L-functions over function fields: A cohomological approach

(report on joint work with Richard Pink)

Gebhard Böckle

June 22, 1999

Notation X/F_{q} a scheme of finite type σ = σ_{X} = Frob_{X} |X| the closed points of X for x ∈ |X|: k_{x} the residue field at x d_{x} = [k_{x} : F_{q}] its degree

1 Motivation

The Hasse-Weil zeta-function

Z(X, T) = ∏_{x∈|X|}(1 − T^{d_{x}})^{−1} ∈ ℤ[[T]] (1)

provides an ingeneous way of encoding #X(F_{q^{n}}), n ∈ ℕ because of the

Weil conjecture (1949). There exists a cohomology theory X ↦ H_{c}^{∗}(X) such that H_{c}^{i} are finite-dimensional vector spaces over a field of characteristic zero carrying an endomorphism Frob_{q}; H_{c}^{i}(X) = 0 for i > dim X and

Z(X, T) = ∏_{i=0}^{2 dim X} det(1 − T Frob_{q}|H_{c}^{i}(X))^{(−1)^{i+1}}.

In particular, Z(X, T) is a rational function of T.

A rationality proof of Z(X, T) was given by Hasse in ’33 for elliptic curves, by Weil in ’40 for all curves, by Dwork in ’60 in general. Artin and Grothendieck gave a general proof in the sixties along the lines foreseen by Weil using the formalism of l-adic cohomology.

For a constructible ℚ_{l}-sheaf F one obtains a morphism

ϕ : σ_{X}^{∗}F → F,

This is used to define the L-function of F. Formula (1) is then a consequence of the Lefschetz-Verdier trace formula.

Global L-functions. For any global scheme X of finite type over Spec(ℤ) its fibers above p are defined by

X  ⟵  X_{p}
fin. type↓  ↓
Spec ℤ ⟵ Spec F_{q}

One then combines all the local data in the following way to obtain a global zeta-function

Z^{gl}(X , s) = ∏_{x∈|X |}(1 − (#k_{x})^{−s})^{−1} = ∏_{p prime} Z(X_{p}, T)|_{T =p^{−s}}.

(For X = Specℤ, the global zeta-function is the Riemann zeta-function.) For global zetafunctions one expects a meromorphic continuation to all of ℂ (and more).

2 A (global) function field analogue

More notation.

A a normal domain (or a finite ring, or ...)

C = Spec(A) the “coefficient scheme”

K = Frac(A)

Definition 2.1 A τ-sheaf \underline{F} on X over A consists of

(i) a coherent sheaf F of O_{X} ⊗ A-modules

(ii) a morphism τ := τ_{\underline{F}} : (σ_{X} × id_{C})^{∗}F → F.

The category of τ-sheaves on X over A is denoted by Coh_{τ} (X, A).

Examples 2.2 (i) If C is a smooth projective curve, ∞ a closed point of C, and A the coordinate ring of C := C − {∞} then there is a natural map

\{ \begin{matrix}Families of Drinfeld \\ A − modules param. by X \end{matrix} \} → \{ \begin{matrix}τ-sheaves \underline{F} on X over A \\ (\underline{F} locally free) \\ and a map “char”: X → Spec A \end{matrix} \}

where the map “char” is the analogue of X → Spec ℤ.

(ii) For 𝔪 ∈ MaxSpec(A), there is an equivalence of categories

\{ \begin{matrix}Continuous Â_{𝔪}-adic Galois \\ representation of π_{1}(X) \end{matrix} \} → \{ \begin{matrix}lisse formal τ-sheaves \\ with Â_{𝔪}-coefficients \end{matrix} \}

= \{ \begin{matrix}Inverse system (\underline{F}_{n}) \\ \underline{F} locally free/X ⊗ A/𝔪^{n}, \\ and an iso.τ_{n} : (σ × id)^{∗}\underline{F}_{n} → \underline{F}_{n} \end{matrix} \}

David Goss has made two conjectures, wher he assumes that \underline{F} and A are as in Example 2.2.(i) above. (At least for the first conjecture this is not necessary.)

Define, for A normal,

L(\underline{F}, X, τ) := ∏_{x∈|X|} det(1 − T^{d_{x}}τ_{x}^{d_{x}}|F_{x} ⊗_{A} K) ∈ A[[T]],

where F_{x} is the fiber of F at x.

Conjecture 2.3 (Local) L(\underline{F}, Xτ) is a rational function in T.

For the global conjecture, assume that \underline{F} and char : X → A are as in the example above. Let K_{∞} be the completion of K w.r.t. the ∞-valuation and A_{∞} its ring of integers. Let ℂ_{∞} be a completion of an algebraic closure of K_{∞}.

We now want to make sense out of 𝔭^{−s} for 𝔭 a prime ideal in A and s in some suitable domain. For simplicity, assume that A = F_{q}[T], let 𝔭 ∈ A be generated by a monic element f_{𝔭} of A. For s ∈ S_{∞} := ℂ_{∞}^{∗} × ℤ_{p} define

𝔭^{s} := x^{deg 𝔭} · (\frac{f_{𝔭}(t)}{t^{d_{x}}})^{y} ∈ ℂ_{∞}^{∗} · A_{∞}^{∗} = ℂ_{∞}^{∗},

so that 𝔭^{s}/x^{deg 𝔭} is a 1-unit in A_{∞}.

