On the field generated by the periods of a Drinfeld module

TL;DR

This paper investigates the properties of the field generated by the periods of a Drinfeld module over a local function field, showing it can be arbitrarily large while its residue class degree remains bounded.

Contribution

It generalizes Maurischat's results by demonstrating the potential unboundedness of the period field over $K_{ olinebreak}_ olinebreak{ olinebreak}_ olinebreak_ olinebreak$ and establishes a bound on the residue class degree depending only on the rank.

Findings

The period field $K_{ ext{infty}}( ext{Lambda})$ can be arbitrarily large over $K_{ ext{infty}}$.

The residue class degree $f(K_{ ext{infty}}( ext{Lambda})|K_{ ext{infty}})$ is bounded by a constant depending only on the rank $r$.

Generalization of Maurischat's results to broader classes of Drinfeld modules.

Abstract

Generalizing the results of Maurischat in \cite{Maurischatxx}, we show that the field $K_{\infty}(\Lambda)$ of periods of a Drinfeld module $\phi$ of rank $r$ defined over $K_{\infty} = \mathds{F}_{q}((T^{-1}))$ may be arbitrarily large over $K_{\infty}$. We also show that, in contrast, the residue class degree $f( K_{\infty}(\Lambda) | K_{\infty})$ remains bounded by a constant that depends only on $r$.

Equations

$${\varphi }_T(X)=TX+g_1{X}^q+\cdots +g_r{X}^{q^r},g_1,\dots ,g_r\in L,g_r\ne 0$$

$$e_{\Lambda }(Tz)$$

$$e_{\Lambda }(z)$$

$$\Lambda \text{ is contained in the separable closure Lsep of L}.$$

$$L(\Lambda )=\bigcup _{n\in \text{mathds}N}L(\varphi [a^n])=L(\mathrm{tor}(\varphi )).$$

$$G=\mathrm{Gal}(L(\Lambda )|L).$$

$$\Big{\lvert}\sum_{1\leq i\leq r}a_{i}\lambda_{i}\Big{\rvert}=\max_{i}\lvert a_{i}\rvert\lvert\lambda_{i}\rvert$$

$$\underset{{\mathbf{B}}_1}{\underbrace{|{\lambda }_1|=\cdots =|{\lambda }_{r_1}|}}<\underset{{\mathbf{B}}_2}{\underbrace{|{\lambda }_{r_1+1}|=\cdots =|{\lambda }_{r_1+r_2}|}}<\cdots \underset{{\mathbf{B}}_t}{\underbrace{=|{\lambda }_r|}}.$$

$${\overline{\mathbf{B}}}_{\tau }={\mathbf{B}}_1\cup {\mathbf{B}}_2\cup \cdots \cup {\mathbf{B}}_{\tau }\text{and}{\Lambda }_{\tau }=\sum _{\lambda \in {\overline{\mathbf{B}}}_{\tau }}A\lambda (0\le \tau \le t),$$

$$\sigma ({\lambda }_j)=\sum _{1\le i\le r}{a}_{i,j}{\lambda }_i(1\le j\le r)$$

$$|a_{i,j}|\le |{\lambda }_i/{\lambda }_j|.$$

$$\left\{\left(\begin{array}{cc}a& 0\\ c& d\end{array}\right)\right|a,d\in \text{mathds}{F}^{\ast },c\in A\text{ such that ∣c∣≤∣λ2/λ1∣}\}.$$

$${\overline{r}}_{\tau }=r_1+r_2+\cdots +r_{\tau }.$$

$$L_{\tau }:=L({\Lambda }_{\tau })$$

$$G_{\tau }^{\prime }:=\{\sigma \in G\mid a_{i,j}={\delta }_{i,j}\text{ if }1\le i,j\le {\overline{r}}_{\tau }\}$$

$$G_{\tau }:=G/G_{\tau }^{\prime }=\mathrm{Gal}(L_{\tau }|L)$$

$$G_{\tau }=\mathrm{Gal}(L_{\tau }|L)⟶G_{\tau -1}=\mathrm{Gal}(L_{\tau -1}|L)$$

$$\begin{array}[]{|c|c|}\hline\cr P&0\\ \hline\cr Q&R\\ \hline\cr\end{array}\longmapsto P,$$

$$U_{\tau }=\{\sigma \in G_{\tau }\mid a_{i,j}={\delta }_{i,j}\text{ for }i\le {\overline{r}}_{\tau -1}\text{ or }j>{\overline{r}}_{\tau -1}\}.$$

$${\tilde{L}}_{\tau }:=L_{\tau }^{U_{\tau }}=\text{fixed field of Uτ},$$

$$H_{\tau }\text{ be the group Gal(Lτ∣Lτ−1)/Uτ=Gal(L~τ∣Lτ−1)}.$$

$$\begin{array}[]{|c|c|}\hline\cr P&0\\ \hline\cr Q&R\\ \hline\cr\end{array}\longmapsto R,$$

