Higher rank case of Dwork’s conjecture.

*(English)*Zbl 1086.11030Let \(X\) be an algebraic variety over a finite field \(\mathbb{F}_q\) of characteristic \(p> 0\). One of the famous Weil conjectures was that the zeta function \(\zeta(X,T)\) of \(X\), which is a priori defined as a certain power series, is in fact a rational function in \(T\). This conjecture was first proved by B. Dwork [Am. J. Math. 82, 631–648 (1960; Zbl 0173.48501)] via \(p\)-adic analysis. Somewhat later, A. Grothendieck gave a different proof based on \(\ell\)-adic cohomology for \(\ell\neq p\).

In the sequel, B. Dwork began the study of the variation of the zeta function within an algebraic family of varieties over \(\mathbb{F}_q\), which led him to the investigation of certain other zeta functions, the so-called unit root zeta functions. These functions are analytic in nature, and generally no longer rational functions, which makes the understanding of their \(p\)-adic analytic properties especially important. In this context, B. Dwork conjectured that a unit root zeta function must be \(p\)-adically meromorphic, i.e., it should be expressible as a quotient of two power series which are convergent everywhere on the completion of \(\mathbb{Q}_p\). In the early 1970s, Dwork himself was able to prove this conjecture of his for special families of curves and surfaces, but a general proof of this hard conjecture had to wait until D. Wan’s work under review was completed.

In the meantime, it was figured out that unit root zeta functions can be interpreted as the \(L\)-functions of \(F\)-crystals on the base space of the respective family of varieties, and this has finally proved to be the right approach to tackle Dwork’s conjecture in full generality. The paper under review, which is the first of two consecutive articles, provides a complete proof of Dwork’s conjecture in the so-called higher rank case, whereas the subsequent paper [cf.: D. Wan, J. Am. Math. Soc. 13, No. 4, 853–908 (2000; Zbl 1086.11031)] accomplishes the remaining case, the so-called rank one case.

The author has outlined his ingenious approach to Dwork’s conjecture in two foregoing articles entitled “A quick introduction to Dwork’s conjecture” [Fried, Michael D. (ed.), Applications of curves over finite fields. 1997 AMS-IMS-SIAM joint summer research conference, July 27–31, 1997, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 245, 147–163 (1999; Zbl 0977.11028)] and “Dwork’s conjecture on unit root zeta functions” [Ann. Math. (2) 150, No. 3, 867–927 (1999; Zbl 1013.11031)], respectively, in which he formulated this conjecture in a more general context, namely in terms of his newly established “\(\sigma\)-modules” and “overconvergent power series rings over complete discrete valuation rings”. Wan’s \(\sigma\)-modules are suitable generalizations of \(F\)-crystals, and the \(p\)-adic analytic properties of their associated \(L\)-functions form the crucial part of his subtle investigations, in the present paper.

The main theorem, whose complete proof is carried out in the two consecutive papers under review, is formulated in this generalized framework and reads as follows:

Theorem 1.1. Let \(X\) be a smooth affine variety defined over a finite field \(\mathbb{F}_q\) of characteristic \(p> 0\). Let \((M,\Phi)\) be a finite-rank overconvergent \(\sigma\)-module over \(X\). Then, for any rational number \(s\), the pure slope-\(s\) \(L\)-function \(L_s(\Phi,T)\) attached to \((M,\Phi)\) is \(p\)-adically meromorphic everywhere.

This theorem gives an affirmative answer to Dwork’s long-standing conjecture, and it even shows that this conjecture holds in greater generality.

As to the contents of the present first paper, which is merely algebraic in nature, there are 10 sections discussing in full detail the following subjects: (1) Introduction, and main results; (2) \(\sigma\)-modules and their \(L\)-functions; (3) Monsky’s trace formula; (4) Hodge-Newton decomposition of a \(\sigma\)-module and Dwork’s conjecture; (5) An easier case of Dwork’s conjecture; (6) The ordinary case of Dwork’s conjecture; (7) The non-ordinary case of Dwork’s conjecture; (8) The general case of Dwork’s conjecture; (9) Reduction to the base scheme \(\mathbb{A}^n\); (10) Appendix: Proof of the extended Monsky trace formula.

Basically, the author reduces Dwork’s conjecture from the higher rank case over any smooth affine variety to the rank one case over the affine space \(\mathbb{A}^n\). In the subsequent second part, the proof of the main theorem (Theorem 1.1) is finished by proving just that rank one case of Dwork’s conjecture over \(\mathbb{A}^n\).

