Proposition 1.1.2. The statement is false as written. For a counterexample, fix a prime
$p$
, take
$V$
to be of dimension
$p$
, and take
$G$
to be the semidirect product of the trace-zero torus
$\mathbb{G}_{m}^{p-1}=\ker (\times :\mathbb{G}_{m}^{p}\rightarrow \mathbb{G}_{m})$
by the group
$\mathbb{Z}/p\mathbb{Z}$
acting via the standard permutation representation on the ambient torus
$\mathbb{G}_{m}^{p}$
. There is a unique ascending filtration
$H^{0},H^{1},\ldots \,$
on the group
$(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{p-1}\cong (\unicode[STIX]{x1D707}_{p^{\infty }})^{p-1}\subseteq \mathbb{G}_{m}^{p-1}$
characterized by the properties that
$H^{n(p-1)}=(p^{-n}\mathbb{Z}_{p}/\mathbb{Z}_{p})^{p-1}$
and the groups
$H^{n(p-1)+i}/H^{n/(p-1)}$
for
$i=0,\ldots ,p-1$
form a unipotent flag for the action of
$\mathbb{Z}/p\mathbb{Z}$
; put
$G^{r}=H^{n}$
for
$n=\max \{0,\lfloor -r\rfloor \}$
.
For a variant of this counterexample, take
$G$
as before, but define
$G^{r}$
as follows: for
$r>0$
take
$G^{r}=H_{n}$
for
$n=\max \{0,\lfloor 1/r\rfloor \}$
, and for
$r\leqslant 0$
take
$G^{r}=G$
.
The error in the proof occurs in the second paragraph, where it is asserted that ‘By applying (d) finitely many times …we can split
$X$
as a direct sum of
$G$
-stable summands, each of which is
$G^{r+}$
-isotypical’. This fails in the given examples because the action of
$\mathbb{Z}/p\mathbb{Z}$
on
$\mathbb{Z}_{p}^{p-1}$
is unipotent mod
$p$
.
To correct the statement, we add two additional conditions. The first of these is:
- (g)
-
For all nonnegative integers
$g,h$
and every
$G$
-stable subspace
$W$
of
$(V^{\vee })^{\otimes g}\otimes V^{\otimes h}$
, if
$G^{-\infty +}$
acts trivially on
$W$
and
$G$
acts on
$W$
via a finite abelian group, then
$G$
acts trivially on
$W$
itself.
This eliminates the original counterexample: the action of
$G$
on the trace-zero subspace of
$V^{\vee }\otimes V$
factors through
$\mathbb{Z}/p\mathbb{Z}$
but is not trivial. If
$F$
contains all roots of unity (or in the context of Lemma 1.1.4, just a primitive
$p$
th root of unity), then the following condition is equivalent:
- (g
$^{\prime }$
)
-
For all nonnegative integers
$g,h$
and every one-dimensional
$G$
-stable subspace
$W$
of
$(V^{\vee })^{\otimes g}\otimes V^{\otimes h}$
, if
$G^{-\infty +}$
acts trivially on
$W$
and
$G$
acts trivially on some power of
$W$
, then
$G$
acts trivially on
$W$
itself.
The second additional condition is:
- (h)
-
For each
$r\in \mathbb{R}$
, there exists
$s_{0}\in (r,+\infty )$
with the following property: for any
$s\in (r,s_{0}]$
, if
$G^{t}/G^{s}$
is finite and abelian for all
$t\in (r,s]$
, then
$G^{r+}/G^{s}$
is finite.
This eliminates the variant counterexample.
For later use (see the discussion of Theorem 2.3.17 below), we observe that without condition (g), it is already possible to show that the groups
$G^{r}$
are finite for all
$r\in \mathbb{R}$
. To wit, let
$S$
be the set of
$r\in \mathbb{R}$
for which
$G^{r}$
is finite. By (a), the set
$S$
is up-closed. By (b), the set
$S$
does not contain its infimum. If this infimum is finite, we obtain a contradiction by combining Lemma 1.1.1 with (h); it follows that
$G^{r}$
is finite for all
$r\in \mathbb{R}$
.
We now prove the modified version of Proposition 1.1.2.
Proof of (modified) Proposition 1.1.2.
There is no harm in assuming at once that
$F$
is algebraically closed. As noted above, the current hypotheses suffice to imply that
$G^{r}$
is finite for all
$r\in \mathbb{R}$
; it thus remains to derive a contradiction under the assumption that
$G^{-\infty +}$
is infinite.
