Moment Zeta Functions in Arithmetic Geometry

Daqing Wan

July 12, 2002

Department of Mathematics University of California, Irvine, CA 92697 Email: dwan@math.uci.edu

1 Introduction

In this expository paper, we discuss several fundamental properties on variation of the Hasse-Weil zeta function attached to an algebraic variety. The zeta function is a suitable generating function which counts the (closed) points of the algebraic variety. Among the most important properties are the conjectural meromorphic continuation and the generalized Riemann hypothesis. In characteristic zero case, both conjectures are wide open in general. In positive characteristic case, both conjectures are well understood by the results of Dwork and Deligne on the Weil conjectures.

Our main concern here is how the zeta function varies when the algebraic variety moves in an algebraic family. This leads to the introduction of a higher moment zeta function. Again, among the most important properties are the conjectural meromorphic continuation and the generalized Riemann hypothesis. In characteristic zero case, everything remains largely conjectural even for an elliptic curve. In positive characteristic case, much can be proved in view of the various tools developed for the Weil conjectures.

A further basic question is to understand how the moment zeta function varies when the moment parameter goes to infinity. This naturally leads to the possible introduction of a limiting moment zeta function, which measures the arithmetic variation of the moment zeta function. In characteristic zero case, we do not yet know how to rigorously define such a limiting moment zeta function, which would presumably be related to the conjectural p-adic zeta function and higher dimensional non-abelian Iwasawa theory. In positive characteristic case, we can indeed define such a limiting moment zeta function. This raises new fundamental open problems (in terms of meromorphic continuation and a suitable Riemann hypothesis) concerning the limiting moment zeta function, generalizing Dwork’s p-adic meromorphy conjecture. We shall give an elementary formulation of some of these problems.

2 Riemann zeta function

Let ℤ = {0, ±1, ±2, ⋯, } be the ring of integers. One of the most important questions in number theory is to understand the distribution of prime numbers. For a positive real number t, let π(t) denote the number of primes p such that p ≤ t. The function π(t) is the counting function for prime numbers. To understand its nature as t → ∞, one attaches a suitable generating function to it. In this case, the right generating function is the Riemann zeta function defined by the formal Euler product

ζ(ℤ, s) = ∏_{p prime}(1 − p^{−s})^{−1}.  (1)

By the unique factorization of integers, one deduces the Dirichlet expansion

ζ(ℤ, s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} .  (2)

This series converges absolutely for all complex numbers s ∈ ℂ whose real part is greater than 1. In particular, it defines a complex analytic function in the half plane Re(s) > 1. The first fundamental property about ζ(ℤ, s) is given by

Theorem 2.1 (Riemann, 1859). The function ζ(ℤ, s) extends to a meromorphic function in the whole complex plane s ∈ ℂ with a simple pole at s = 1.

By the Weierstrass factorization of entire functions, the meaning of the above theorem is that there exists an explicit formula for π(t) in terms of zeros and poles of ζ(ℤ, s). More precisely, let Λ(n) be the von-Mangoldt function which equals ln p if n is a positive integral power of p and zero otherwise. The logarithmic derivative of ζ(ℤ, s) gives the Dirichlet series of Λ(n):

−\frac{ζ′(ℤ, s)}{ζ(ℤ, s)} = ∑_{n=1}^{∞} \frac{Λ(n)}{n^{s}}.

Let

ψ(t) = ∑_{1≤n≤t} Λ(n).

This is a weighted counting of primary numbers, each positive power of p is counted with weight ln p. It is equivalent and more convenient to work with ψ(t) instead of π(t). If t > 0 is not a prime power, then von Mangoldt (1895) obtained the explicit formula

ψ(t) = t − ∑_{ρ} \frac{t^{ρ}}{ρ} − \frac{ζ′(ℤ, 0)}{ζ(ℤ, 0)} − \frac{1}{2} ln(1 − t^{−2}),

where ρ runs over all non-trivial zeros of ζ(ℤ, s) in the symmetric critical strip 0 < Re(s) < 1. Note that the two terms for ρ and ρ̄ need to be summed together.

The main term t arises from the simple pole at s = 1. The term \frac{1}{2} ln(1 − t^{−2}) arises from the trivial zeros at negative even integers of ζ(ℤ, s).

To get further information about π(t) and ψ(t), one needs to know the nature of the zeros and poles of ζ(ℤ, s). This is given by the following well known Riemann Hypothesis.

Conjecture 2.2 (RH, 1859). If ζ(ℤ, s) = 0 or ∞ for some s ∈ ℂ, then Re(s) ∈ \frac{1}{2}ℤ. That is, all non-trivial zeros lie on the critical line Re(s) = \frac{1}{2}.

The meaning of this conjecture is that there is a sharp asymptotic formula for π(t) and ψ(t). Namely, the Riemann Hypothesis implies that for any ϵ > 0, we have the estimate

ψ(t) = t + O_{ϵ}(t^{\frac{1}{2} +ϵ}).

Equivalently, for any ϵ > 0, we have

π(t) = ∫_{2}^{t} \frac{1}{ln u}du + O_{ϵ}(t^{\frac{1}{2} +ϵ}).  (3)

By the prime number theorem as proved by Hadamard and de la Vallé Poussin in 1899, the probability that a randomly chosen positive integer u is a prime is given by 1/ ln(u). This explains the above main term which was first suggested by Gauss in 1792.

The RH is undoubtedly very profound. No promising ideas of proving it have emerged yet although there are many conjectural approaches. In terms of our formulation of RH as above, one could try to prove the RH in three stages. The first stage is to prove that Re(s) for ζ(ℤ, s) = 0 is a rational number. The second stage is to prove that the denominator of Re(s) is uniformly bounded. The third stage is to prove that the common denominator of the Re(s) is bounded by 2. One reason to cut the RH into three stages comes from certain p-adic analogues [19], where the corresponding p-adic RH analogue fits naturally in the three stages.

At negative odd integer s = 1 − 2k, one can compute that

ζ(ℤ, 1 − 2k) = −\frac{B_{2k}}{2k} ,

where the B_{k} are the Bernoulli numbers given by

\frac{t}{e^{t} − 1} = ∑_{k=0}^{∞} B_{k} \frac{t^{k}}{k} .