Altogether, this is a very important contribution toward the developments around Dwork’s conjecture, which finally provides a long and fascinating story with a happy end. The paper is extremely rich of new ideas, concepts, constructions, and results, thereby utmost detailed, lucid and comprehensible.

In the sequel, B. Dwork began the study of the variation of the zeta function within an algebraic family of varieties over \(\mathbb{F}_q\), which led him to the investigation of certain other zeta functions, the so-called unit root zeta functions. These functions are analytic in nature, and generally no longer rational functions, which makes the understanding of their \(p\)-adic analytic properties especially important. In this context, B. Dwork conjectured that a unit root zeta function must be \(p\)-adically meromorphic, i.e., it should be expressible as a quotient of two power series which are convergent everywhere on the completion of \(\mathbb{Q}_p\). In the early 1970s, Dwork himself was able to prove this conjecture of his for special families of curves and surfaces, but a general proof of this hard conjecture had to wait until D. Wan’s work under review was completed.

In the meantime, it was figured out that unit root zeta functions can be interpreted as the \(L\)-functions of \(F\)-crystals on the base space of the respective family of varieties, and this has finally proved to be the right approach to tackle Dwork’s conjecture in full generality. The paper under review, which is the first of two consecutive articles, provides a complete proof of Dwork’s conjecture in the so-called higher rank case, whereas the subsequent paper [cf.: D. Wan, J. Am. Math. Soc. 13, No. 4, 853–908 (2000; Zbl 1086.11031)] accomplishes the remaining case, the so-called rank one case.

The author has outlined his ingenious approach to Dwork’s conjecture in two foregoing articles entitled “A quick introduction to Dwork’s conjecture” [Fried, Michael D. (ed.), Applications of curves over finite fields. 1997 AMS-IMS-SIAM joint summer research conference, July 27–31, 1997, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 245, 147–163 (1999; Zbl 0977.11028)] and “Dwork’s conjecture on unit root zeta functions” [Ann. Math. (2) 150, No. 3, 867–927 (1999; Zbl 1013.11031)], respectively, in which he formulated this conjecture in a more general context, namely in terms of his newly established “\(\sigma\)-modules” and “overconvergent power series rings over complete discrete valuation rings”. Wan’s \(\sigma\)-modules are suitable generalizations of \(F\)-crystals, and the \(p\)-adic analytic properties of their associated \(L\)-functions form the crucial part of his subtle investigations, in the present paper.

The main theorem, whose complete proof is carried out in the two consecutive papers under review, is formulated in this generalized framework and reads as follows:

Theorem 1.1. Let \(X\) be a smooth affine variety defined over a finite field \(\mathbb{F}_q\) of characteristic \(p> 0\). Let \((M,\Phi)\) be a finite-rank overconvergent \(\sigma\)-module over \(X\). Then, for any rational number \(s\), the pure slope-\(s\) \(L\)-function \(L_s(\Phi,T)\) attached to \((M,\Phi)\) is \(p\)-adically meromorphic everywhere.

This theorem gives an affirmative answer to Dwork’s long-standing conjecture, and it even shows that this conjecture holds in greater generality.

As to the contents of the present first paper, which is merely algebraic in nature, there are 10 sections discussing in full detail the following subjects: (1) Introduction, and main results; (2) \(\sigma\)-modules and their \(L\)-functions; (3) Monsky’s trace formula; (4) Hodge-Newton decomposition of a \(\sigma\)-module and Dwork’s conjecture; (5) An easier case of Dwork’s conjecture; (6) The ordinary case of Dwork’s conjecture; (7) The non-ordinary case of Dwork’s conjecture; (8) The general case of Dwork’s conjecture; (9) Reduction to the base scheme \(\mathbb{A}^n\); (10) Appendix: Proof of the extended Monsky trace formula.

Basically, the author reduces Dwork’s conjecture from the higher rank case over any smooth affine variety to the rank one case over the affine space \(\mathbb{A}^n\). In the subsequent second part, the proof of the main theorem (Theorem 1.1) is finished by proving just that rank one case of Dwork’s conjecture over \(\mathbb{A}^n\).

Altogether, this is a very important contribution toward the developments around Dwork’s conjecture, which finally provides a long and fascinating story with a happy end. The paper is extremely rich of new ideas, concepts, constructions, and results, thereby utmost detailed, lucid and comprehensible.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11S40 | Zeta functions and \(L\)-functions |

11M41 | Other Dirichlet series and zeta functions |

11G25 | Varieties over finite and local fields |

11G15 | Complex multiplication and moduli of abelian varieties |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

##### Keywords:

algebraic varieties over finite fields; \(L\)-functions; zeta functions; \(p\)-adic analysis; Dwork’s conjecture**OpenURL**

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