Consider the pullback map i_{𝔭} given by

X  ⟵^{i_{𝔭}} X_{𝔭}
char↓  ↓
Spec A ⟵ 𝔭

Define for |x| > c (c sufficiently large) and all y ∈ ℤ_{p}

L^{gl}(\underline{F}, X, s) := ∏_{𝔭}L(i_{𝔭}^{∗}\underline{F}, X_{𝔭}, T)|_{T =𝔭^{−s}}.

Conjecture 2.4 (Global) L^{gl}(\underline{F}, X, s) is a “continuous family of meromorphic functions on ℂ_{∞} parametrized by ℤ_{p}”

The local and global conjecture for A = F_{q}[T] were proved in ’96 by Taguchi and Wan using Dwork-type methods.

3 A cohomology theory of ‘τ-sheaves’: Crystals

We now want to construct a cohomology theory with ‘τ-sheaves’ as coefficients with as many of the usual six functors f_{∗}, f^{∗}, f_{!}, f^{!}, ⊗, Hom as possible, as well as a trace formula. We will be able to construct f_{!}, f^{∗}, ⊗, i.e., half of the usual six functors, and prove a trace formula. The absence of half of the functors should not come as too much of a surprise, as there are no known functional equations for the L-functions our theory will describe, and thus one cannot hope for a duality theory.

Known results for τ-sheaves: For a smooth projective curve X over F_{q}, one has the Woodshole fixed point formula for coherent cohomology which gives a trace formula for τ-sheaves. Furthermore Anderson proves, by elementary means, a trace formula for X smooth (not necessarily proper), and \underline{F} locally free, however without any cohomological interpretation.

Missing for τ-sheaves: j_{!}: A definition of j_{!} for an open immersion j. (The usual j_{!} of a coherent sheaf is in general no longer coherent.) Rigidity: Example 2.2.(ii) above suggests that τ-sheaves should behave like étale sheaves, and in particular they should posess no non-trivial infinitesimal deformations:

{‘τ-sheaves’ on X} ≅ {‘τ-sheaves’ on X_{red}}

3.1 Crystals

The ‘τ-sheaves’, which from now on are called crystals, are defined as follows.

Definition 3.1 (i) \underline{F} ∈ Coh_{τ} (X, A) is called nilpotent iff

∃n : τ^{n}: (σ^{n} × id)^{∗}F → F is zero.

(ii) ϕ: \underline{F} → \underline{G} is a nil-isomorphism (we write \underline{F} ⇒ \underline{G}) iff ker(ϕ), coker(ϕ) are nilpotent. Fact: The nil-isomorphism form a saturated multiplicative system S_{nil} in Coh_{τ} (X, A).

(iii) Crys(X, A) is defined to be the localization of Coh_{τ} (X, A) at S_{nil}. The objects of Crys(X, A) are called crystals.

Crys(X, A) has the same objects as Coh_{τ} (X, A), however, the morphism sets are different. For crystals \underline{F}, \underline{G}, thought of as τ-sheaves, a morphism is a diagram

\underline{F} ⇐ \underline{F}′ → \underline{G}.

Fact: \underline{F} ⇒ \underline{G} implies that L(\underline{F}, X, T) = L(\underline{G}, X, T).

Theorem 3.2 If A is finite, then Crys(X, A) is equivalent to the category of constructible étale A-torsion sheaves on X.

3.2 Construction of j_{!}

Suppose we are given U ↪^{j} X ↩^{i} Y, j an open immersion, i the embedding of a closed complement whose ideal sheaf is given by J. Let \underline{F} be a crystal on U (over A).

Let F̃ be any coherent extension of F to X ⊗ C. Then for n ≫ 0 there exists

τ̃: (σ × id)^{∗}(J^{n}F̃) ≅ F^{qn}(σ × id)^{∗}F̃ → J^{n}F̃.

Thus τ̃ is nilpotent on the restriction of J^{n+1}F̃ to Y. We find:

Theorem 3.3 Given \underline{F}, a crystal on U (over A), there exists a crystal j_{!}\underline{F} on X unique up to unique isomorphism, such that j^{∗}j_{!}\underline{F} ≅ \underline{F} and i^{∗}j_{!}\underline{F} = 0. Furthermore, j_{!} : Crys(U, A) → Crys(X, A) is an exact functor.

Proposition 3.4 With the above notation, the following is a short exact sequence of functors

0 → j_{!}j^{∗} → id → i_{∗}i^{∗} → 0.

3.3 Functors on Crystals

We fix f : X → Y. To define f^{∗} on crystals, we first define it on τ-sheaves using the usual formalsm of f^{∗} for coherent sheaves. One easily checks that f^{∗} preserves nil-isomorphisms, and thus one has defined f^{∗} on crystals.