For a prime number p, Kummer’s congruence shows that the modified special value (1 − p^{2k−1})ζ(ℤ, 1 − 2k) is p-adically continuous in k when 2k varies in a fixed congruence class modulo (p − 1). This gives rise to a p-adic zeta function ζ_{p}(ℤ, s) of ℤ for each congruence class modulo (p−1). Such p-adic zeta functions motivated much of the modern study in number theory and p-adic modular forms, see [13].

3 Hasse-Weil zeta functions

From algebraic geometric point of view, instead of the ring ℤ of integers, we can more generally consider a finitely generated commutative ℤ-algebra A with 1. By the Hilbert basis theorem, such an algebra can be written explicitly in the form

A = ℤ[X_{1}, ⋯, X_{n}]/(f_{1}, ⋯, f_{m}),  (4)

where the f_{i} are polynomials in integer coefficients. The algebra A defines an affine arithmetic scheme X = Spec(A), whose points are simply the set of prime ideals in A. Let |X| denote the subset of closed points of X, namely, the subset of maximal ideals in A. Generalizing the counting of prime numbers (closed points in Spec(ℤ)), we would like to count the closed points in |X|.

If x ∈ |X| is a closed point, then the quotient ring A/x is a finite field, called the residue field of X at x. Denote the number of elements in the finite field A/x by N(x). For a positive real number t, we define π_{X}(t) to be the number of closed points x ∈ |X| such that N(x) ≤ t. This is a finite integer for each finite t. The function π_{X}(t) is our counting function for closed points of X. To understand its behavior as t → ∞, we similarly attach a generating function. The right generating function is the Hasse-Weil zeta function defined by the following formal Euler product

ζ(X, s) = ∏_{x∈|X|}(1 − N(x)^{−s})^{−1}.  (5)

It can be written as a Dirichlet series

ζ(X, s) = ∑_{n=1}^{∞} \frac{a_{n}(X)}{n^{s}},  (6)

where a_{n}(X) are some non-negative integers depending on X. This series ζ(X, s) converges absolutely for all complex numbers s ∈ ℂ whose real part is sufficiently large (say, larger than the Krull dimension of X). In particular, it defines a complex analytic function in the half plane Re(s) > dim(X). The first fundamental property one would like to have is

Conjecture 3.1 (Hasse-Weil). The function ζ(X, s) extends to a meromorphic function in the whole complex plane s ∈ ℂ.

The meaning of this conjecture is that there exists an explicit formula for π_{X}(t) in terms of zeros and poles of ζ(X, s). Without loss of generality, one can assume that X is irreducible. The above meromorphy conjecture is still wide open in general. It is true if dim(X) = 1 by the work of Hecke (1917) on the Dedekind zeta functions. If dim(X) = 2, the conjecture is still open but true in the case that the generic fibre X ⊗ ℚ is an elliptic curve:

y^{2} = x^{3} + ax + b, a, b ∈ ℤ.

This follows from the work of Wiles, Taylor and others (2001 [2]) on the modularity of elliptic curves. It is then easy to prove that if the generic fibre X ⊗ℚ is an elliptic curve, then for every positive integer k, ζ(Sym^{k}X, s) is meromorphic everywhere, where Sym^{k}X is the k-th symmetric power of X, whose generic fibre is a k-dimensional variety over ℚ. In fact, in this case, Sym^{k}(X ⊗ ℚ) is a projective bundle of rank k − 1 over X ⊗ ℚ (the Jacobian of the elliptic curve X ⊗ ℚ). Thus, up to possibly finitely many Euler factors, we have the relation

ζ(Sym^{k}X, s) = ∏_{i=0}^{k−1} ζ(X, s − i).

More generally, we have

Proposition 3.2 Let the generic fibre X ⊗ ℚ be a smooth projective curve of genus g. Assume that ζ(Sym^{k}X, s) is meromorphic for the finitely many integers 1 ≤ k ≤ g. Then, ζ(Sym^{k}X, s) is meromorphic for every positive integer k.

This finiteness result can be generalized to certain higher dimensional varieties. Let the generic fibre X ⊗ℚ be a smooth projective variety. Let B^{even}(X) (resp. B^{odd}(X)) denote the even (resp. odd) Betti number of X ⊗ℚ. Let R(X) denote the rank of the group of algebraic cycles on X ⊗ ℚ modulo homological equivalence. The cycle class map shows that we have the inequality

1 + dim(X ⊗ ℚ) ≤ R(X) ≤ B^{even}(X).

These inequalities become equality if X ⊗ ℚ is a curve or an odd dimensional complete intersection. Using either the Weil conjectures or Deligne’s symmetric Künneth formula, one can quickly prove the following finiteness result. We leave the proofs to the reader as an exercise.

Theorem 3.3 Let X ⊗ ℚ be a smooth projective variety over ℚ. Assume R(X) = B^{even}(X). Assume that ζ(Sym^{k}X, s) is meromorphic for the finitely many integers 1 ≤ k ≤ B^{odd}/2. Then, ζ(Sym^{k}X, s) is meromorphic for every positive integer k.

If R(X) < B^{even}(X) such as the case of a general surface X ⊗ ℚ, we do not know if there is a similar finiteness property for the sequence of zeta functions ζ(Sym^{k}X, s) parametrized by positive integers k. The above theorem suggests that under the assumption R(X) = B^{even}(X), the geometry of Sym^{k}X for large k can probably be determined from the first few terms Sym^{k}X for small k, as in the case of curves. If so, it would be interesting to make this precise geometrically.

Assume the meromorphic continuation of ζ(X, s). To get further information, one needs to know the nature of the zeros and poles. This is given by the following Generalized Riemann Hypothesis.

Conjecture 3.4 (GRH). If ζ(X, s) = 0 or ∞ for some complex number s ∈ ℂ, then Re(s) ∈ \frac{1}{2}ℤ.

The meaning of this conjecture is that there is a sharp asymptotic formula for π_{X}(t). Although the above two conjectures are apparently extremely difficult in general, they have been proved completely if A has positive characteristic. We shall review this geometric case in next section.