$${\tilde{\Lambda }}_{\tau }:=e_{\tau -1}({\Lambda }_{\tau })$$

$$e_{\tau -1}:{\Lambda }_{\tau }/{\Lambda }_{\tau -1}$$

$$|e_{\tau -1}(\lambda )|=|\lambda |{\prod _{\mu \in {\Lambda }_{\tau }}}^{\prime }|1-\lambda /\mu |=|\lambda |{\prod _{\mu \in {\Lambda }_{\tau },|\mu |\le |\lambda |}}^{\prime }|\lambda /\mu |,$$

$$\sigma ({\lambda }_j)=\sum _{1\le i\le {\overline{r}}_{\tau }}{a}_{i,j}{\lambda }_i$$

$${\tilde{L}}_{\tau }=L_{\tau -1}({\tilde{\Lambda }}_{\tau -1})=L_{\tau -1}(V_{\tau }),$$

$$\mathrm{GL}(V_{\tau })\stackrel{\cong }{⟶}\mathrm{GL}(r_{\tau },\text{mathds}F),$$

$$L_{\tau -1}^{\prime }(V_{\tau })=L_{\tau -1}^{\prime }(V_{\tau }^{\prime }),\text{where }w(V_{\tau }^{\prime })=0.$$

$$V_{\tau }^{\prime }$$

Full text

0. Introduction

The present paper is inspired by the recent note [Maurischatxx] of Andreas Maurischat. Let $\phi$ be a Drinfeld $A$-module of rank $r\in\mathds{N}$ over $K_{\infty}$, where $A=\mathds{F}_{q}[T]$ and $K_{\infty}=\mathds{F}_{q}((T^{-1}))$. The field $K_{\infty}(\Lambda)$ generated by the period lattice $\Lambda$ of $\phi$ agrees with its torsion field $K_{\infty}(\operatorname{tor}(\phi))$, and even with $K_{\infty}(\bigcup_{n\in\mathds{N}}\phi[T^{n}])$, where $\phi[T^{n}]$ is the module of $T^{n}$-torsion of $\phi$.

In the case where $r=2$ (the case $r=1$ is almost trivial), Maurischat shows through investigation of the Newton polygon of the $T$-division polynomial $\phi_{T}(X)$ of $\phi$ and its iterates that the ramification index $e(K_{\infty}(\Lambda)|K_{\infty})$ may become arbitrarily large. We generalize this fact to arbitrary ranks $r\geq 2$ (and to finite extensions $L$ of $K_{\infty})$, see Theorem 3.5. We show moreover that the residue class degree $f(L(\Lambda)|L)$ is bounded through a constant depending only on $r$ (Theorem 4.1), thereby answering a question raised by several mathematicians, among which Bjorn Poonen, Chantal David, and Mihran Papikian.

Instead of the Newton polygon, we use the spectral filtration $\Lambda$ as our basic tool. Let $\mathbf{B}=\{\lambda_{1},\lambda_{2},\dots,\lambda_{r}\}$ be a successive minimum basis of $\Lambda$ (see [Gekeler2017] Section 3), grouped $\mathbf{B}=\mathbf{B}_{1}\cup\mathbf{B}_{2}\cup\dots\cup\mathbf{B}_{t}$ into subsets $\mathbf{B}_{t}$ of basis vectors of equal length, and let $\Lambda_{\tau}$ be the sublattice of $\Lambda$ generated by $\mathbf{B}_{1}\cup\dots\cup\mathbf{B}_{\tau}$, where $1\leq\tau\leq t$. Put $L_{\tau}\vcentcolon=L(\Lambda_{\tau})$, so $L_{0}=L$, and $L_{t}=L(\Lambda)$. By means of an analysis of the intermediate extensions $L_{\tau}|L_{\tau-1}$ carried out in Section 2, we get control on the behavior of the ramification index and residue class degree along the tower $(L_{\tau})_{0\leq\tau\leq t}$. Together with the determination of the spectrum $\operatorname{spec}_{A}(\Lambda)=(\lvert\lambda_{1}\rvert,\lvert\lambda_{2}\rvert,\dots,\lvert\lambda_{r}\rvert)$ of $\Lambda$ out of the Newton polygon $\mathrm{NP}(\phi_{\tau}(X))$ in a special case, performed in Section 3, this yields our results Theorem 3.5 and Theorem 4.1.

Notation**:**

• •

\mathds{F}=\mathds{F}_{q}=\text{finite field with q$elements, of characteristic$p},

• •

$A=\mathds{F}[T]$, $K=\mathds{F}(T)$,

• •

$K_{\infty}=\mathds{F}((T^{-1}))=$ completion of $K$ w.r.t. the valuation $v=v_{\infty}$ of $K$ at infinity, with absolute value $\lvert\cdot\rvert$ normalized by $\lvert T\rvert=q$;

• •

$C=\widehat{\overline{K}}_{\infty}=$ completed algebraic closure of $K_{\infty}$, provided with the unique extension of $\lvert\cdot\rvert$ to $C$.

