Rank one case of Dwork’s conjecture.

*(English)*Zbl 1086.11031This paper is the immediate continuation of the author’s foregoing work “Higher rank case of Dwork’s conjecture” [J. Am. Math. Soc. 13, No. 4, 807–852 (2000; Zbl 1086.11028)]. Recall that, in this foregoing paper, the author has elaborated the first complete proof of B. Dwork’s long-standing conjecture predicting that the unit root zeta function of a family of algebraic varieties over a finite field \(\mathbb{F}_q\) of characteristic \(p>0\) is \(p\)-adically meromorphic on the completion of \(\mathbb{Q}_p\). In fact, the author has proved an even stronger version of Dwork’s conjecture in terms of his new conceptual framework of \(\sigma\)-modules [cf., D. Wan, Ann. Math. (2) 150, No. 3, 867–927 (1999; Zbl 1013.11031)], for families over a smooth affine variety \(X/ \mathbb{F}_q\). More precisely, he has shown that the case of finite-rank \(\sigma\)-modules over \(X\) can be reduced to the rank-one case over an affine space \(\mathbb{A}^n/\mathbb{F}_q\), and the present, subsequent paper is exclusively devoted to the study of this case, just in order to complete the whole proof of the (generalized) Dwork conjecture.

Whilst the reduction principle discussed in the foregoing paper was almost purely algebraic, the tackling of the rank-one case of \(\sigma\)-modules over an affine space is merely \(p\)-analytic in nature, and that explains why the author has split his entire work on Dwork’s conjecture into two separate, but strongly interacting parts.

As it turns out, in order to handle the rank-one case, one is forced to work in the much more difficult infinite-rank case of \(\sigma\)-modules. This requires an appropriate setup, i.e., an efficient machinery for treating infinite-rank \(\sigma\)-modules, which is built up in Section 2 of the present paper by extending the easier finite-rank setting established in the foregoing paper. The concepts introduced to this end are Banach modules over certain rings, various bases for them, nuclear \(\sigma\)-modules, overconvergent \(\sigma\)-modules, nuclear overconvergent \(\sigma\)-modules, and various tensor operations. Also, the concept of the \(L\)-function of a nuclear \(\sigma\)-module is derived in this context, and the remaining part of the present paper is then devoted to the proof of the fact that certain \(L\)-functions of this type are \(p\)-adically meromorphic. Along this line, Section 3 discusses a uniform family version for overconvergent \(\sigma\)-modules, thereby revising several notions and constructions from the author’s earlier paper (1999; loc. cit.), which may be regarded as both the proper forerunner of the present work and the essential reference for it. Section 4 deals with a generalization of the classical Dwork operators, the so-called overconvergent Dwork operators in the present context, and with explicit estimates for those. In Section 5, the author extends the generalized Monsky trace formula from the finite-rank case over \(\mathbb{A}^n\) to the infinite-rank case, whereas Section 6 reviews the Newton polygon, the basis polygon, and the concept of ordinariness of nuclear \(\sigma\)-modules. Sections 7 and 8 are the core of the entire treatise, in that they provide the proof of the author’s main theorem establishing the correctness of the (generalized) Dwork conjecture in the rank-one case over \(\mathbb{A}^n\). Putting everything from the previous sections together, the author proves that certain \(L\)-functions derived from nuclear overconvergent \(\sigma\)-modules with rank-one unit root part over an affine space are \(p\)-adically meromorphic. This main theorem of the paper under review implies the author’s crucial “key lemma” in his foregoing paper on the finite-rank case, and therefore completes the proof of Theorem 1.1. stated there.

As this theorem confirms the correctness of the classical Dwork conjecture, among other conclusions, this long-standing and challenging conjecture is now a well-established theorem, after more than 30 years of hard struggling! It is needless to emphasize that the author’s brilliant work must be seen as one of the major recent contributions to contemporary mathematics as a whole.

Whilst the reduction principle discussed in the foregoing paper was almost purely algebraic, the tackling of the rank-one case of \(\sigma\)-modules over an affine space is merely \(p\)-analytic in nature, and that explains why the author has split his entire work on Dwork’s conjecture into two separate, but strongly interacting parts.

As it turns out, in order to handle the rank-one case, one is forced to work in the much more difficult infinite-rank case of \(\sigma\)-modules. This requires an appropriate setup, i.e., an efficient machinery for treating infinite-rank \(\sigma\)-modules, which is built up in Section 2 of the present paper by extending the easier finite-rank setting established in the foregoing paper. The concepts introduced to this end are Banach modules over certain rings, various bases for them, nuclear \(\sigma\)-modules, overconvergent \(\sigma\)-modules, nuclear overconvergent \(\sigma\)-modules, and various tensor operations. Also, the concept of the \(L\)-function of a nuclear \(\sigma\)-module is derived in this context, and the remaining part of the present paper is then devoted to the proof of the fact that certain \(L\)-functions of this type are \(p\)-adically meromorphic. Along this line, Section 3 discusses a uniform family version for overconvergent \(\sigma\)-modules, thereby revising several notions and constructions from the author’s earlier paper (1999; loc. cit.), which may be regarded as both the proper forerunner of the present work and the essential reference for it. Section 4 deals with a generalization of the classical Dwork operators, the so-called overconvergent Dwork operators in the present context, and with explicit estimates for those. In Section 5, the author extends the generalized Monsky trace formula from the finite-rank case over \(\mathbb{A}^n\) to the infinite-rank case, whereas Section 6 reviews the Newton polygon, the basis polygon, and the concept of ordinariness of nuclear \(\sigma\)-modules. Sections 7 and 8 are the core of the entire treatise, in that they provide the proof of the author’s main theorem establishing the correctness of the (generalized) Dwork conjecture in the rank-one case over \(\mathbb{A}^n\). Putting everything from the previous sections together, the author proves that certain \(L\)-functions derived from nuclear overconvergent \(\sigma\)-modules with rank-one unit root part over an affine space are \(p\)-adically meromorphic. This main theorem of the paper under review implies the author’s crucial “key lemma” in his foregoing paper on the finite-rank case, and therefore completes the proof of Theorem 1.1. stated there.

As this theorem confirms the correctness of the classical Dwork conjecture, among other conclusions, this long-standing and challenging conjecture is now a well-established theorem, after more than 30 years of hard struggling! It is needless to emphasize that the author’s brilliant work must be seen as one of the major recent contributions to contemporary mathematics as a whole.

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 |

14G15 | Finite ground fields in algebraic geometry |

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

##### Keywords:

zeta functions; \(p\)-adic analysis; algebraic varieties over a finite field; \(\sigma\)-modules##### References:

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