There are many general conjectures on the orders of zeros (resp. poles) of ζ(X, s) and the special values ζ(X, k) at various integers or half integers k. These include the classical BSD conjecture and Tate conjectures [17]. For each prime number p, conjecturally there is a p-adic zeta function ζ_{p}(X, s) attached to X, which interpolates certain special values of ζ(X, s) at negative integers. If dim(X) = 1, the p-adic zeta function has been constructed. If dim(X) = 2, the p-adic zeta function has not been constructed in general, but in the case that X ⊗ ℚ is an ellliptic curve, this has been done. It is a major open problem to construct the p-adic zeta function ζ_{p}(X, s) of a scheme of finite type over ℤ, see [16].

4 Zeta functions over finite fields

First, we assume that the characteristic of A is a prime number p. Then, A contains the finite prime field 𝔽_{p} of p elements and it becomes a finitely generated commutative 𝔽_{p}-algebra with 1. The affine scheme X then becomes an affine algebraic variety defined by finitely many polynomial equations

f_{1}(X_{1}, ⋯, X_{n}) = ⋯ = f_{m}(X_{1}, ⋯, X_{n}) = 0,

where the f_{i} are polynomials over 𝔽_{p}.

If x ∈ |X| is a closed point, the residue field A/x is a finite field containing 𝔽_{p} as a subfield. Thus, A/x is a finite extension of 𝔽_{p} of degree denoted by deg(x). We have

N(x) = p^{deg(x)}.

The Hasse-Weil zeta function becomes

ζ(X, s) = ∏_{x∈|X|}(1 − (p^{−s})^{deg(x)})^{−1}.  (7)

Let T = p^{−s} and define the zeta function Z(X, T) of X over 𝔽_{p} by

Z(X, T) = ∏_{x∈|X|}(1 − T^{deg(x)})^{−1} ∈ 1 + Tℤ[[T]].  (8)

Then, we have the relation

ζ(X, s) = Z(X, p^{−s}).

Since the exponential function p^{−s} is an entire function with no zeros, it suffices to study Z(X, T).

We shall consider a slightly more general situation. Instead of the finite prime field 𝔽_{p}, we shall consider a general finite field 𝔽_{q}. Let 𝔽_{q} be a finite field of q elements of characteristic p. Let X be a scheme of finite type over 𝔽_{q}. One can similarly define

Z(X, T) = ∏_{x∈|X|}(1 − T^{deg(x)})^{−1} ∈ 1 + Tℤ[[T]],  (9)

where deg(x) denotes the extension degree of the finite residue field at x over the ground field 𝔽_{q} of X (instead of 𝔽_{p}). An alternative definition for Z(X, T) is in terms of counting rational points of X over various finite extensions of 𝔽_{q}. Let 𝔽^{}_{¯ q} denote a fixed algebraic closure of 𝔽_{q}. The Galois group of 𝔽^{}_{¯ q} over 𝔽_{q} is topologically generated by the q-th power Frobenius map

Frob : a → a^{q}, a ∈ 𝔽^{}_{¯ q}.

For each positive integer k, 𝔽^{}_{¯ q} contains a unique subfield of q^{k} elements, denoted by 𝔽_{q^{k}}, given explicitly by

𝔽_{q^{k}} = Fix(Frob^{k}) = {a ∈ 𝔽^{}_{¯ q}|a^{q^{k}} = a}.

We have

𝔽^{}_{¯ q} = ∪_{k=1}^{∞}𝔽_{q^{k}}.

Let X(𝔽^{}_{¯ q}) be the set of geometric points of X. For each positive integer k, let X(𝔽_{q^{k}}) denote the set of 𝔽_{q^{k}}-rational points. Clearly,

X(𝔽_{q^{k}}) = Fix(Frob^{k}|X(𝔽^{}_{¯ q})).

The alternative definition of Z(X, T) is

Z(X, T) = exp(∑_{k=1}^{∞} \frac{T^{k}}{k} #X(𝔽_{q^{k}})),  (10)

where #X(𝔽_{q^{k}}) denotes the number of 𝔽_{q^{k}}-rational points of X.

The rationality of Z(X, T) was conjectured by Weil (1949 [26]) and first proved by Dwork [7] using p-adic analysis. That is, we have

Theorem 4.1 (Dwork, 1960) Let X be a scheme of finite type over 𝔽_{q}. Then, Z(X, T) ∈ ℚ(T).

Corollary 4.2 Let X be a scheme of finite type over 𝔽_{q}. The Hasse-Weil zeta function ζ(X, s) = Z(X, q^{−s}) is a meromorphic function in s ∈ ℂ.

Write

Z(X, T) = \frac{∏_{i=1}^{s}(1 − α_{i}T)}{∏_{j=1}^{r}(1 − β_{j}T)},

where the α_{i}’s and β_{j}’s are algebraic integers. Taking logarithmic derivative, one obtains an explicit formula for rational point counting:

#X(𝔽_{q^{k}}) = ∑_{j=1}^{r} β_{j}^{k} − ∑_{i=1}^{s} α_{i}^{k}, k = 1, 2, · · · .

The Riemann hypothesis for Z(X, T) was conjectured by Weil (1949) and proved by Deligne [5] in the general form. In terms of ζ(X, s), this says

Theorem 4.3 (Deligne, 1980). Let X be a scheme of finite type over 𝔽_{q}. If ζ(X, s) = Z(X, q^{−s}) = 0 or ∞ for some s ∈ ℂ, then Re(s) ∈ \frac{1}{2}ℤ.

In terms of the zeros and poles of the rational function Z(X, T), Deligne’s theorem says that we can write

|α_{i}| = q^{u_{i}/2}, |β_{j}| = q^{v_{j}/2}, u_{i} ∈ ℤ ∩ [0, 2n], v_{j} ∈ ℤ ∩ [0, 2n],  (11)

where ℤ ∩ [0, 2n] denotes the set of integers in the interval [0, 2n]. Furthermore, each α_{i} (resp. each β_{j}) and its Galois conjugates over ℚ have the same complex absolute value. That is, the α_{i} and β_{j} are Weil q-integers. This result immediately gives the above Riemann hypothesis consequence for ζ(X, s).

As the α_{i} and β_{j} are algebraic integers, one can also ask for their ℓ-adic absolute values if we fix an embedding of ℚ^{}¯ to ℚ^{}_{¯ ℓ} of ℓ-adic numbers, where ℓ is a prime number. If ℓ ≠ p, the α_{i} and the β_{j} are eigenvalues of ℓ-adic representations and hence they are ℓ-adic units:

|α_{i}|_{ℓ} = |β_{j}|_{ℓ} = 1.

