On finite Drinfeld modules.

*(English)*Zbl 0731.11034The 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.

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.

Reviewer: David Goss (Columbus/Ohio)

##### 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 |

##### Keywords:

abelian varieties over finite fields; \(\ell\)-adic Tate modules; Riemann Hypothesis; Drinfeld module; isogeny; global zeta-functions
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##### References:

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