By Lemma 1.1.1, there exists
$s_{0}\in \mathbb{R}$
such that
$G^{-\infty +}/G^{s_{0}}$
embeds into a product of finitely many copies of
$\mathbb{Q}/\mathbb{Z}$
. By decreasing
$s_{0}$
suitably, we may ensure in addition that
$G^{-\infty +}/G^{s_{0}}$
is nontrivial and has no nontrivial finite quotients.
We next verify that
$G$
centralizes
$G^{-\infty +}$
. For each
$s\in \mathbb{R}$
, the conjugation action of
$G$
on the finite group
$G^{s}$
must be trivial on the identity connected component of
$G$
, so this action factors through the component group
$\unicode[STIX]{x1D70B}_{0}(G)$
of
$G$
. For each prime
$p$
, let
$H_{p}$
be the union of the
$p$
-Sylow subgroups of
$G^{s}/G^{s_{0}}$
for
$s\leqslant s_{0}$
; it then suffices to check that
$G$
centralizes
$H_{p}$
provided that the latter is nonzero. Since
$H_{p}$
has no nontrivial finite quotients, it is isomorphic to
$(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{n}$
for some positive integer
$n$
, and the action of
$\unicode[STIX]{x1D70B}_{0}(G)$
corresponds to a homomorphism
$\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}_{0}(G)\rightarrow \text{GL}_{n}(\mathbb{Z}_{p})$
, which we wish to show is trivial. By (d) plus Tannaka–Krein duality, the image of
$\unicode[STIX]{x1D70B}_{0}(G)$
in
$\text{GL}_{n}(\mathbb{F}_{p})$
is unipotent; consequently, the image of
$\unicode[STIX]{x1D70C}$
is a
$p$
-group. Suppose by way of contradiction that this image is nontrivial; then the kernel of
$\unicode[STIX]{x1D70C}$
is contained in a normal subgroup
$G^{\prime }$
of
$G$
such that
$G/G^{\prime }\cong \mathbb{Z}/p\mathbb{Z}$
. By duality again, for some
$g,h$
, there exists a
$G$
-stable subspace
$W$
of
$(V^{\vee })^{\otimes g}\otimes V^{\otimes h}$
on which
$G^{\prime }$
acts trivially but
$G$
acts (completely reducibly and) nontrivially. However, since
$G^{-\infty +}\subseteq \ker (\unicode[STIX]{x1D70C})\subseteq G^{\prime }$
, this constitutes a violation of (g); this yields a contradiction and shows that indeed
$G$
centralizes
$G^{-\infty +}$
.
By duality yet again, we can choose
$g,h$
so that
$(V^{\vee })^{\otimes g}\otimes V^{\otimes h}$
contains a
$G$
-stable subspace
$X$
on which
$G^{s_{0}}$
acts trivially but
$G^{s}$
acts nontrivially for some
$s<s_{0}$
. By Maschke’s theorem,
$X$
admits a
$G^{-\infty +}$
-isotypical decomposition, which is respected by
$G$
since
$G$
centralizes
$G^{-\infty +}$
; since
$G^{-\infty +}$
is infinite, this decomposition admits a summand
$Y$
for which the image of the action of
$G^{-\infty +}$
on
$Y$
is isomorphic to an infinite subgroup of
$\mathbb{Q}/\mathbb{Z}$
. Put
$W=\wedge ^{\dim (Y)}Y$
; this space occurs as a
$G$
-invariant subspace of
$(V^{\vee })^{\otimes g}\otimes V^{\otimes h}$
for some
$g,h$
. However, the image of
$G^{-\infty +}$
in
$\text{GL}(W)$
is again isomorphic to an infinite subgroup of
$\mathbb{Q}/\mathbb{Z}$
, which is a contradiction. This yields that
$G^{-\infty +}$
is finite, as claimed.◻
Remark 1.1.3. The fiber functor
$\unicode[STIX]{x1D714}$
should be assumed to be neutral. More seriously, additional conditions are needed in order to imply the added hypotheses (g) and (h) of Proposition 1.1.2.
- (iv)
-
For each
$V\in {\mathcal{C}}$
with
$r(V)=-\infty$
, if
$G(V)$
is finite abelian, then it is trivial. (This implies (g).) - (v)
-
For each
$r\in \mathbb{R}$
, there exists
$s_{0}\in (r,+\infty )$
with the following property: for any
$s\in (r,s_{0}]$
and any
$V\in {\mathcal{C}}$
with
$r(V)<s$
, if
$G^{t}(V)$
is finite and abelian for all
$t\in (r,s]$
, then
$G^{r+}(V)$
is finite. (This implies (h).)