If $K_{\infty}\subset L\subset L^{\prime}$ are finite extensions, $e(L^{\prime}|L)$ and $f(L^{\prime}|L)$ denote the ramification index and the residue class degree, respectively. Throughout, we fix one such $L$.

1. Preliminaries

1.1.

The letter $\phi$ always denotes a Drinfeld $A$-module of rank $r\geq 2$ defined over $L$. It is described through its $\boldsymbol{T}$-operator polynomial.

[TABLE]

or its period lattice $\Lambda$, a discrete free $A$-submodule of $C$ of rank $r$. These are related by

[TABLE]

is the exponential function of $\Lambda$. See e.g. [Goss96], Chapter 4 for the elementary theory of Drinfeld modules and their uniformizations. Given $0\neq a\in A$, we write $\phi[a]$ for the $A$-module of $a$-division points of $\phi$, which is isomorphic with $(A/(a))^{r}$. It is easy to see that in fact $e_{\Lambda}$ as a power series in $z$ has coefficients in $L$, thus:

[TABLE]

Hence $L(\Lambda)$ is a finite Galois extension of $L$. Our starting point is

Proposition 1.2**:**

Let $a$ be any non-constant element of $A$. Then

[TABLE]

Here $\operatorname{tor}(\phi)=\bigcup_{0\neq a\in A}\phi[a]\cong(K/A)^{r}$ is the torsion submodule of $\phi$, and $L(*)$ is the field extension of $L$ generated by $(*)$.

Proof.

The first equality for $L=K_{\infty}$ and $a=T$ is shown in [Maurischatxx] Proposition 2.1; the generalization to arbitrary $L$ and $a$ is obvious, as then is the second equality. ∎

In the sequel, we study the extension $L(\Lambda)|L$ that occurs in 1.2, and its Galois group

[TABLE]

1.3.

Let $\mathbf{B}=\{\lambda_{1},\lambda_{2},\dots,\lambda_{r}\}$ be a successive minimum basis (SMB) of the lattice $\Lambda$. It is characterized by:

1.3.1.

For each $i\in\{1,2,\dots,r\}$, $\lambda_{i}\in\Lambda$ has minimal absolute value $\lvert\lambda_{i}\rvert$ among the lattice vectors $\lambda\in\Lambda\smallsetminus\sum_{1\leq j<i}A\lambda_{j}$. (The condition for $i=1$ means that $\lambda_{1}$ has minimal absolute value $>0$.)

Choosing $\mathbf{B}$ according to 1.3.1, it is in fact a basis of $\Lambda$, which is orthogonal, that is

[TABLE]

for each family $a_{1},\dots,a_{r}$ of coefficients in $K_{\infty}$, and:

1.3.3.

The series $\lvert\lambda_{1}\rvert\leq\lvert\lambda_{2}\rvert\leq\dots\leq\lvert\lambda_{r}\rvert$ of values of the $\lambda_{i}$ is an invariant of $\Lambda$, that is, independent of choices made. (See [Gekeler2017] Section 3 for a proof of the assertions.)

We call the $r$-tuple $(\lvert\lambda_{1}\rvert,\lvert\lambda_{2}\rvert,\dots,\lvert\lambda_{r}\rvert)$ the spectrum $\operatorname{spec}_{A}(\Lambda)$ of the $A$-lattice $\Lambda$. For later use, we cite the following result, which is discussed in [Gekeler2018], Observation 4.3. (Note the different numbering there, in which $\{\lambda_{r},\lambda_{r-1},\dots,\lambda_{1}\}$ forms an SMB of $\Lambda$!)

Proposition 1.4**:**

1. $\mathrm{(i)}$

The spectrum $\operatorname{spec}_{A}(\Lambda)$ and the Newton polygon $\mathrm{NP}(\phi_{T}(X))$ of $\phi_{T}(X)$ determine each other. 2. $\mathrm{(ii)}$

The point $(q^{i},v(g_{i}))$, where $1\leq i<r$, is a break point of $\mathrm{NP}(\phi_{T}(X))$ if and only if $\lvert\lambda_{i}\rvert<\lvert\lambda_{i+1}\rvert$.

The break points of $\mathrm{NP}(\phi_{T}(X))$, i.e., the sizes of the roots of $\phi_{T}(X)$, may be found from $\operatorname{spec}_{A}(\Lambda)$ by [Gekeler2017] Lemma 3.4. However, the inverse map which describes $\operatorname{spec}_{A}(\Lambda)$ in terms of $\mathrm{NP}(\phi_{T}(X))$, is less explicit. See Section 3 for a comparatively simple special case.

1.5.

