Euler factors of equivariant $L$–functions
of Drinfeld modules and beyond

Let $E/F/k$ be a tower of finite field extensions, where $k:=\mathbb{F}_{q}(t)$, $q$ is a power of a prime $p$, and $E/F$ is Galois, of abelian Galois group $G$. Further, let $\mathcal{O}_{F}$ and $\mathcal{O}_{K}$ denote the integral closure of $A:=\mathbb{F}_{q}[t]$ in $F$ and $K$, respectively, and consider a Drinfeld module $E$ defined on $A:=\mathbb{F}_{q}[t]$ and with coefficients in $\mathcal{O}_{F}$.

In [2], the authors associated to the set of data $(K/F/k,E)$ a $G$–equivariant Goss–type $L$–function

where $\mathbb{S}_{\infty}$ is a certain Goss space of mixed characteristic containing naturally a copy of $\mathbb{Z}_{\geq 0}$ and $\mathbb{C}_{\infty}$ is the completion of the algebraic closure of $k_{\infty}:=\mathbb{F}_{q}((t^{-1}))$ with respect to the unique extension of the valuation $v_{\infty}$ of $k$ of uniformizer $1/t.$ (See the Introduction in [2] for the precise definitions.)

As a natural $G$–equivariant generalization of the Goss $\zeta$–function $\zeta_{F}^{E}$ associated to $E$ and studied by Taelman in [13], the $L$–function $\Theta_{K/F}^{E}$ is defined as an infinite product of Euler factors

where $v$ runs over all the primes in the maximal spectrum ${\rm MSpec}(\mathcal{O}_{F})$ of $\mathcal{O}_{F}$ which are tamely ramified in $K/F$ and of good reduction for $E$. For each such prime $v$, $P_{v}^{\ast,G}(X)$ is a polynomial with coefficients in $A[G]$, very closely related to the well understood characteristic polynomial

of the action of a Frobenius morphism $\sigma(v)\in{\rm Gal}(\overline{F}/F)$ associated to $v$ on the $v_{0}$–adic Tate module $T_{v_{0}}(E)$ associated to the rank $r$ Drinfeld module $E$ at any prime $v_{0}\in{\rm MSpec}(A)\setminus\{v\cap A\}$, viewed as a free, rank $r$ module over the completion $A_{v_{0}}$ of $A$ at $v_{0}$. In fact, one has equalities (see [2], Introduction):

where the infinite product converges in $\mathbb{F}_{q}((t^{-1}))[G]$, $\sigma_{v}\in G$ is a choice of Frobenius for $v$ in $G$, and $e_{v}$ is the idempotent in $A[G]$ associated to the trivial character of the inertia group of $v$ in $G$.

The main result of [2] is the proof of a Tamagawa number formula of the type

where $\Theta_{K/F}^{E,\mathcal{M}}(0)$ is a certain Euler product completion of $\Theta_{K/F}^{E}(0)$ at primes $v\in{\rm MSpec}(\mathcal{O}_{F})$ which are either wildly ramified in $K/F$ or of bad reduction for $E$, constructed out of some additional arithmetic data $\mathcal{M}$ (called a taming module) for $K/F$. The numerator and denominator of the right side in the above formula are both volumes (with values in $\mathbb{F}_{q}((t^{-1}))[G]$ and defined precisely in [2]) of certain compact $A[G]$–modules of Arakelov type $E(K_{\infty}/\mathcal{M})$ and $K_{\infty}/\mathcal{M}$.

The above formula, proved in [2] for Drinfeld modules and extended in [4] for higher dimensional abelian $t$–modules, generalizes to the $G$–equivariant setting Taelman’s celebrated class–number formula [13] for the value $\zeta_{F}^{E}(0)$ of the Goss zeta–function associated to $E$. The interested reader should also consult [1] for a slightly different proof in the abelian $t$–module case, using deformation theory, which only gives the desired result under certain restrictive conditions.

However, the proof of the above formula in [2] and [4] (and the same applies to the proofs given in [13] and [1]) hinges upon an essential equality at the level of Euler factors, namely

for all good and tame primes $v$ as above, where the numerator and denominator of the right side are certain special generators of the Fitting ideals of the finite $A[G]$–modules of finite projective dimension $E(\mathcal{O}_{K}/v)$ and $\mathcal{O}_{K}/v$, respectively. In the Appendix of [2], we prove the above equality only in the simplest case where $E:=\mathcal{C}$ is the Carlitz module, and Taelman does the same in [13], in the case where $G$ is trivial.

In this paper, we develop techniques which allow us to prove the above equality for arbitrary Drinfeld modules (see Theorem 2.3 and its proofs in the unramified case, given in §5 and tamely ramified case, given in §6.) In §8, we also indicate how our techniques can be easily extended to prove the above formula for higher dimensional abelian $t$–modules, satisfying a certain purity condition. The case of arbitrary abelian $t$–modules will be treated in a separate, upcoming paper.

