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Some Results on Finite Drinfeld Modules 1957 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 1955{1961 S 0002-9939(98)04337-8 SOME RESULTS ON FINITE DRINFELD MODULES CHIH-NUNG HSU (Communicated by William W. Adams) Abstract. Let K be a global function field, a degree one prime divisor of K and let A be the Dedekind domain of functions1 in K regular outside .Let 1 H be the Hilbert class field of A, B the integral closure of A in H. Let be a rank one normalized Drinfeld A -module and let P be a prime ideal in B. We explicitly determine the finite A-module structure of (B =PN ). In particular, if K = Fq(t), q is an odd prime number and is the Carlitz Fq[t]-module, then N the finite Fq[t]-module (Fq [t]=P ) is always cyclic. 1. Introduction N N Recall that Gm(Z=p Z)=(Z=p Z)× is always cyclic except for the case that N p =2andN 3; if N 3, then (Z=2 Z)× is the direct product of two cyclic ≥ ≥ N 2 groups, one of order 2, the other of order 2 − .LetXbe a smooth, projective, geometrically connected curve defined over the finite field Fq with q elements and let be a rational point on X. We set K to be the function field of X over Fq and∞ A K to be the Dedekind domain of functions regular outside . We will consider⊂ the Drinfeld A-modules. From the view point of class field theory,∞ these modules are interesting arithmetic objects over function fields. In particular, the rank one Drinfeld A-modules play a role entirely analogous to the important role played by Gm over number fields. This naturally leads us to explore an analogous phenomenon for rank one Drinfeld A-modules. Let K be the completion of K with respective to and let C be the comple- ∞ ∞ ∞ tion of the algebraic closure of K with respect to .LetC τ =End (Ga=C ) ∞ ∞ ∞{ } Fq ∞ be the twisted polynomial ring in the qth power Frobenius mapping τ. A rank one Drinfeld A-module over C is an injective ring homomorphism :A C τ ∞ → ∞{ } such that the constant coefficient of a(τ)isequaltoaand degτ a(τ)= Ord a − ∞ for all a A. A sign-function (cf. [5] and [3]) sgn : K× Fq× is a co-section of the ∈ ∞ → inclusion map Fq× , K× such that sgn(α) = 1 for all α K× with α 1 vanishing at . A Drinfeld→ A-module∞ of rank one over C is∈ said∞ to be sgn-normalized− ∞ ∞ if the leading coefficient of a(τ)inτis equal to sgn(a) for all 0 = a A. It is known [5] that any Drinfeld A-module of rank one over C is isomorphic6 ∈ to a sgn-normalized A-module over H, where H is the Hilbert∞ class field of A, i.e., H is the maximal abelian extension of K such that the extension H = K completely splits over and is unramified over every finite place of K. ∞ Received by the editors July 23, 1996 and, in revised form, December 26, 1996. 1991 Mathematics Subject Classification. Primary 11G09; Secondary 11A05. Key words and phrases. Drinfeld modules, Hilbert class field. c 1998 American Mathematical Society 1955 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1956 CHIH-NUNG HSU Let B be the integral closure of A in H and let be a rank one sgn-normalized Drinfeld A-module. Let M be an ideal in B. Via the action of (mod M), B =M becomes a finite A-module. We denote this by (B =M). This finite A-module N plays the role of Gm(Z=p Z). The purpose of this note is to determine the A- module structure of (B =M). It is sufficient to consider the case (B =PN ), where P is a prime ideal in B. The structure of (B =PN ) is obtained in Theorems 2:1 and 2:2. In particular, if is the Carlitz Fq[t]-module ([3], chapter 3) and q =2, then (B =PN ) is always cyclic (cf. Corollaries 2:1and2:2). The Carlitz case6 is completely analogous to the classical case. N In section 3, we discuss the relations between BP and lim (B =P ) via the exponential and logarithm functions of the sgn-normalized Drinfeld←− module . 