For the remaining prime p, it is easy to prove

|α_{i}|_{p} = q^{−r_{i}}, |β_{j}|_{p} = q^{−s_{j}}, r_{i} ∈ ℚ ∩ [0, 2n], s_{j} ∈ ℚ ∩ [0, 2n],

where we have normalized the p-adic absolute value by |q|_{p} = q^{−1}. Deligne’s integrality theorem [6] implies the following improved information:

r_{i} ∈ ℚ ∩ [0, n], s_{j} ∈ ℚ ∩ [0, n].

Note that each α_{i} (resp. each β_{j}) and its Galois conjugates over ℚ may have different p-adic absolute values. The rational number r_{i} (resp. s_{j}) is called the slope of α_{i} (resp. β_{j}). Its denominator can be greater than 2, but can be effectively bounded depending on X. The nature of the slopes r_{i} and s_{j} can be viewed as the p-adic Riemann hypothesis for Z(X, T). It is a very interesting subject with many open problems. A classical example is the Stickelberger theorem on Gauss sum. We shall not discuss it further here, see [25] for a systematic introduction.

An immediate consequence of the rationality, to be used later, is

Corollary 4.4 For each positive integer k, we have

Z(X ⊗ 𝔽_{q^{k}}, T) = exp(∑_{d=1}^{∞} \frac{T^{d}}{d} #X(𝔽_{q^{kd}}))
= \frac{∏_{i=1}^{s}(1 − α_{i}^{k}T)}{∏_{j=1}^{r}(1 − β_{j}^{k}T)}.  (12)

The next question we would like to study is how Z(X, T) varies when X moves in an algebraic family. This is called the geometric variation problem. We will discuss it in next section and introduce a moment zeta function.

To finish this section, let us briefly comment on the general X case, not necessarily of characteristic p. Let X be a scheme of finite type over ℤ. Each closed point x ∈ |X| contains a unique prime p, where N(x) = p^{deg(x)}. The prime number p is called the residue characteristic of x. The subset of all closed points of X of a fixed residue characteristic p coincide with the set of closed point of the reduction X ⊗ 𝔽_{p}. Thus, the Hasse-Weil zeta function can be decomposed as the following infinite Euler product

ζ(X, s) = ∏_{p prime} Z(X ⊗ 𝔽_{p}, p^{−s}).  (13)

This product is a finite product, if the characteristic h of X is positive. Let h = p_{1}^{k_{1}} ⋯ p_{m}^{k_{m}} be the standard factorization of h. Then,

ζ(X, s) = ∏_{i=1}^{m} Z(X ⊗ 𝔽_{p_{i}}, p_{i}^{−s}).  (14)

We obtain

Corollary 4.5 Let X be a scheme of finite type over ℤ. Assume that the characteristic h of X is positive. Then the Hasse-Weil zeta function ζ(X, s) is a meromorphic function in s ∈ ℂ. Furthermore, the real parts of its zeros and poles are in \frac{1}{2}ℤ.

For the Hasse-Weil conjecture and GRH, it remains to study the case when X has characteristic zero. The Hasse-Weil zeta function contains critical information about the arithmetic variation of the Euler factors Z(X ⊗ 𝔽_{p}, p^{−s}) as the arithmetic parameter p varies. We shall return to this question at the end of the paper.

5 Moment zeta functions: geometric case

Let f : X ⟶ Y be a morphism of schemes of finite type over 𝔽_{q}. We may view f as a family of algebraic varieties over finite fields parametrized by Y. If y ∈ |Y |, the fibre X_{y} = f^{−1}(y) is an algebraic variety defined over the finite residue field 𝔽_{q^{deg(y)}} of y. By the results in the previous section, we know that each zeta function Z(X_{y}, T) is a rational function and satisfies the Riemann hypothesis. We would like to understand

Question 5.1 How Z(X_{y}, T) varies when y varies in |Y |? How the zeros and poles of Z(X_{y}, T) vary when y varies in |Y |?

To make this variation question more precise, a classical procedure is to understand all the high moments of this family of rational functions. The k-th moment of each rational function Z(X_{y}, T) is obtained by raising each reciprocal zeros (resp. reciprocal poles) to its k-th power. By Corollary 4.4, this is simply given by Z(X_{y} ⊗ 𝔽_{q^{deg(y)k}}, T). Thus, we have

Definition 5.2 For each positive integer k, the k-th moment zeta function of the family f is defined to be the product

Z(k, f, T) = ∏_{y∈|Y |} Z(X_{y} ⊗ 𝔽_{q^{deg(y)k}}, T^{deg(y)}) ∈ 1 + Tℤ[[T]].

Arithmetically, the k-th moment zeta function can be viewed as the zeta function associated to a partial rational point counting problem. For a positive integer d, let

M_{d}(k, f) = #{x ∈ X(𝔽_{q^{dk}})|f(x) ∈ Y (𝔽_{q^{d}})}.

Then, it is easy to check that

Z(k, f, T) = exp(∑_{d=1}^{∞} \frac{T^{d}}{d} M_{d}(k, f)).

According to an observation of Faltings (see [24]), the number M_{d}(k, f) is the number of fixed points of a certain twisted Frobenius. Let f^{⊗k} be the k-th fibre product of X over Y. Let σ be the right shifting map on the coordinates of f^{⊗k}. That is, for a point (x^{(1)}, ⋯, x^{(k)}) ∈ f^{⊗k},

σ(x^{(1)}, ⋯, x^{(k)}) = (x^{(k)}, x^{(1)}, ⋯, x^{(k−1)}).

The map σ is an automorphism of f^{⊗k} of order k. It commutes with the Frobenius map. One checks that

M_{d}(k, f) = #Fix(σ ∘ Frob^{d}|f^{⊗k}(𝔽^{}_{¯ q})).

This formula implies that Z(k, f, T) is nearly rational in the sense of [24], as noted by Faltings.

The sequence of moment zeta functions Z(k, f, T) (k = 1, 2, ⋯,) contains critical information about the variation of the family Z(X_{y}, T) parametrized by y. Our standard questions are then the meromorphic continuation and Riemann hypothesis for these moment zeta functions. The meromorphic continuation implies the existence of good structural distribution law. The Riemann hypothesis gives sharp information about the distribution law. As in the previous section, we have the following general result which is a special case from [24] on partial zeta functions. It can be proved quickly from the results of Grothnedieck-Deligne and the theory of virtual representations.