See the discussions of Theorems 2.3.17 and 3.8.16 below.
Corollary 2.2.7. In the statement of (b),
$\log \unicode[STIX]{x1D714}-\log \unicode[STIX]{x1D70C}+s$
should be
$\log \unicode[STIX]{x1D714}-\log \unicode[STIX]{x1D70C}+c$
. More seriously, the application of [Reference KedlayaKed10, Theorem 6.7.4] is not sufficient; see the erratum for [Reference KedlayaKed10, Lemma 6.8.1] for an alternate approach that gives the value
$\unicode[STIX]{x1D6FF}=s(\unicode[STIX]{x1D714}/s)^{a(n_{1},n_{2},m)}$
for some
$a(n_{1},n_{2},m)$
.
Lemma 2.3.9. Given the inclusion
$V_{2}^{\prime }\rightarrow \bigoplus _{m=0}^{p-1}(V_{1}^{\prime }\otimes X_{m})$
, it is asserted that ‘since
$V_{2}^{\prime }$
is indecomposable, we have
$V_{2}^{\prime }\subseteq V_{1}^{\prime }\otimes X_{m}$
for some
$m$
’. We clarify this point here.
For each irreducible constituent
$Z$
of
$V_{2}^{\prime }$
, there must be an index
$m$
for which
$Z$
also occurs as an irreducible constituent of
$V_{1}^{\prime }\otimes X_{m}$
. In fact, this index must be unique because
$V_{1}^{\prime }$
is refined. (For
$m^{\prime }\neq m$
, the intrinsic subsidiary radii of
$(V_{1}^{\prime }\otimes X_{m})^{\vee }\otimes (V_{1}^{\prime }\otimes X_{m^{\prime }})\cong ((V_{1}^{\prime })^{\vee }\otimes V_{1})\otimes X_{m^{\prime }-m}$
are all equal to
$\unicode[STIX]{x1D714}^{p}$
, so this differential module cannot contain a trivial constituent.)
For
$m\in \{0,\ldots ,p-1\}$
, let
$Y_{m}$
be the kernel of
$V_{2}^{\prime }\rightarrow \bigoplus _{m^{\prime }\neq m}(V_{1}^{\prime }\otimes X_{m^{\prime }})$
. The map
$\bigoplus _{m=0}^{p-1}Y_{m}\rightarrow V_{2}^{\prime }$
is injective: if
$Z$
were an irreducible submodule of the kernel, then
$Z$
would project nontrivially to
$Y_{m}$
for at least one value of
$m$
, but the previous paragraph would then imply that this value is unique and this would contradict the manifest injectivity of
$Y_{m}\rightarrow V_{2}^{\prime }$
. The map
$\bigoplus _{m=0}^{p-1}Y_{m}\rightarrow V_{2}^{\prime }$
is also surjective: if
$Z$
were a constituent of the cokernel, then by the previous paragraph
$Z$
would occur in
$V_{1}^{\prime }\otimes X_{m}$
for some
$m$
, but then
$V_{2}^{\prime }/Y_{m}\rightarrow \bigoplus _{m^{\prime }\neq m}(V_{1}^{\prime }\otimes X_{m^{\prime }})$
would be an injective map for which
$Z$
occurs in the source but not the target.
We thus conclude that
$V_{2}^{\prime }=\bigoplus _{m=0}^{p-1}Y_{m}$
. Since
$V_{2}^{\prime }$
is indecomposable, there can only be one index
$m$
for which
$Y_{m}\neq 0$
. For that
$m$
, the map
$V_{2}^{\prime }\rightarrow V_{1}^{\prime }\otimes X_{m}$
is injective.