We collect basis vectors of equal lengths in packets $\mathbf{B}_{\tau}$ of sizes $r_{\tau}$, and ordered according to size:

[TABLE]

That is \mathbf{B}=\mathop{\vphantom{\bigcup}\mathchoice{\leavevmode\vtop{\halign{\hfil\m@th\displaystyle#\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil\m@th\textstyle#\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil\m@th\scriptstyle#\hfil\cr\bigcup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil\m@th\scriptscriptstyle#\hfil\cr\bigcup\cr\cdot\crcr}}}}\displaylimits_{1\leq\tau\leq t}\mathbf{B}_{\tau}, and $r_{\tau}$ is the multiplicity by which the $\tau$-th of the values $\lvert\lambda_{i}\rvert$ counted without multiplicity occurs. We further let

[TABLE]

a direct summand of $\Lambda$ with $\operatorname{rk}_{A}(\Lambda_{\tau}|\Lambda_{\tau-1})=r_{\tau}(1\leq\tau\leq t)$. The sequence of sublattices $\{0\}=\Lambda_{0}\subset\Lambda_{1}\subset\dots\subset\Lambda_{t}=\Lambda$ is intrinsically defined and is called the spectral filtration on $\Lambda$. As the Galois group $G=\operatorname{Gal}(L(\Lambda)|L)$ acts $A$-linearly and length-preserving on $\Lambda$, it follows from (1.3.2) that each $\sigma\in G$ satisfies

[TABLE]

with $a_{i,j}\in A$, where

[TABLE]

That is, $\sigma$ is represented by a $r\times r$-matrix of shape

[TABLE]

with matrices $B_{\tau}\in\operatorname{GL}(r_{\tau},\mathds{F})$ along the diagonal, zeroes above the blocks $B_{\tau}$, and entries $a_{i,j}\in A$ bounded by (1.5.3) below the blocks. We conclude:

Proposition 1.6**:**

$G=\operatorname{Gal}(L(\Lambda)|L)$* is a finite group whose order is bounded by a constant that depends only on the spectrum $\operatorname{spec}_{A}(\Lambda)$ of $\Lambda$.*

Henceforth, we identify $G$ with the subgroup of $\operatorname{GL}(r,A)$ given by (1.5.2).

Example 1.7** (see [Maurischatxx], Theorem 3.1):**

Suppose that $r=2$, $\phi_{T}(X)=TX+gX^{q}+\Delta X^{q^{2}}$. There are two possibilities for the Newton polygon:

1. (a)

$\mathrm{NP}(\phi_{T}(X))$ is a straight line, that is, the point $(q,v(g))$ lies above or on the line determined by $(1,-1)$ and $(q^{2},v(\Delta))$. Then, according to (1.4), $\operatorname{spec}_{A}(\Lambda)=(\lvert\lambda_{1}\rvert,\lvert\lambda_{1}\rvert)$ and $G\hookrightarrow\operatorname{GL}(2,\mathds{F})$. 2. (b)

$(q,v(g))$ lies below the line through $(1,-1)$ and $(q^{2},v(\Delta))$, and is a break point of $\mathrm{NP}(\phi_{T}(X))$. Then $\operatorname{spec}_{A}(\Lambda)=(\lvert\lambda_{1}\rvert,\lvert\lambda_{2}\rvert)$ with $\lvert\lambda_{1}\rvert<\lvert\lambda_{2}\rvert$, and $G$ is a subgroup of

[TABLE]

In case (b) we still have to determine $\lvert\lambda_{1}\rvert$ and $\lvert\lambda_{2}\rvert$ from $\mathrm{NP}(\phi_{T}(X))$ (see Section 3), and, in both cases, the actual size of $G$ inside the corresponding matrix group.

2. The structure of the field extension $L(\Lambda)|L$

We describe the tower of subfields of $L(\Lambda)|L$ that corresponds to the block structure (1.5.4) of $G$.

2.1.

Having fixed some number $\tau$ with $1\leq\tau\leq t$, we let $e_{\tau}=e_{\Lambda_{\tau}}$ be the exponential function associated with $\Lambda_{\tau}$ and $\phi^{(\tau)}$ the corresponding Drinfeld module, of rank

[TABLE]

We also set $r_{0}=\overline{r}_{0}=0$, $\Lambda_{0}=\{0\}$, $e_{0}(z)=z$, and $\phi^{(0)}$ is the trivial Drinfeld module of rank [math], $\phi^{(0)}_{T}(X)=TX$. Further,

[TABLE]

the field generated over $L$ by $\Lambda_{\tau}$, which by Galois theory equals the fixed field of the normal subgroup

[TABLE]

of $G$. Here we have used the representation (1.5.2) of $\sigma\in G$, and $\delta_{i,j}$ is the Kronecker delta. Then

[TABLE]

may be seen as a group of $\overline{r}_{\tau}\times\overline{r}_{\tau}$-matrices with a block structure similar to (1.5.4). The canonical map