Theorem 5.3 (Wan[24]). Let f : X ⟶ Y be a morphism of scheme of finite type over 𝔽_{q}. For each positive integer k, the k-th moment zeta function Z(k, f, T) is a rational function in ℚ(T), whose reciprocal zeros and reciprocal poles are Weil q-integers. In particular, the k-th moment Hasse-Weil zeta function Z(k, f, q^{−s}) is a meromorphic function. The real parts of the zeros and poles of Z(k, f, q^{−s}) are in \frac{1}{2}ℤ.

A very interesting question is to understand the slopes of the zeros and poles of Z(k, f, T), corresponding to the p-adic Riemann hypothesis for Z(k, f, T). This is in general quite mysterious, already so in the special case of the unversal family of elliptic curves where the slopes reflect crucial arithmetic information about modular forms [19].

We can further ask how the sequence Z(k, f, T) varies when the integer k varies. This is treated in next section. It is related to Dwork’s conjecture.

6 Limiting moment zeta functions

Let f : X ⟶ Y be a morphism of schemes of finite types over 𝔽_{q}. For each positive integer k, we have the k-th moment zeta function Z(k, f, T) which is a rational function satisfying the Riemann hypothesis. We would like to understand how the moment zeta function Z(k, f, T) varies as the integer k varies arithmetically.

To be more specific, we fix a prime number ℓ (which may be equal to the characteristic p). We let k vary ℓ-adically and want to understand how Z(k, f, T) varies ℓ-adically. An initial property would be possible ℓ-adic continuity.

Let K_{f,ℓ} = ℚ_{ℓ}(α_{i}(y), β_{j}(y)), the extension field of ℚ_{ℓ} obtained by adjoining all ℓ-adic unit zeros α_{i}(y) and all ℓ-adic unit poles β_{j}(y) of Z(X_{y}, T) for all y ∈ |Y |. The uniform upper bound [1] for the total degree of Z(X_{y}, T) together with a standard algebraic number theory argument implies

Proposition 6.1 The extension K_{f,ℓ} over ℚ_{ℓ} is finite.

Denote the uniformizer of K_{f,ℓ} by π. Let d_{f,ℓ} be the unramified degree of K_{f,ℓ} over ℚ_{ℓ}. Let

D_{f,ℓ} = ℓ^{d_{f,ℓ}} − 1.  (15)

This is simply the order of the multiplicative group of the residue field of K_{f,ℓ}.

Then, for each ℓ-adic unit α in K_{f,ℓ} (such as those ℓ-adic unit α_{i}(y) and those ℓ-adic unit β_{j}(y)), the power α^{D_{f,ℓ}} is an ℓ-adic 1-unit in K_{f,ℓ}. In particular, we have the limiting formula

lim_{m→∞} α^{D_{f,ℓ}ℓ^{m}} = 1

and the congruences

α^{k_{1}} ≡ α^{k_{2}} (mod π^{m})

for any two integers k_{1} and k_{2} satisfying k_{1} ≡ k_{2} (modD_{f,ℓ}ℓ^{m−1}), where m is any positive integer.

This ℓ-adic continuity result and the Euler product definition of Z(k, f, T), show that the limit in the following definition exists as a ℓ-adic formal power series.

Definition 6.2 Given a morphism f : X ⟶ Y over 𝔽_{q}, a prime number ℓ, an integer k, we define the limiting moment zeta function by

Z_{ℓ}(k, f, T) = lim_{m→∞} Z(k + D_{f,ℓ}ℓ^{m}, f, T) ∈ 1 + Tℤ_{ℓ}[[T]].  (16)

Note that Z_{ℓ}(k, f, T) is very different from Z(k, f, T) in general. Let

M_{d,ℓ}(k, f) = lim_{m→∞} #{x ∈ X(𝔽_{q^{d(k+D_{f,ℓ}ℓ^{m})}})|f(x) ∈ Y (𝔽_{q^{d}})}.

This is an ℓ-adic integer. It can be viewed as the number of 𝔽_{q^{d}}-rational points on a certain infinite dimensional variety defined over 𝔽_{q}. The additive definition of the limiting moment zeta function can be written as

Z_{ℓ}(k, f, T) = exp(∑_{d=1}^{∞} \frac{T^{d}}{d} M_{d,ℓ}(k, f)).

The series Z_{ℓ}(k, f, T) is clearly ℓ-adic analytic in the open unit disk |T |_{ℓ} < 1. It can be viewed in certain sense as the zeta function of a certain infinite dimensional variety over 𝔽_{q} counted in certain direction. Our first fundamental question is

Question 6.3 For k ∈ ℤ, Z_{ℓ}(k, f, T) is an ℓ-adic meromorphic function on the whole ℓ-adic plane |T |_{ℓ} < ∞.

If Z_{ℓ}(k, f, T) is indeed ℓ-adic meromorphic everywhere, then there are ℓ-adic numbers α_{i} and β_{j} approaching to zero such that

Z_{ℓ}(k, f, T) = \frac{∏_{i=1}^{∞}(1 − α_{i}T)}{∏_{j=1}^{∞}(1 − β_{j}T)}.

In terms of the ℓ-adic sequence M_{d,ℓ}(d, f) parametrized by d, this means that for each integer d > 0, we have the formula

M_{d,ℓ}(k, f) = ∑_{j=1}^{∞} β_{j}^{d} − ∑_{i=1}^{∞} α_{i}^{d}.

Conversely, the existence of such a formula is equivalent to the ℓ-adic meromorphic continuation of Z_{ℓ}(k, f, T) to the whole ℓ-adic plane.

If ℓ ≠ p, the α_{i}(y), β_{j}(y) are always ℓ-adic units. In this case, we deduce that if k is a positive integer, then

Z_{ℓ}(k, f, T) = Z(k, f, T),

which is a rational function in ℚ(T) whose reciprocal zeros and reciprocal poles are Weil q-integers, by the results in the previous section. If k is negative, it can be proved in a similar way using contragradient representations.