Acknowledgement. We would like to thank Urs Hartl for his expert and generous help with the material on local shtukas in §7. His private e-mail exchanges with the first author [6] were invaluable to us.

Let $p$ be a prime number and let $q$ be a power of $p$. In what follows, $k:=\mathbb{F}_{q}(t)$ denotes the rational function field in one variable over $\mathbb{F}_{q}.$ For any commutative $\mathbb{F}_{q}$-algebra $R,$ we denote by $\tau$ the $q$-power Frobenius endomorphism of $R$. We denote by $R\{\tau\}$ the twisted polynomial ring in $\tau$, with the property that

Let $F$ be a finite, separable extension of $\mathbb{F}_{q}(t)$ and let $K$ be a finite abelian extension of $F$ with Galois group $G.$ We also assume that the field of constants in $K$ is $\mathbb{F}_{q},$ i.e.

Let us denote $\mathbb{F}_{q}[t]$ by $A.$ Note that if $v$ denotes an arbitrary normalized valuation on $\mathbb{F}_{q}(t)$ and $\infty$ denotes the normalized valuation of uniformizer $1/t$, then

Let $\mathcal{O}_{F}$ and $\mathcal{O}_{K}$ denote the integral closures of $A$ in $F$ and $K$, respectively. In what follows, we abuse notation and use the same letter for normalized valuations and the associated maximal ideals of elements of strictly positive valuation.

Next, we consider a Drinfeld module $E$ of rank $r\in\mathbb{N}$ defined on $A$ with values in $\mathcal{O}_{F}\{\tau\}.$ More precisely, $E$ is given by an $\mathbb{F}_{q}$-algebra morphism

where $a_{i}\in\mathcal{O}_{F}$, for all $i$ and $e_{r}\neq 0.$ This gives rise to a functor

In other words, for any $\mathcal{O}_{F}\{\tau\}[G]$-module $M,$ we denote by $E(M)$ the $A[G]-$module whose underlying $\mathbb{F}_{q}[G]$–module is $M$ and the $A$-action is given by

Let $v_{0}\in{\rm MSpec}(A)$ and let $A_{v_{0}}$ and $k_{v_{0}}$ denote the completions of $A$ and $k$ with respect to the valuation $v_{0}$. For all $n\in\mathbb{N}$, we denote by $E[v_{0}^{n}]$ the $A_{v_{0}}$-module of $v_{0}^{n}$-torsion points of $E,$ i.e.

The $v_{0}$-adic Tate module of $E$ is defined as

Since $A$ is a PID, we also have

where the transition maps in the projective limit are given by multiplication with a generator of $v_{0}$, while $E[v_{0}^{\infty}]=\bigcup_{n\geq 1}E[v_{0}^{n}].$ Recall that $E[v_{0}^{n}]$ and $T_{v_{0}}(E)$ are free modules of rank $r$ over $A/v_{0}^{n}$ and $A_{v_{0}}$, respectively, and are endowed with obvious $A_{v_{0}}$-linear, continuous $G_{F}$-actions, where $G_{F}=Gal(\overline{F}/F).$

Let $v\in$ MSpec($\mathcal{O}_{F}$), such that $v\nmid v_{0}.$ Fix a choice of decomposition group $G(v)\subset G_{F},$ and a Frobenius morphism $\sigma(v)\in G(v).$ Then, it is known (see [2] and the references therein) that if $E$ has good reduction at $v$ (i.e. $v\nmid e_{r}$), the $G_{F}$-representation $T_{v_{0}}(E)$ is unramified at $v$ and the polynomial

is independent of $v_{0}$ and actually lies in $A[X].$ Above, $I_{r}$ denotes the $r\times r$ identity matrix.

The following is Proposition A5.1. from the Appendix in [2]:

Let $I_{v}\subset G_{v}\subset G$ denote the inertia and decomposition groups of $v$ in $G$, respectively. Let $\sigma_{v}$ be the image of ${\sigma(v)}$ via the Galois restriction map $G(v)\twoheadrightarrow G_{v}$. Our main goal in this paper is the proof of the following.

A proof of the above statement in the case where $E$ is the Carlitz module $C$, defined by $\phi_{C}(t)=t+\tau$, was given in the Appendix of [2]. Below, we develop techniques which settle the above theorem for a general Drinfeld module $E$. Proposition 1.2(2) gives us a good understanding of $|\mathcal{O}_{K}/v|_{G}$. Therefore a major portion of our work is directed towards understanding the relation between $|E(\mathcal{O}_{K}/v)|_{G}$ and $P_{v}(\sigma_{v}e_{v}).$