[TABLE]

corresponds to

[TABLE]

where $P$, $Q$, $R$ is an $\overline{r}_{\tau-1}\times\overline{r}_{\tau-1}$-matrix, $r_{\tau}\times\overline{r}_{\tau-1}$-matrix, $r_{\tau}\times r_{\tau}$-matrix, respectively. Let $U_{\tau}\subset G_{\tau}$ be the subgroup of matrices with trivial $P$- and $R$-parts, i.e.,

[TABLE]

In view of the given block structure, it is abelian and in fact an elementary abelian $p$-group. The kernel $\operatorname{Gal}(L_{\tau}|L_{\tau-1})$ contains $U_{\tau}$ as a normal subgroup. Define

[TABLE]

and let

[TABLE]

It equals the image of $G_{\tau}$ under the projection

[TABLE]

and is a subgroup of $\operatorname{GL}(r_{\tau},A)$. Finally, define the subgroup

[TABLE]

of $L_{\tau}$.

Proposition 2.2**:**

1. $\mathrm{(i)}$

$\tilde{\Lambda}_{\tau}$* is an $A$-submodule of $(L_{\tau},\phi^{(\tau-1)})$, the additive group $L_{\tau}$ provided with the $A$-module structure via $\phi^{(\tau-1)}$.* 2. $\mathrm{(ii)}$

As such, $\tilde{\Lambda}_{\tau}$ is free of rank $r_{\tau}$. 3. $\mathrm{(iii)}$

Let $\mathbf{B}_{\tau}=\{\lambda_{i}\in\mathbf{B}\mid\overline{r}_{\tau-1}<i\leq\overline{r}_{\tau}\}$ be as in 1.5. Then $\mathbf{B}_{\tau}$ maps under $e_{\tau-1}$ bijectively to an $A$-basis of $\tilde{\Lambda}_{\tau}$, and all the elements of $e_{\tau-1}(\mathbf{B}_{\tau})$ have the same absolute value in $L_{\tau}$, which is minimal among $\{\lvert\lambda\rvert\mid 0\neq\lambda\in\tilde{\Lambda}_{\tau}\}$.

Proof.

$\tilde{\Lambda}_{\tau}$ is a well-defined subgroup of $L_{\tau}$, as $\Lambda_{\tau}\subset L_{\tau}$ and $e_{\tau-1}$ has coefficients in $L_{\tau-1}\subset L_{\tau}$. By definition, $e_{\tau-1}$ provides a morphism of $A$-modules from $L_{\tau-1}$ to $(L_{\tau-1},\phi^{(\tau-1)})$, compatible with base extension to $L_{\tau}$, and

[TABLE]

where the left hand side is isomorphic with $A^{r_{\tau}}$ by (1.5.1). This shows (i) and (ii) and also that $e_{\tau-1}(\mathbf{B}_{\tau})$ is a basis. Let $\lambda\in\mathbf{B}_{\tau}$. From the orthogonality property (1.3.2),

[TABLE]

which depends only on $\lvert\lambda\rvert$. Therefore, all the $\lvert e_{\tau-1}(\lambda)\rvert$ are equal and, in fact, minimal among $\{\lvert e_{\tau-1}(\lambda)\rvert\mid\lambda\in\Lambda_{\tau}\smallsetminus\Lambda_{\tau-1}\}$. ∎

Proposition 2.3**:**

The fixed field $\tilde{L}_{\tau}$ of $U_{\tau}$ equals the field $L_{\tau-1}(\tilde{\Lambda}_{\tau})$ generated over $L_{\tau-1}$ by $\tilde{\Lambda}_{\tau}$.

Proof.

By (1.5.2) and the $\Lambda_{\tau-1}$-periodicity of $e_{\tau-1}$, $U_{\tau}$ fixes $\tilde{\Lambda}_{\tau}$. On the other hand, suppose that $\sigma\in G_{\tau}$ fixes $L_{\tau-1}(\tilde{\Lambda}_{\tau})$. Then $\sigma(\lambda_{j})=\lambda_{j}$ for $j\leq\overline{r}_{\tau-1}$ and

[TABLE]

for $\overline{r}_{\tau-1}<j\leq\overline{r}_{\tau}$ with $a_{i,j}=\delta_{i,j}$ if $\overline{r}_{\tau-1}<i\leq\overline{r}_{\tau}$, as $\sigma(\lambda_{j})\equiv\lambda_{j}\pmod{\Lambda_{\tau-1}}$. Hence $\sigma\in U_{\tau}$. ∎

2.4.

