Zp-曲线塔的Zeta函数

Zeta Functions of Zp-Towers of Curves

WAN Daqing

【摘 要】Abstract：In this note, we explore possible stable properties for the zeta function of a geometric Zp-tower of curves over a finite field of characteristic p, in the spirit of Iwasawa theory. A number of fundamental questions and conjectures are proposed for those Zp-towers coming from algebraic geometry. 【期刊名称】四川师范大学学报（自然科学版） 【年(卷),期】2018(041)004 【总页数】12

【关键词】Keywords：Zp-tower; algebraic geometry; Zeta function; genus; slope stable; genus stable

1 Zp-towers of curves

Let Fq be a finite field of q elements with characteristic p>0 and let Zp denote the ring of p-adic integers. Consider a Zp-tower C∞:…→Cn →…→C1 →C0

of smooth projective geometrically irreducible curves defined over Fq. The Zp-tower gives a continuous group isomorphism ρ: G∞:=Gal(C∞/C0) ? Zp.

For each integer n≥0, reduction modulo pn gives an isomorphism Gn:=Gal(Cn/C0) ? Z/pnZ.

Let S be the ramification locus of the tower, which is a subset S of closet

points of C0. The tower is un-ramified on its complement U=C0-S. We shall assume that S is finite.

By class field theory, the ramification locus S is non-empty, and for each non-empty finite S, there are uncountably many such Zp-towers over Fq. In fact, all Zp-towers over C0 can be explicitly classified by the Artin-Schreier-Witt theory, see [1]. In contrast, if l is a prime different from p, there are no Zl-towers over Fq. Note that constant extensions do not give a tower in our sense since that would produce curves which are geometrically reducible. Thus, the Zp-towers we consider are all geometric Zp-towers.

The most important class of Zp-towers naturally comes from algebraic geometry. We describe this briefly now. Let GU be the Galois group of the maximal abelian extension of the function field of U which is unramified on U. Let be a continuous rank one surjective p-adic representation. Let

This is a subgroup of The composition of φ and the reduction homomorphism give a surjective homomorphism (if p>2) ?Zp/pnZp.

These surjective homomorphisms naturally produce a Zp-tower of C0, unramified on U, which we assume to be geometric in our sense, that is, there is no constantsubextensions. This geometric tower is further called arising (or coming) from algebraic geometry if φ arises from a

relative p-adic ètale cohomology of an ordinary family of smooth proper variety X parameterized by U, or more generally in the sense of Dwork’s unit root conjecture[2] as proved in [3], that is, φ comes from the unit root part of an ordinary over convergent F-crystal on U. A classical example is the Igusa Zp-tower arising from the universal family of ordinary elliptic curves over Fp, more generally, the Zp-tower arising from the following Dwork family of Calabi-Yau hyper surfaces over Fp parametrized by λ:

2 Zeta functions

For integer n≥0, let Z(Cn,s) denote the zeta function of Cn. It is defined by

where |Cn| denotes the set of closed points of Cn. The Riemann-Roch theorem implies that the zeta function is a rational function in s of the form

P(Cn,s)∈1+sZ[s],

where P(Cn,s) is a polynomial of degree 2gn and gn=g(Cn)

denotes the genus of the curve Cn. By the celebrated theorem of Weil, the polynomial P(Cn,s) is pure of q-weight 1, that is, the reciprocal roots of Z(Cn,s) all have complex absolute value equal to

The q-adic valuations (also called q-slopes) of the reciprocal roots of Z(Cn,s) remain quite mysterious in general. Our aim is to study this