2. The structure of (B =PN ) Let the notation X; Fq; ; K; A; H; B and sgn be as in the introduction. If M is ∞ a commutative Fq-algebra, we let M τ denote the composition ring of Frobenius polynomials in τ,whereτis the qth {power} mapping. From now on, we let be a sgn-normalized rank one Drinfeld A-module over H, i.e., :A H τ is a rank one Drinfeld A-module over H such that for any a A, the leading→ { coefficient} of ∈ a(τ) is equal to sgn(a). It is known that a(τ) B τ for all a A. Thus, via , B becomes an A-module. We denote this∈ module{ } by (B)∈ and denote a the action a(τ)(b)byb for all a A;b B. Let M be an ideal in B. Since (A) B τ , it follows that, via ∈(mod M∈), B =M becomes a finite A-module. We denote⊂ { this} finite module by (B =M). If the decomposition of M is equal to N1 N2 NL P1 P2 PL ,whereP1;P2; ;PL are prime ideals in B, then, by the Chinese remainder··· theorem, we have ··· L (B = ) (B = Ni ): M ∼= Pi i=1 M Thus to determine the A-module structure of (B =M), it is sufficient to consider the case (B =PN ), where P is a prime ideal in B. Let } = P A and let f be the dimension of the vector space B =P over A =}. It follows from∩ class field theory H f f that NormKP = } is a principal ideal in A. We let } =(π}) for the unique element π} A with sgn(π}) = 1. It is known that (B =P) is a cyclic A-module with Euler-Poincar´∈ e characteristic π 1 (cf. [3], chapter 4), i.e., as A-module, } − (B =P) = A =(π 1): ∼ } − We let }(τ) be the monic generator of the left ideal of H τ generated by a(τ) for all a }. We also denote the polynomial (τ)(x)inxby{ x}} for all x H. The ∈ } ∈ important property of the polynomial }(τ), } A a prime ideal, is the following (cf. [5], Proposition 11.4): ⊂ } f(x)= }(τ)(x)=x = x =x is an Eisenstein polynomial over B at any prime ideal P above }.Letc =f(0) B. } ∈ Then we have degP c} =1.If}is a rational point on the curve X (i.e., } is a 0 1 prime ideal in A of degree one) defined over F2,then }(τ)=c}τ +τ B τ , i.e., x} = (τ)(x)=c x+x2,wherec Pbut c P2. ∈ { } } } }∈ } 6∈ Lemma 2.1. Suppose that N is a positive integer, P and } are as above. Then N } } N+1 for any b1;b2 (B),ifb1 b2 (mod P ),thenb b (mod P ): ∈ ≡ 1 ≡ 2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use SOME RESULTS ON FINITE DRINFELD MODULES 1957 N Proof. We may write b1 = b 2 + x for some x P . Then, by the Eisenstein polynomial property, ∈ } } } } } N+1 b =(b2+x) =b +x b (mod P ): 1 2 ≡ 2 Lemma 2.2. Suppose that P, } and c} are as above. Then: N 2 } − N 2 N (1) If N 2 is a positive integer, then x x c − (mod P ) for all x P ≥ ≡ · } ∈ except for the case that } is a rational point on the curve X defined over F2. (2) If } is a rational point on the curve X defined over F2 (i.e., } is a prime ideal of A of degree one) and N 3 is a positive integer, then ≥ N 3 } − N 3 N x x c − (mod P ) ≡ · } for all x P2. ∈ (3) If } is a rational point on the curve X defined over F2, then for all x P;x 0;c (mod P2), we have x} P2 but x} P3. ∈ 6≡ } ∈ 6∈ Proof. Assertion (1) is obvious for N = 2. Now suppose that it is true for N 2, i.e., ≥ N 2 } − N 2 N x x c − (mod P ): ≡ · } We show that it is true for N + 1. Applying Lemma 2:1 and the Eisenstein poly- nomial property, we have N 2 deg } } − } N 2 } N 2 N 2 q N+1 (x ) (x c − ) c (x c − )+(x c − ) (mod P ): ≡ · } ≡ } · · } · } If q =2orq=2and}is not a rational point on the curve X over F2 (i.e., deg }6 N 2 qdeg } N+1 q 3),thenwehave(x c − ) P . Hence ≥ · } ∈ N 1 } − N 1 N+1 x xc − (mod P ): ≡ } This completes the proof of (1). The proof of (2) is similar. To prove (3), since x P and deg c =1,x} =c x+x2 =x(x+c ) P2. ∈ P } }· } ∈ The assertion x} P3 follows from the facts that the characteristic of B is 2 and x 0;c (mod P6∈2). 6≡ } Lemma 2.3. Suppose that P and } are as above. Then: (1) If N 2 is a positive integer and x P;x P2, then the submodule x of (B≥=PN ) generated by x = x (mod∈ PN ) 6∈ (B =PN ) is isomorphich toi N 1 ∈ A =} − except for the case when } is a rational point on the curve X defined over F2.
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