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**Dwork’s conjecture on unit root zeta functions.**
*(English)*
Zbl 1013.11031

From the introduction: In this article, we introduce a systematic new method to investigate the conjectural \(p\)-adic meromorphic continuation of B. Dwork’s unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic \(p\).

After his pioneer \(p\)-adic investigation of the Weil conjectures on the zeta function of an algebraic variety over a finite field, Dwork went on to study the \(p\)-adic analytic variation of a family of such zeta functions when the variety moves through an algebraic family. In the course of doing so, he was led to a new zeta function called the unit root zeta function, which goes beyond the reach of the existing theory. He conjectured [Ann. Math. (2) 98, 1-57 (1973; Zbl 0265.14008)] that such a unit root zeta function is \(p\)-adic meromorphic everywhere. These unit root zeta functions contain important arithmetic information about a family of algebraic varieties. They are truly \(p\)-adic in nature and are transcendental functions, sometimes seeming quite mysterious. In fact, no single “nontrivial” example has been proved to be true about this conjecture, other than the “trivial” overconvergent (or \(\infty\) log-convergent) case for which Dwork’s classical \(p\)-adic theory already applies.

In this artiele, we introduce a systematic new method to study such unit root zeta functions. Our method can be used to prove the conjecture in the case when the involved unit root \(F\)-crystal has rank one. In particular, this settles the first “nontrivial” case, the rank one unit root \(F\)-crystal coming from the family of higher dimensional Kloosterman sums. Our method further allows us to understand reasonably well about analytic variation of an arithmetic family of such rank one unit root zeta functions, motivated by the Gouvêa-Mazur conjecture about dimension variation of classical and \(p\)-adic modular forms. We introduce another systematic method in another paper [J. Am. Math. Soc. 13, 853-908 (2000; Zbl 1086.11030)] which combined with the method in the present paper will be able to prove Dwork’s conjecture in the higher rank case.

After his pioneer \(p\)-adic investigation of the Weil conjectures on the zeta function of an algebraic variety over a finite field, Dwork went on to study the \(p\)-adic analytic variation of a family of such zeta functions when the variety moves through an algebraic family. In the course of doing so, he was led to a new zeta function called the unit root zeta function, which goes beyond the reach of the existing theory. He conjectured [Ann. Math. (2) 98, 1-57 (1973; Zbl 0265.14008)] that such a unit root zeta function is \(p\)-adic meromorphic everywhere. These unit root zeta functions contain important arithmetic information about a family of algebraic varieties. They are truly \(p\)-adic in nature and are transcendental functions, sometimes seeming quite mysterious. In fact, no single “nontrivial” example has been proved to be true about this conjecture, other than the “trivial” overconvergent (or \(\infty\) log-convergent) case for which Dwork’s classical \(p\)-adic theory already applies.

In this artiele, we introduce a systematic new method to study such unit root zeta functions. Our method can be used to prove the conjecture in the case when the involved unit root \(F\)-crystal has rank one. In particular, this settles the first “nontrivial” case, the rank one unit root \(F\)-crystal coming from the family of higher dimensional Kloosterman sums. Our method further allows us to understand reasonably well about analytic variation of an arithmetic family of such rank one unit root zeta functions, motivated by the Gouvêa-Mazur conjecture about dimension variation of classical and \(p\)-adic modular forms. We introduce another systematic method in another paper [J. Am. Math. Soc. 13, 853-908 (2000; Zbl 1086.11030)] which combined with the method in the present paper will be able to prove Dwork’s conjecture in the higher rank case.

### MSC:

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

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

11G25 | Varieties over finite and local fields |

14G15 | Finite ground fields in algebraic geometry |

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