If ℓ = p, the situation is more complicated. Assume that k is a positive integer. Then, Z_{p}(k, f, T) is exactly the k-th power unit root zeta function of Dwork attached to the family f. Denote this k-th power unit root zeta function by Z_{p−unit}(k, f, T). This is a p-adic power series, whose coefficients are not contained in a fixed number field and hence not a rational function any more in general. Dwork [9] conjectured that Z_{p−unit}(k, f, T) is p-adic meromorphic in the whole p-adic plane |T |_{p} < ∞. This was proved recently in [20][21][22]. Note that here we only consider the slope zero (unit root) part. Similar results hold for higher slopes, see [23] for a simple introduction.

The possible p-adic Riemann hypothesis for Z_{p−unit}(k, f, T) is completely mysterious at the moment. One can further ask how Z_{p−unit}(k, f, T) varies padically as k varies p-adically. This is not well understood at all, but in the so-called rank one case it is much better understood [22], see also next section.

We can extend the above results from positive integers to all integers. Thus, the above question has a positive answer and we have

Theorem 6.4 (Wan [21][22]). Let ℓ be a prime. Let k be an integer. Then, the limiting moment zeta function Z_{ℓ}(k, f, T) is ℓ-adic meromorphic everywhere. If ℓ ≠ p, it is rational over ℚ, whose reciprocal zeros and reciprocal poles are Weil q-integers.

The integer domain of the variable k in the function Z_{ℓ}(k, f, T) can be extended to a larger ℓ-adic domain as follows. Write k = k_{1} + D_{f,ℓ}k_{2}. Then, it is easy to check that the formal power series Z_{ℓ}(k_{1} + D_{f,ℓ}k_{2}, f, T) is ℓ-adically continuous in k_{2}. This continuity implies that we can define

Definition 6.5 Let k_{1} be a fixed integer. Let k_{2} be any ℓ-adic integer. We define

Z_{ℓ}(k_{1}, k_{2}, f, T) = lim_{m→∞} Z_{ℓ}(k_{1} + D_{f,ℓ}k_{2}(m), f, T),

where k_{2}(m) is a sequence of strictly increasing positive integers which converges ℓ-adically to k_{2}.

This is a well defined ℓ-adic power series for any integer k_{1} ∈ ℤ and any ℓ-adic integer k_{2} ∈ ℤ_{ℓ}. It is independent of the choice of the sequence k_{2}(m) converging ℓ-adically to k_{2}.

The function Z_{ℓ}(k_{1}, k_{2}, f, T) in the two variables (k_{2}, T) should be viewed as the ℓ-adic zeta function attached to the morphism f. It is then natural to ask

Question 6.6 For k_{1} ∈ ℤ and k_{2} ∈ ℤ_{ℓ}, is Z_{ℓ}(k_{1}, k_{2}, f, T) an ℓ-adic meromorphic function on the closed unit disk |T |_{ℓ} ≤ 1? or even on the whole ℓ-adic plane |T |_{ℓ} < ∞?

In general, the ℓ-adic meromorphic continuation of Z_{ℓ}(k_{1}, k_{2}, f, T) to the closed unit disk |T |_{ℓ} ≤ 1 is already not clear, even for the universal family of elliptic curves if ℓ ≠ p. It would be premature to conjecture the ℓ-adic meromorphic continuation to the entire ℓ-adic plane |T |_{ℓ} < ∞. For this reason, we simply state the above problem as a question instead of a conjecture as the answer could be negative in general (there seems to have a little negative feeling in the general case). It is however interesting to find out when the answer is positive.

The previous theorem shows that the answer is positive if k_{2} ∈ ℤ. Another positive result in this direction is the following theorem which says that the answer is aslo positive for some special k_{2} ∈ ℤ_{ℓ} ∩ ℚ.

For each integer n > m for some fixed large integer m depending on D_{f,ℓ}, the integer ℓ^{n!}(ℓ^{(n+1)!−n!} − 1) is clearly divisible by D_{f,ℓ}. Thus, for an integer d, we can write the formal symbal dℓ^{∞} (the zero element in ℤ_{ℓ}) in the form

dℓ^{∞} = (d + ∑_{n=0}^{m} dℓ^{n!}(ℓ^{(n+1)!−n!} − 1)) + ∑_{n=m+1}^{∞} dℓ^{n!}(ℓ^{(n+1)!−n!} − 1) = k_{1} + D_{f,ℓ}k_{2},

where k_{2} ∈ ℤ_{ℓ} ∩ ℚ.

Theorem 6.7 (Lenstra-Wan [15]) Let ℓ be a prime. Let d be an integer. Let k_{1} and k_{2} be defined as above. Then, Z_{ℓ}(k_{1}, k_{2}, f, T) ∈ ℚ_{ℓ}(T), an ℓ-adic rational function whose reciprocal zeros and reciprocal poles are Weil q-integers.

This theorem can be proved quickly using the Grothendieck-Deligne results and Brauer’s virtual lifting theorem for modulo ℓ representations.

Let us now look at an important special case.

Example 6.8 Let n ≥ 2 be an integer. Let f be the family of Calabi-Yau projective hypersurfaces over 𝔽_{q} defined by the equation

X_{0}^{n+1} + X_{1}^{n+1} + ⋯ + X_{n}^{n+1} + yX_{0}X_{1} ⋯ X_{n} = 0,

parametrized by the affine line y ∈ 𝔸^{1}. In the case n = 2, this is a family of elliptic curves. In the case n = 3, this is a family of K − 3 surfaces. In the case n = 4, this is a family of Calabi-Yau quintic hypersurfaces.

If ℓ ≠ p, the zeta function Z_{ℓ}(k_{1}, k_{2}, f, T) is in general not known to be ℓ-adic meromorphic on the closed unit disk |T |_{ℓ} ≤ 1 if k_{2} ∈ ℤ_{ℓ}. If in addition, either k_{2} ∈ ℤ or k_{1} + D_{f,ℓ}k_{2} is equal to the formal symbal dℓ^{∞} as defined above, then Z_{ℓ}(k_{1}, k_{2}, f, T) is rational.

If ℓ = p, then there is only one (or none) non-trivial p-adic unit root for the zeta function Z(X_{y}, T) and one can show that D_{f,p} = p − 1. Our result implies that the limiting moment zeta function Z_{p}(k_{1}, k_{2}, f, T) is always p-adic meromorphic everywhere for all k_{1} ∈ ℤ and k_{2} ∈ ℤ_{p}. This follows from the rank one case of Dwork’s conjecture as given in [22]. The cases n = 2, 3 with k_{2} ∈ Z had been proved previously by Dwork (1971[8], 1973[9]). Such p-adic meromorphic continuation should be related to deep p-adic properties of the mirror map.