As $\tilde{\Lambda}_{\tau}$ is an $A$-sublattice in $(L_{\tau-1},\phi^{(\tau-1)})$,

[TABLE]

where $V_{\tau}$ is the $\mathds{F}$-vector space of dimension $r_{\tau}$ generated by $e_{\tau-1}(\mathbf{B}_{\tau})$. Further, $V_{\tau}$ is stable under $H_{\tau}=\operatorname{Gal}(\tilde{L}_{\tau}|L_{\tau-1})$, and thus $H_{\tau}$ embeds into

[TABLE]

where the last isomorphism is via the choice of the ordered basis $e_{\tau-1}(\mathbf{B}_{\tau})$, i.e., the choice of $\mathbf{B}_{\tau}$. This is in keeping with the fact the projection of $\sigma\in G$ onto its $\tau$-part (see (1.5.4)) is an element of $\operatorname{GL}(r_{\tau},\mathds{F})$.

Definition 2.5**:**

A finite $\mathds{F}$-vector subspace $V$ of $L^{\mathrm{sep}}$ is called pure of weight $w(V)\in\mathds{Q}$ if all the non-vanishing $x\in V$ satisfy $\log x=\log_{q}\lvert x\rvert=w(V)$.

We summarize the preceding considerations as follows:

Proposition 2.6**:**

For each $\tau$ with $1\leq\tau\leq t$ the intermediate field $\tilde{L}_{\tau}$ of $L_{\tau}|L_{\tau-1}$ satisfies:

1. $\mathrm{(i)}$

$\tilde{L}_{\tau}=L_{\tau-1}(\tilde{\Lambda}_{\tau})=L_{\tau-1}(V_{\tau})$* is obtained by adjoining the $H_{\tau}$-stable pure $\mathds{F}$-subspace $V_{\tau}$ of dimension $r_{\tau}$ (and a certain weight given by (2.2.1)).* 2. $\mathrm{(ii)}$

The Galois group $U_{\tau}$ of $L_{\tau}$ over $\tilde{L}_{\tau}$ is an elementary abelian $p$-group.

Corollary 2.7**:**

With notations as before, the ramification index $e(L_{\tau}|\tilde{L}_{\tau})$ equals $\#U_{\tau}$ or $p^{-1}\#U_{\tau}$.

Proof.

Consider the filtration of $U_{\tau}$ given by the higher ramification groups $U_{\tau,i}$ (see [Serre68] Chapitre 4), $i=-1,0,1,\dots$. Then

• $U_{\tau,-1}=U_{\tau}$; $U_{\tau}/U_{\tau,0}$ is cyclic;
• $U_{\tau,0}/U_{\tau,1}$ is cyclic of order prime to $p$, hence trivial, and
• all the higher $U_{\tau,i}/U_{\tau,i+1}$ are $p$-groups.

Hence, as $U_{\tau}$ is $p$-elementary abelian, either $U_{\tau}=U_{\tau,0}$ and $L_{\tau}|\tilde{L}_{\tau}$ is completely ramified of degree $\#U_{\tau}$ or the residue class degree $[U_{\tau}:U_{\tau,0}]$ is $p$ and the ramification index equals $p^{-1}\#U_{\tau}$. ∎

2.8.

Let’s have a closer look at the extension $\tilde{L}_{\tau}=L_{\tau-1}(V_{\tau})$ of $L_{\tau-1}$. We assume that all the field extensions considered are subfields of the separable closure $L^{\mathrm{sep}}$ of $L$. This identifies the residue class fields with subfields of the residue class field of $L^{\mathrm{sep}}$, which is a separable closure $\mathds{F}^{\mathrm{sep}}$ of $\mathds{F}$.

Let $d\in\mathds{N}$ be the precise denominator of the weight $w(V_{\tau})$, and let $L_{\tau-1}^{\prime}$ be a completely ramified separable extension of $L_{\tau-1}$ of degree $d$. Then there exists $y\in L_{\tau-1}^{\prime}$ with $v(y)=w(V_{\tau})=v(x)$ for each $0\neq x\in V_{\tau}$. Replacing $V_{\tau}$ with $V_{\tau}^{\prime}\vcentcolon=y^{-1}V_{\tau}$,

[TABLE]

Now, since $v(x-x^{\prime})=v(x)=v(x^{\prime})$ for $x\neq x^{\prime}\in V_{\tau}\smallsetminus\{0\}$, different elements of $V_{\tau}^{\prime}$ reduce to different elements in $\mathds{F}^{\mathrm{sep}}$, that is, reduction modulo the maximal ideal yields an $\mathds{F}$-isomorphism

[TABLE]

Let $\mathds{F}_{\tau-1}$, $\mathds{F}_{\tau-1}^{\prime}$ be the residue class fields of $L_{\tau-1}$, $L_{\tau-1}^{\prime}$, respectively. Now:

• $V_{\tau}$ is stable under $H_{\tau}=\operatorname{Gal}(\tilde{L}_{\tau}|L_{\tau-1})$, so
• $V_{\tau}^{\prime}$ is stable under $H^{\prime}_{\tau}\vcentcolon=\operatorname{Gal}(L_{\tau-1}^{\prime}(V_{\tau}^{\prime})|L_{\tau-1}^{\prime})$, and
• $\overline{V}_{\tau}^{\prime}$ is stable under $\operatorname{Gal}(\mathds{F}_{\tau-1}^{\prime}(\overline{V}_{\tau}^{\prime})|\mathds{F}_{\tau-1}^{\prime})$.