In this section, we fix a prime $v\in\text{MSpec}(\mathcal{O}_{F})$ such that $E$ has good reduction at $v.$ We are not assuming that $v$ is necessarily tamely ramified in $K/F$. Let us denote by $w_{0}$ the prime in $A$ that lies below $v.$ After reduction of $E\mod v$ (i.e. reduction of the coefficients of $E$ modulo $v$), we obtain the rank $r$ Drinfeld module $\overline{E},$ defined over $\mathcal{O}_{F}/v$, given by the $\mathbb{F}_{q}$-algebra morphism

where $\phi_{\overline{E}}(t)=i(t)\cdot\tau^{0}+...+i(e_{r})\cdot\tau^{r}$ with $i:A\xhookrightarrow{}\mathcal{O}_{F}\twoheadrightarrow\mathcal{O}_{F}/v$ being the obvious map. The Drinfeld module $\overline{E}$ has rank $r$, characteristic $w_{0}:=\ker(i)$, and height $h$. By definition, $h$ is the unique integer $0<h\leq r$ which, for any generator $\pi_{w_{0}}$ of $w_{0}$, satisfies the equality

where $d_{0}:=[A/w_{0}:\mathbb{F}_{q}]$, $b_{i}\in\mathcal{O}_{F}/v$, and $b_{d_{0}h}\neq 0$. (See [3], Section 4.5 for the existence of $h$.)

Recall that, by the notation introduced above, we have a field isomorphism $\mathcal{O}_{F}/v\simeq\mathbb{F}_{q^{n_{v}}}.$ Consequently, the Frobenius morphism ${\rm Frob}_{q^{n_{v}}}:=\tau^{n_{v}}$ is an endomorphism of $\overline{E}$ and it acts naturally and linearly on all the Tate modules associated to $\overline{E}$.

Next, we fix $v_{0}\in\text{MSpec}(A)$, $v_{0}\neq w_{0}$ and consider the characteristic polynomial of the action of the $q^{n_{v}}$-power Frobenius morphism, viewed as an endomorphism of $\overline{E}$, on the free $A_{v_{0}}$–module $T_{v_{0}}(\overline{E})$ of rank $r$:

Then, $f_{\overline{E}}(X)$ is independent of $v_{0}$ and lies in $A[X]$. (See §4.12 in [3].) By Theorem 4.12.15 in [3] and the discussion preceding that, we have the following.

Further, one can consider the characteristic polynomial of ${\rm Frob}_{q^{n_{v}}}$ (viewed as endomorphism of $\overline{E}$) acting on the free $A_{w_{0}}$–module $T_{w_{0}}(\overline{E})$ of rank $(r-h)$ (see [3], Section 4.5 for the calculation of the rank)

This is a monic polynomial in $A_{w_{0}}[X]$ of degree $(r-h).$ In §? below we will prove the following.

Let $\mathcal{O}_{v}$ and $F_{v}$ be the completions at $v$ of $\mathcal{O}_{F}$ and $F$, respectively. Our choice of decomposition group $G(v)$ corresponds to choosing an embedding $\overline{F}\to\overline{F_{v}}$ at the level of separable closures of $F$ and $F_{v}$, such that Galois restriction induces a group isomorphism $G(\overline{F_{v}}/F_{v})\simeq G(v)$. Since $E$ has good reduction at $v$ and the Galois representations $E[v_{0}^{n}]$ are unramified at $v$, it is not difficult to see that we have

where $\mathcal{O}_{v}^{nr}$ is the integral closure of $\mathcal{O}_{v}$ in the maximal unramified extension $F_{v}^{unr}$ of $F_{v}$ in $\overline{F_{v}}$. Moreover, the reduction $\mod v$ map induces isomorphisms of $A_{v_{0}}[[\overline{G(v)}]]$–modules

where

The group isomorphism above sends $\overline{{\sigma(v)}}$ (the image of our choice of Frobenius $\sigma(v)$ in $\overline{G(v)}$) to ${\rm Frob}_{q^{n_{v}}}$. Consequently, we have an equality of characteristic polynomials in $A[X]$:

Consequently, Proposition 3.1 gives us information on the roots of the characteristic polynomial $P_{v}(X)$. The following corollary regarding the coefficients of $P_{v}(X)$ will be particularly useful in what follows.

We begin by stating a general commutative algebra result regarding Fitting ideals of modules over certain rings of equivariant Iwasawa algebra type. For a proof of this result, see Proposition 4.1 in [5].

An immediate consequence of the above proposition is the following.

Next, we fix a prime $w\in{\rm MSpec}(\mathcal{O}_{K})$ lying above $v$. We let $G(w)=G(v)\cap G_{K}$, $I(w)=I(v)\cap G_{K}$ and $\sigma(w):=\sigma(v)^{f}$, where $f:=f(w/v)=[\mathcal{O}_{K}/w:\mathcal{O}_{F}/v]$. Then, $\sigma(w)\in G(w)$ is a choice of Frobenius for $w$ and its image $\overline{\sigma(w)}\in\overline{G(w)}:=G(w)/I(w)$ corresponds via the group isomorphism $\overline{G(w)}\simeq G_{\mathcal{O}_{K}/w}$ to ${\rm Frob}_{q^{fn_{v}}}$, viewed as an endomorphism of $\overline{E}$.