## KUMMER THEORY OF DIVISION POINTS OVER DRINFELD MODULES OF RANK ONE (1999)

### BibTeX

@MISC{Chi99kummertheory,

author = {Wen-chen Chi and Anly Li},

title = {KUMMER THEORY OF DIVISION POINTS OVER DRINFELD MODULES OF RANK ONE},

year = {1999}

}

### OpenURL

### Abstract

Abstract. A Kummer theory of division points over rank one Drinfeld A = Fq[T]- modules defined over global function fields was given. The results are in complete analogy with the classical Kummer theory of division points over the multiplicative algebraic group Gm defined over number fields. Let K be a number field and let ¯ K be a fixed algebraic closure of K. For any positive integer n, let µn be the group of n-th roots of unity in ¯ K. Let G(n) = Gal(K(µn)/K). For K = Q, G(n) ∼ = (Z/nZ) ∗ , and for any number field K, G(l) ∼ = (Z/lZ) ∗ for almost all prime numbers l.

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Citation Context ...× ... × ΛM ( r-copies). (ii). If the orders of ΛM and G(M) are relatively prime, for example, M = ∏ P is a product of distinct irreducible polynomials P, P |M then H 2 (G(M), ΛM) = 0 by Cor.(10.2) in =-=[2]-=-. In this case, the orders of HΓ(M) and G(M) are also relatively prime, so H 2 (G(M), HΓ(M)) = 1, where10 WEN-CHEN CHI † AND ANLY LI ‡ G(M) acts on HΓ(M) by conjugation. In particular, the exact sequ... |

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Citation Context ...ixed algebraic closure of k. In this section, we will briefly review some definitions and basic properties of Drinfeld modules of rank one. For a general reference, we should refer to Chapter 3, 4 of =-=[5]-=- and [7]. First, recall that Carlitz makes A act as a ring of endomorphisms on the additive group of ¯ k as follows: Let τ : ¯ k → ¯ k be the Frobenius automorphism defined by τ(α) = αq and let µT be ... |

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Citation Context ...ebraic closure of k. In this section, we will briefly review some definitions and basic properties of Drinfeld modules of rank one. For a general reference, we should refer to Chapter 3, 4 of [5] and =-=[7]-=-. First, recall that Carlitz makes A act as a ring of endomorphisms on the additive group of ¯ k as follows: Let τ : ¯ k → ¯ k be the Frobenius automorphism defined by τ(α) = αq and let µT be the map ... |

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Citation Context ... This proves the assertion. Remarks: (i). Let Γ be a finitely generated A-submodule of (k, +) of rank r. By general theory of modules over principal ideal rings (see [1, Ch.VII, §4]) and Theorem 1 of =-=[11]-=-, Γ is isomorphic to a direct sum of the form A ⊕ ... ⊕ A or A ⊕ ... ⊕ A ⊕ A/(N), where N is a nonzero polynomial. If eM(Γ) = 1, by Corollary 2.11, we have a noncanonical A/(M)-module isomorphism betw... |

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Citation Context ...the whole theory. This establishes the above result (i). For a general rank one Drinfeld A-module, using the A-module structure of HΓ(l) together with the independence property given by L.Denis ( see =-=[4]-=-, Theorem 5), the above result (ii) can be established easily. Our proof is essentially the same as that of Denis in [4] except the above A-module is naturally equipped throughout the whole theory. As... |

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Algebra (3rd
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(Show Context)
Citation Context ...idea is to show that an A-module structure is naturally equipped on HΓ(M) ( or particularly, on HΓ(l)). Then, for the Carlitz module case, a Kummer theory along the line of the classical theory ( see =-=[9]-=-, Ch.VI, Section 11) can be developed with this A-module structure naturally equipped throughout the whole theory. This establishes the above result (i). For a general rank one Drinfeld A-module, usin... |