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On finite Drinfeld modules. (English) Zbl 0731.11034
The theory of abelian varieties over finite fields is a cornerstone of modern arithmetic. As is well-known, one has a very satisfactory theory of the number of points of the abelian variety as well as its isogeny class in terms of the characteristic polynomial of the action of Frobenius on \(\ell\)-adic Tate modules (or, \(H^ 1_{\acute et})\). Moreover, each such polynomial is constrained by the fact that its roots must satisfy the Riemann Hypothesis; results established through a deep study of the endomorphism ring of the variety.
Let \(A\) be the affine ring of a function field \(k\) (defined over \(\mathbb F_ r\), \(r=p^ m)\) minus a place \(\infty\). Let \(\wp \in Spec(A)\) and let \(L\) be a finite extension of \(A/\wp\). Let \(\phi\) be a Drinfeld module over \(L\) – a “finite Drinfeld module” in the author’s parlance. The group of \(L\)-rational points of \(\phi\) is \(\mathbb G_ a(L)\simeq L\) with an exotic \(A\)-action; thus it is a finite \(A\)-module. The author gives a self-contained study of such Drinfeld modules based on the analogy with abelian varieties. For instance, one has analogues of the Riemann Hypothesis, Tate’s isogeny theory etc.. In particular, the author uses Euler-Poincaré characteristic ideals as a suitable replacement for the order of a finite abelian group. To such characteristics the author attaches a zeta-function \(Z_{\phi}(t)\in k(t)\). The characteristic polynomial of Frobenius gives an expression for these E-P characteristics; the author has recently shown that this expression, together with \(\text{rank}(\phi)\), determines the \(Z\)-function and thus the isogeny class of \(\phi\). (The subtlety is that one cannot simply exponentiate in characteristic \(p\)!).
Finally the author shows how to use these “local” zeta-functions as Euler factors to obtain global zeta-functions. (N.B: The author mentions that there may be some difficulty when \(A\) is not a polynomial ring. This is not the case. As “\(f^ s\)” takes values in characteristic \(p\), one is able to lift (in a perhaps non-canonical way) exponentiation of monics to arbitrary fractional ideals.) The theory of Drinfeld modules over global bases does not obviously follow the analogy with classical theory (even the naive form of the weak Mordell-Weil theorem is false!). Thus it is encouraging that the finite theory works so well.

MSC:
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
14K15 Arithmetic ground fields for abelian varieties
14K22 Complex multiplication and abelian varieties
11G20 Curves over finite and local fields
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G15 Finite ground fields in algebraic geometry
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References:
[1] Bourbaki, N., Algèbre, (), Chap. 8 · Zbl 0455.18010
[2] Deligne, P.; Husemöller, D., Survey of Drinfeld modules, Contemp. math., 67, 25-91, (1987)
[3] Denert, M., Affine and projective orders in central simple algebras over global function fields, Ph.D. thesis, (1987), Gent
[4] Denert, M.; Van Geel, J., The class number of hereditary orders in non-eichler algebras over global function fields, Math. ann., 282, 379-393, (1988) · Zbl 0627.16003
[5] Deuring, M., Die typen der multiplikatorenringe elliptischer funktionenkörper, (), 197-272 · JFM 67.0107.01
[6] Drinfeld, V.G.; Drinfeld, V.G., Elliptic modules, Math. sb., Math. USSR-sb., 23, 561-592, (1976), English translation: · Zbl 0321.14014
[7] Drinfeld, V.G., Elliptic modules, II, Math. USSR-sb., 31, 159-170, (1977) · Zbl 0386.20022
[8] Gekeler, E.-U., Zur arithmetik von Drinfeld-moduln, Math. ann., 262, 167-182, (1983) · Zbl 0536.14028
[9] Gekeler, E.-U., Über Drinfeld’sche modulkurven vom Hecke-typ, Compositio math., 57, 219-236, (1986) · Zbl 0599.14032
[10] Gekeler, E.-U., Drinfeld modular curves, () · Zbl 0848.11029
[11] Goss, D., On a new type of L-function for algebraic curves over finite fields, Pacific J. math., 105, 143-181, (1983) · Zbl 0571.14010
[12] Hayes, D., Explicit class field theory in global function fields, () · Zbl 0476.12010
[13] Honda, T., Isogeny classes of abelian varieties over finite fields, J. math. soc. Japan, 20, 83-95, (1968) · Zbl 0203.53302
[14] Reiner, I., Maximal orders, (1975), Academic Press London/New York/San Francisco · Zbl 0305.16001
[15] Rosen, M., The Hilbert class field in function fields, Expositio math., 5, 365-378, (1987) · Zbl 0632.12017
[16] Takahashi, T., Good reduction of elliptic modules, J. math. soc. Japan, 34, 475-487, (1982) · Zbl 0476.14010
[17] Tate, J., Endomorphisms of abelian varieties over finite fields, Invent. math., 2, 134-144, (1966) · Zbl 0147.20303
[18] Gekeler, E.-U., Sur la géométrie de certaines algèbres de quaternions, Sém. théorie nombres, Bordeaux, 2, 143-153, (1990) · Zbl 0716.11056
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