The latter is generated by a suitable Frobenius element (i.e., $x\mapsto x^{\#(\mathds{F}_{\tau-1}^{\prime})}$), which by our choice of the basis $\mathbf{B}_{\tau}$ for $V_{\tau}$ (thus bases for $V_{\tau}^{\prime}$ and $\overline{V}_{\tau}^{\prime}$) corresponds to a matrix in $\operatorname{GL}(r_{\tau},\mathds{F})$. Let $f_{\tau}$ be its multiplicative order. Then

[TABLE]

where the last equality comes from Hensel’s Lemma. In particular, $L_{\tau-1}^{\prime}(V_{\tau}^{\prime})=L_{\tau-1}^{\prime}(V_{\tau})$ is unramified of degree $f_{\tau}$ over $L_{\tau-1}^{\prime}$. Consider the diagram

[TABLE]

where $M_{\tau}\vcentcolon=L_{\tau-1}(V_{\tau})\cap L_{\tau-1}^{\prime}$. We see from the above:

Proposition 2.9**:**

[TABLE]

where the groups are cyclic of order $f_{\tau}$, $L_{\tau-1}(V_{\tau})|M_{\tau}$ is unramified and $M_{\tau}|L_{\tau-1}$ is completely ramified.

Remark 2.9.1**:**

Note that neither $L_{\tau-1}^{\prime}$ nor $M_{\tau}$ is canonically determined; only their degrees over $L_{\tau-1}$ are intrinsically characterized by $[L_{\tau-1}^{\prime}:L_{\tau-1}]=d_{\tau}=\mathrm{denom}(w(V_{\tau}))$ and $[M_{\tau}:L_{\tau-1}]=e(\tilde{L}_{\tau}|L_{\tau-1})$.

3. An example

3.1.

We consider in more detail the special case where the spectrum $\operatorname{spec}_{A}(\Lambda)$ of $\Lambda$ is $(\lvert\lambda_{1}\rvert<\lvert\lambda_{2}\rvert=\dots=\lvert\lambda_{r}\rvert$). By Proposition 1.4, this corresponds to the following behavior of the Newton polygon:

[TABLE]

Such behavior is realized in particular by the Drinfeld module $\phi$ with

[TABLE]

where the intermediate terms $g_{i}X^{q^{i}}$ ($1<i<r$) vanish and $(q,v(g))$ lies below the line joining $(1,-1)$ and $(q^{r},v(\Delta))$. As $\operatorname{spec}_{A}(\Lambda)$ and the properties derived from it depend only on $\mathrm{NP}(\phi_{T}(X))$, we assume for the rest of this section that $\phi$ is given by (3.1.2). Our aim is to determine $\lvert\lambda_{1}\rvert$ and $\lvert\lambda_{2}\rvert$ from $v(g)$ and $v(\Delta)$, and to draw conclusions. Also, for psychological reasons, we mainly work with $\log x=\log_{q}\lvert x\rvert=-v(x)$ instead of the valuation $v$.

Remark 3.1.3**:**

Rank-$r$ Drinfeld modules given by (3.1.2) are called $\boldsymbol{1}$-sparse in [Gekeler2017].

Proposition 3.2**:**

In the given situation, put $j\vcentcolon=g^{(q^{r}-1)/(q-1)}/\Delta$. Let $\{\lambda_{1},\lambda_{2},\dots,\lambda_{r}\}$ be an SMB of the period lattice $\Lambda$ of $\phi$, $\lvert\lambda_{1}\rvert<\lvert\lambda_{2}\rvert=\dots=\lvert\lambda_{r}\rvert$. Define $s\vcentcolon=\log(\lambda_{2}/\lambda_{1})$, with integral part $n=[s]$. Then the formula

[TABLE]

holds.

Proof.

This is formula (i) in Corollary 5.7 of [Gekeler2017]. (Note the different numbering in [Gekeler2017], where $\lambda_{r},\lambda_{r-1},\dots,\lambda_{1}$ in this order is an SMB.) ∎

We will use 3.2 to find lower bounds for the ramification index of $L(\lambda_{1},\lambda_{2})$ over $L(\lambda_{1})$, and thus for the degree of $L(\Lambda)$ over $L$.

3.3.