7 Moment L-functions: geometric case

To give a further and more precise exposition, we need to work with the languages of ℓ-adic representations and ℓ-adic ètale cohomology.

Let Y be a geometrically connected smooth affine scheme of finite type over 𝔽_{q} with function field 𝔽_{q}(Y). Let π_{1}(Y) be the arithmetic fundamental group of Y. It is the absolute Galois group of 𝔽_{q}(Y) modulo the closed subgroup generated by the inertial subgroups at closed points of Y. Let ℓ be a prime number. Let

ψ_{ℓ} : π_{1}(Y) ⟶ GL_{n}(ℤ_{ℓ})  (17)

be a continuous ℓ-adic representation. Equivalently, ψ_{ℓ} defines a lisse ℓ-adic ètale sheaf on Y. The L-function of ψ_{ℓ} is defined in a standard manner as follow:

L(ψ_{ℓ}, T) = ∏_{y∈|Y |} \frac{1}{det(I − ψ_{ℓ}(Frob_{y})T^{deg(y)})} ∈ 1 + Tℤ_{ℓ}[[T]],  (18)

where Frob_{y} denotes the Frobenius conjugacy class of π_{1}(Y) at y. If ψ_{ℓ} = 1 is the trivial representation, then L(ψ_{ℓ}, T) = Z(Y, T) is rational. More generally, if ψ_{ℓ} is of finite order, then L(ψ_{ℓ}, T) is also rational. This follows from the following general result.

Theorem 7.1 (Grothendieck[11]). Let ψ_{ℓ} be a continuous ℓ-adic representation of π_{1}(Y) as above. If ℓ ≠ p, then L(ψ_{ℓ}, T) is a rational function over ℚ_{ℓ}.

In the remaining case that ℓ = p, the situation is more complicated and the Lfunction is not rational in general. Katz [12] conjectured that L(ψ_{p}, T) is p-adic meromorphic everywhere. This turned out to be false in general [18]. Dwork’s original conjecture [9] says that the L-function L(ψ_{p}, T) is p-adic meromorphic if the p-adic representation ψ_{p} is geometric in some sense (coming from the relative p-adic ètale cohomology of a family of smooth proper varieties for instance). This conjecture of Dwork has been proved to be true. Thus, we have

Theorem 7.2 (Wan [21][22]). Let ψ_{p} be a continuous p-adic representation of π_{1}(Y). If ψ_{p} is geometric in the sense of [21], then L(ψ_{p}, T) is p-adic meromorphic.

If ℓ ≠ p, the geometric Langlands conjecture as proved by Lafforgue [14] shows that up to a constant twist, every irreducible ℓ-adic representation of π_{1}(Y) comes from algebraic geometry. On the other hand, if ℓ = p, there are a lot more non-geometric highly transcendental p-adic representations because of very wild ramifications. It is not clear how to characterize the geometric p-adic representations of π_{1}(Y). This can be viewed as some sort of p-adic geometric Langlands conjecture, which has not been formulated yet.

From now on, we assume that ψ_{ℓ} is a geometric ℓ-adic representation, where ℓ may be equal to p. From the above results, we know that the L-function L(ψ_{ℓ}, T) is ℓ-adic meromorphic everywhere (in fact, rational if ℓ ≠ p). For an integer k, we can define the k-th moment L-function of ψ_{ℓ} by

L(ψ_{ℓ}^{k}, T) = ∏_{y∈|Y |} \frac{1}{det(I − ψ_{ℓ}(Frob_{y})^{k}T^{deg(y)})} ∈ 1 + Tℤ_{ℓ}[[T]].  (19)

This L-function is the k-th moment of the Euler factors of L(ψ_{ℓ}, T). It is also the L-function of the k-th Adams operation ψ_{ℓ}^{k}, which is a virtual ℓ-adic representation. Similarly, we have

Theorem 7.3 (Wan [22][24]). Let ψ_{ℓ} be a geometric ℓ-adic representation of π_{1}(Y). Then, for each integer k, L(ψ_{ℓ}^{k}, T) is ℓ-adic meromorphic everywhere (rational if ℓ ≠ p).

A further question is to understand the ℓ-adic variation of L(ψ_{ℓ}^{k}, T) as the integer k varies ℓ-adically. Just as in the zeta function case, this leads to a suitable limiting moment L-function.

Let D = D_{ψ_{ℓ}} be the order of the image of the residue repsentation ψ̄_{ℓ}. Let k_{1} and k_{2} be integers. As in the previous section, it is easy to see that the power series L(ψ_{ℓ}^{k_{1}+Dk_{2}}, T) is ℓ-adically continuous in k_{2}. This continuity shows that L(ψ_{ℓ}^{k_{1}+Dk_{2}}, T) is a well defined ℓ-adic power series in ℤ_{ℓ}[[T]] for all k_{1} ∈ ℤ and k_{2} ∈ ℤ_{ℓ}. This power series is clearly ℓ-adic analytic in the open unit disk |T |_{ℓ} < 1. It can be viewed as the ℓ-adic L-function attached to the representation ψ_{ℓ}. We can ask if this limiting moment L-function is ℓ-adic meromorphic on the closed unit disk or even everywhere. This is unknown if the rank of ψ_{ℓ} is greater than 1. But, if the rank of ψ_{ℓ} is 1, it is true.

Theorem 7.4 (Wan [22]) Let ψ_{ℓ} be a rank one geometric ℓ-adic representation of π_{1}(Y), where ℓ may be equal to p. For all integers k_{1} and ℓ-adic integers k_{2} ∈ ℤ_{ℓ}, the limiting moment L-function L(ψ_{ℓ}^{k_{1}+Dk_{2}}, T) is ℓ-adic meromorphic everywhere (rational if ℓ ≠ p).

This result is easy if ℓ ≠ p since ψ_{ℓ} is then geometrically of finite order by geometric class field theory. If ℓ = p, the result is non-trivial and follows from our work on the rank one case [22] of Dwork’s conjecture. For higher rank geometric representations, just as in the previous section on limiting moment zeta functions, L(ψ_{ℓ}^{k_{1}+Dk_{2}}, T) is known to be ℓ-adic meromorphic if either k_{2} is a integer (Wan [21]) or k_{1} + Dk_{2} is the formal symbol dℓ^{∞} (Lenstra-Wan [15]).