Theorem 3.5 f^{∗} is exact on crystals.

The main point of the proof is that L_{1}f^{∗}\underline{F} is nilpotent for any τ-sheaf \underline{F}. A useful tool is the following pointwise characterization of nilpotency.

Theorem 3.6 \underline{F} is nilpotent iff for all x ∈ X (or even for all x ∈ |X|), i_{x}^{∗}\underline{F} is nilpotent (i_{x} : {x} → X).

Theorem 3.7 (Rigidity) If f is radicial surjective, then f^{∗} and f_{∗} are mutually inverse isomorphisms between Crys(X, A) and Crys(Y, A).

In particular Crys(X, A) ≅ Crys(X_{red}, A)

Theorem 3.8 If f is proper, then f_{∗}: Crys(X, A) → Crys(Y, A) is right adjoint to f^{∗}.

Properness is needed so that f^{∗} maps coherent sheaves to coherent ones. We could use coherent cohomology to define higher derived functors of f^{∗}. However we would really like to have Rf_{∗} in the derived context.

3.4 Derived categories

Let QCrys(X, A) be the localization of the category of quasi-coherent τ-sheaves at nilisomorphisms. (In the quasi-coherent context, a τ-sheaf is nilpotent iff it is the union of coherent nilpotent subsheaves.) As in general not all quasi-coherent τ-sheaves can be represented as filtered direct limits of coherent ones, the following theorem requires some work.

Theorem 3.9 D^{b}(Crys(X, A)) →^{≅} D_{crys}^{b}(QCrys(X, A)).

Theorem 3.10 For f proper, there exists a right derived functor

Rf_{∗}: D_{crys}^{b}(QCrys(X, A)) → D_{crys}^{b}(QCrys(Y, A)),

which can be computed in via Čech-resolutions.

As j_{!} and f^{∗} are exact on crystals, they obviously give rise to fonctors in the derived setting. Furthermore, for f compactifiable, we define f_{!} := Rf̄_{∗} ∘ j_{!}, where f = f̄∘ j, j is an open immersion and f̄ is proper.

In the derived setting, one can prove the proper base change theorem, the projection formula, and various other expected compatibilities.

4 Trace formula

For (\underline{F}^{•}) ∈ D^{b}(Crys(X, A)), we define

L(X, (\underline{F}^{•}), T) := ∏_{i} L(X, \underline{F}^{i}, T)^{(−1)^{i}} ∈ A[[T]].

Theorem 4.1 For f: X → Y compactifiable (which is always the case, if X, Y are of finite type over F_{q}):

L(X, (\underline{F}^{•}), T) = L(Y, f_{!}(\underline{F}^{•}), T).

Proof: We may assume that A = K, X smooth, affine, quasi-projective, (\underline{F}^{•}) concentrated in degree zero, \underline{F}^{0} locally free. Then we can apply Anderson’s trace formula, and the above result follows upon properly interpreting Anderson’s nucleus.

Corollary 4.2 Conjecture 2.3 holds for any X/F_{q} of finite type, any normal domain A and any crystal on X over A.

5 Global L-functions

Let A = F_{q}[t], let \underline{F} be as in Example 2.2.(ii). Thus A_{∞} = F_{q}[[1/T]] and K_{∞} = F_{q}((1/T)). Using our global theory, we can define ‘motivic pieces’ of \underline{F}:

M^{(i)}\underline{F} := R^{i}char_{!}\underline{F} ∈ Crys(SpecA, A).

For |x| > c one finds L^{gl}(X, \underline{F}, s) = ∏_{i} L^{gl}(SpecA, M^{(i)}\underline{F}, s). Thus to prove ‘meromorphy’ it suffices to understand the M^{(i)}\underline{F}. One may further reduce this to understanding L^{gl}(X, \underline{G}, s) for \underline{G} locally free in Crys(SpecA, A).

Let \underline{C} be the Carlitz module (see Goss’ book on function fields). This is a particular locally free rank one τ-sheaf on SpecA over A. Then for |x| > c|π|^{−n}, one has

L^{gl}(SpecA, \underline{G}, (xT^{−n}, −n)) = L(SpecA, \underline{G} ⊗ \underline{C}^{n}, T)_{|T =x^{−1}}.

The cohomology theory of crystals shows that the right hand side is a polynomial in A[x^{−1}] AND it can be used to show that for n → ∞ the degree of this polynomial in the variable x^{−1} has order of magnitude O(log_{q} n). Thus one has good control over special values of Goss’ global L-function.

The latter together with Goss’ interpolation methods (modelled at those used in the construction of p-adic L-functions), suffices to prove the holomorphy on S_{∞} of L^{gl}(SpecA, \underline{G}, s).

Corollary 5.1 Conjecture 2.4 holds for A = F_{q}[t] and any X/F_{q} of finite type.

Address: Lehrstuhl für Mathematik (Prof. Pink) Universität Mannheim D 7, 27 68131 Mannheim Germany

email: boeckle@math.uni-mannheim.de