For $1\leq i\leq r$ let

[TABLE]

Then $\{\mu_{1},\dots,\mu_{r}\}$ is an $\mathds{F}$-basis of $\phi[T]$ and

[TABLE]

holds, as well as

[TABLE]

for coefficients $a_{i}\in\mathds{F}$, not all zero, see [Gekeler2017] Lemma 3.4. One easly derives from (3.3.3) and (1.3.2) that

[TABLE]

and

[TABLE]

that is

[TABLE]

Hence (3.2.1) together with (3.3.4) yields $\log g$ and $\log\Delta$ in terms of $\lvert\lambda_{1}\rvert$ and $\lvert\lambda_{2}\rvert$.

3.4.

Let $\varphi$ be the real function

[TABLE]

that occurs in (3.2.1), i.e., $\log j=\varphi(s)$. It is continuous, monotonically increasing, convex, piecewise linear, and bijective from the real interval $[0,\infty)$ to $[\varphi(0),\infty)$, with $\varphi(0)=q(q^{r-1}-1)/(q-1)$. Let

[TABLE]

be its inverse; it has similar properties, with “convex” replaced with “concave”. Then $s=\psi(\log j)$, i.e.,

[TABLE]

In particular, as for fixed $g$, $\log j$ can be made arbitrarily large letting $\Delta$ tend to zero, also $\log\lambda_{2}$ can be made arbitrarily large.

Assuming $\log g=0$ and $\log\Delta$ very small with numerator coprime to $p$, the formula derived from (3.2.1)

[TABLE]

shows that $q^{n+1}$ (with $n\gg 0$) divides the denominator of $s=\log(\lambda_{2}/\lambda_{1})$. Hence the ramification index of $L(\lambda_{1},\lambda_{2})$ over $K_{\infty}$, and in particular the extension degree $[L(\lambda_{1},\lambda_{2}):K_{\infty}]$ is divisible by $q^{n+1}$. Thus we have proved:

Theorem 3.5**:**

The field $L(\Lambda)=L(\operatorname{tor}(\phi))$ of periods of $\phi$, where $\phi$ is a Drinfeld $A$-module of rank $r\geq 2$ over $L$, may have arbitrarily large ramification index over $L$. In particular, its degree $[L(\Lambda):L]$ is unbounded.

4. The residue class degree

In contrast with the ramification index, the residue class degree $f(L(\Lambda)|L)$ is bounded by a constant that depends only on the rank $r$ of the Drinfeld module.

Theorem 4.1**:**

Let $\phi$ be a rank-$r$ Drinfeld $A$-module over $L$ with period lattice $\Lambda$, provided with a successive minimum basis $\mathbf{B}=\{\lambda_{1},\lambda_{2},\dots,\lambda_{r}\}$. Split $\mathbf{B}=\mathbf{B}_{1}\cup\dots\cup\mathbf{B}_{t}$ as in 1.5, where $\#\mathbf{B}_{\tau}=r_{\tau}$ and $\sum_{1\leq\tau\leq t}r_{\tau}=r$. The residue class degree $f(L(\Lambda)|L)$ is bounded by

[TABLE]

Proof.

Consider the tower of subfields described in Section 2:

[TABLE]

Each step $L_{\tau}|L_{\tau-1}$ is subdivided

[TABLE]

Now:

• •

$M_{\tau}|L_{\tau-1}$ is completely ramified (Proposition 2.9);

• •

$\tilde{L}_{\tau}|M_{\tau}$ is unramified of degree $f_{\tau}$, where $f_{\tau}=\mathrm{ord}(x)$ with some $x\in\operatorname{GL}(r_{\tau},\mathds{F})$ (Proposition 2.9);

• •

$L_{\tau}|\tilde{L}_{\tau}$ has residue class degree 1 or $p$ (Corollary 2.7) and $\tilde{L}_{1}=L_{1}$, since $\tilde{\Lambda}_{1}=\Lambda_{1}$.

The result now follows by the multiplicativity of the residue class degree in towers and from the elementary fact that

[TABLE]

Remarks 4.2**:**

1. (i)

As $\sum r_{\tau}=r$, $\prod(q^{r_{\tau}}-1)\leq q^{r}-1$, so $f(L(\Lambda)|L)$ is always less or equal to $(q^{r}-1)p^{t-1}$. 2. (ii)

As $L(\Lambda)=L(\operatorname{tor}(\phi))$, the constant field extension in the torsion field $L(\operatorname{tor}(\phi))$, i.e., the algebraic closure $\overline{\mathds{F}}$ of $\mathds{F}$ in $L(\operatorname{tor}(\phi))$, is finite. Its degree $[\overline{\mathds{F}}:\mathds{F}]$ is bounded by $f(L|K_{\infty})(q^{r}-1)p^{t-1}$. In particular, if $\phi$ is defined over a finite extension $K^{\prime}$ of $K$, and $\overline{\mathds{F}}\vcentcolon=$ algebraic closure of $\mathds{F}$ in $K^{\prime}(\operatorname{tor}(\phi))$, then

[TABLE]

where $w$ runs through the places of $K^{\prime}$ above the place $\infty$ of $K$.