In the case that ℓ = p, the above rank one theorem and its proof [22] can be pushed further as first observed by Coleman in his private notes on the author’s proof of the Dwork conjecture. The full form of this refinement, which we now describe, has been carried out by Grosse-Klönne [10].

Let ψ_{p} be a continuous rank one geometric p-adic representation

ψ_{p} : π_{1}(Y) ⟶ GL_{1}(ℤ_{p}) = ℤ_{p}^{∗}.

Let ℂ_{p} be the completion of an algebraic closure of ℚ_{p}. Let W_{p} be the set of continuous p-adic homomorphisms from ℤ_{p}^{∗} to ℂ_{p}^{∗}. It is called the weight space. It is a rigid analytic space. It can be identified with a finite disjoint union of open unit disks. For χ ∈ W_{p}, the composition χ ∘ ψ_{p} is a continuous group homomorphism from π_{1}(Y) to ℂ_{p}^{∗}. For example, for each k ∈ ℤ, the map a → a^{k} is a continuous group homomorphism from ℤ_{p}^{∗} to ℂ_{p}^{∗}. One can define an L-function as above by

L(χ ∘ ψ_{p}, T) = ∏_{y∈|Y |} \frac{1}{1 − (χ ∘ ψ_{p})(Frob_{y})T^{deg(y)}} ∈ 1 + Tℂ_{p}[[T]].  (20)

In this way, L(χ ∘ ψ_{p}, T) becomes a function in the two variables (χ, T). In the special case that χ is the k-th power map, L(χ ∘ ψ_{p}, T) becomes the k-th moment L-function L(ψ_{p}^{k}, T), which is just L(ψ_{p}^{⊗k}, T) since ψ_{p} has rank one.

Theorem 7.5 (Grosse-Klönne [10]) Let ψ_{p} be a rank one geometric p-adic representation of π_{1}(Y). Then, L(χ∘ψ_{p}, T) is a two variable meromorphic function in the domain (χ, T) ∈ W_{p} × ℂ_{p}.

In the special case that ψ_{p} comes from the universal family of elliptic curves over 𝔽_{p}, this result was proved by Coleman [3] and the zero locus of the rigid meromorphic function L(χ ∘ ψ_{p}, T) is closely related to the eigencurve [4]. The special value L(χ∘ψ_{p}, 1), which is a meromorphic function in χ ∈ W_{p}, is related to Iwasawa theory and p-adic L-functions.

8 Moment zeta functions: arithmetic case

In the arithmetic (characteristic zero) case, we can similarly define a k-th moment zeta function. We now briefly describe this. Let X be a scheme of finite type over ℤ as in section 3. We assume that the characteristic is zero. The Hasse-Weil zeta function attempts to understand the arithmetic variation of the family Z(X ⊗ 𝔽_{p}, p^{−s}) as the prime p varies. It corresponds to the first moment zeta function. More generally, for each positive integer k, we define the k-th moment Hasse-Weil zeta function by

ζ(k, X, s) = ∏_{p prime} Z(X ⊗ 𝔽_{p^{k}}, p^{−s}).  (21)

In the case that k = 1, this reduces to the original Hasse-Weil zeta function. For example, one checks that

ζ(k, 𝔸^{n}/ℤ, s) = ζ(ℤ, s − nk).

Generalizing the conjectures in section 3, we have

Conjecture 8.1 Let X be a scheme of finite type over ℤ. For each positive integer k, the k-th moment Hasse-Weil zeta function ζ(k, X, s) extends to a meromorphic function in the whole complex plane s ∈ ℂ.

The meaning of this conjecture is that there exists a good structural distribution law for the family Z(X ⊗ 𝔽_{p}, p^{−s}) as p varies.

To get further information about the conjectural distribution law, one needs to know the nature of the zeros and poles. This is given by the following further generalization of the Riemann Hypothesis.

Conjecture 8.2 Let X be a scheme of finite type over ℤ. For each positive integer k, if ζ(k, X, s) = 0 or ∞ for some s ∈ ℂ, then Re(s) ∈ \frac{1}{2}ℤ.

The meaning of this conjecture is that there is a sharp distribution law for the arithmetic family of meromorphic functions Z(X ⊗ 𝔽_{p}, p^{−s}) parametrized by p. By the previous theorems, the geometric analogue of the above two conjectures hold when we consider a geometric family f : X ⟶ Y over a finite field instead of the arithmetic family X ⟶ Spec(ℤ).

Returning to the characteristic zero case. If dim(X) = 1, it is easy to show that ζ(k, X, s) is meromorphic for all k by Hecke’s theorem. If dim(X) = 2, the meromorphic continuation of ζ(k, X, s) is unknown, even in the case that the generic fibre X ⊗ ℚ is an elliptic curve without CM and k ≥ 10. Assuming the meromorphic continuation of ζ(k, X, s), one can further ask the p-adic variation of ζ(k, X, s) (or its special value at s = 1 or at s = 0) as k varies p-adically. The resulting limiting moment zeta function (if it exists) for X = Y × 𝔸^{n} (n > 0) is presumably related to the conjectural construction of the p-adic zeta function of Y. This is easily seen to be the case for Y = Spec(ℤ), since for any positive integer k, we have the limiting formula

ζ_{p}(ℤ, 1 − 2k) = lim_{m→∞} ζ(k + (p − 1)p^{m}, 𝔸^{2}/ℤ, 1),

where ζ_{p}(ℤ, s) is the p-adic zeta function of ℤ.

Instead of considering a single scheme X over ℤ, we can also consider a morphism f : X ⟶ Y of schemes of finite types over ℤ. The k-th moment zeta function of the morphism f is defined to be

ζ(k, f, s) = ∏_{y∈|Y |} Z(X_{y} ⊗ 𝔽_{y}(k), N(y)^{−s}),

where 𝔽_{y}(k) denotes the degree k extension field of the finite residue field 𝔽_{y} at y. The above two meromorphic continuation and GRH conjectures extend to ζ(k, f, s).

In the characteristic zero case, one can also define moment L-functions and possibly limiting moment L-functions. This would involve additional notions such as compatible systems and motives. We leave such a generalization to the reader.

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