A$-modules of rank $2$ over $K$ is also proven.

scholarly journals Explicit Weil-pairing for Drinfeld modules

2021 ◽  
pp. 1-22
Author(s):  
Jeffrey Katen
Keyword(s):  
Elementary Proof ◽  
Drinfeld Modules ◽  
Weil Pairing ◽  
Explicit Formulas ◽  
Rank 2

The goal of this paper is to define an analogue of the Weil-pairing for Drinfeld modules using explicit formulas and to deduce its main properties from these formulas. Our result generalizes the formula given for rank 2 Drinfeld modules by van der Heiden and works as a more explicit, elementary proof of the Weil-pairing’s existence given by van der Heiden.

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