Lemma 2.3.11. In the last paragraph of the proof, the
$V_{i}^{\prime }$
need not be indecomposable, so Lemma 2.3.9 does not apply directly. However, we may decompose
$V_{i}^{\prime }$
as a direct sum
$\bigoplus _{j}V_{ij}^{\prime }$
of indecomposable summands and then apply Lemma 2.3.9 to produce
$m_{ij}\in \{0,\ldots ,p-1\}$
such that
$\mathit{IR}((V_{0}^{\prime })^{\vee }\otimes V_{ij}^{\prime }\otimes W_{m_{ij}})>\unicode[STIX]{x1D714}^{p}$
. For
$j^{\prime }\neq j$
, we also have
$\mathit{IR}((V_{ij}^{\prime })^{\vee }\otimes V_{ij^{\prime }}^{\prime }\otimes W_{m_{ij^{\prime }}-m_{ij}})>\unicode[STIX]{x1D714}^{p}$
; since
$V_{i}^{\prime }$
is refined, this implies that
$m_{ij^{\prime }}=m_{ij}$
. That is, the
$m_{ij}$
are all equal to a common value
$m_{i}$
, and we may take
$V^{\prime }=\bigoplus _{i=0}^{r}V_{i}^{\prime }\otimes W_{m_{i}}$
, as originally claimed.
Theorem 2.3.17. In light of the correction to Proposition 1.1.2, we no longer can apply it directly to yield this result. As explained in the discussion of Proposition 1.1.2 above, to prove that the groups
$G^{r}(V)$
are finite, it is not necessary to check condition (iv) of Remark 1.1.3; since conditions (i)–(iii) are already confirmed in the original text, we need only supplement them by verifying condition (v). (Note that in light of Remark 2.3.18, the filtration of
$G$
must be truncated at some index
$s^{\prime }\in (s,1)$
in order for condition (iii) to be satisfied.)
Before doing so, however, we should point out that the fiber functor
$\unicode[STIX]{x1D714}$
as defined is not neutral, so Remark 1.1.3 does not apply directly (see above). Instead, one should extend scalars from
$E$
to a larger differential field over which every differential module over
$E$
becomes trivial (e.g., a Picard–Vessiot extension).
To establish condition (v) of Remark 1.1.3, we use the following lemma.
Lemma. With notation as in Theorem 2.3.17, define
$G^{r-}=\bigcup _{s<r}G^{s}$
. For any
$r<\unicode[STIX]{x1D714}$
, if
$G^{s}(V)$
is a finite abelian
$p$
-group for all
$s<r$
, then
$G^{r-}(V)$
is finite.
Proof. We will use throughout the following observation: for
$X\in [V]$
and
$s\in (0,1)$
, the multiplicity of
$s$
among the intrinsic subsidiary radii of
$X$
is equal to the dimension of the space
$\unicode[STIX]{x1D714}(X)^{G(V)^{s-}}/\unicode[STIX]{x1D714}(X)^{G(V)^{s}}$
. In particular, for
$s<r$
this quantity depends only on the isomorphism class of
$\unicode[STIX]{x1D714}(X)$
as a representation of
$G^{r-}(V)$
.
Let
$H$
be the dual group of
$G^{r-}(V)$
; it is a finitely generated
$\mathbb{Z}_{p}$
-module. Choose an irreducible constituent
$V_{i}$
of
$V$
and let
$\unicode[STIX]{x1D712}_{1},\ldots ,\unicode[STIX]{x1D712}_{n}\in H$
be the distinct characters occurring in
$\unicode[STIX]{x1D714}(V_{i})$
; these are all
$G(V)$
-conjugate, and so occur to the same multiplicity
$d$
. For each nonnegative integer
$k$
, let
$W_{k}$
be the differential submodule of
$(\wedge ^{d}V_{i})^{\otimes p^{k}}$
with the property that
$\unicode[STIX]{x1D714}(W_{k})$
is the union of the one-dimensional subspaces of
$\unicode[STIX]{x1D714}((\wedge ^{d}V)^{\otimes p^{k}})$
on which
$G^{r-}(V)$
acts via the characters
$\unicode[STIX]{x1D712}_{1}^{\otimes p^{k}d},\ldots ,\unicode[STIX]{x1D712}_{n}^{\otimes p^{k}d}$
. By the previous paragraph, it will suffice to check that there exists a constant
$c>1$
independent of
$k$
such that
$\mathit{IR}(W_{k+1})\geqslant \max \{r,c\mathit{IR}(W_{k})\}$
; there is nothing to check unless
$\mathit{IR}(W_{k})<r$
.
Let
$T$
be the unique maximal torus inside the group
$\text{GL}(\unicode[STIX]{x1D714}(W_{k}))$
containing the image of
$G^{r-}(V)$
. The Weyl group of
$\text{GL}(\unicode[STIX]{x1D714}(W_{k}))$
is the symmetric group
$S_{n}$
. The highest-weight vectors are the orbits of
$S_{n}$
on the character group of
$T$
; these orbits may be identified with partitions
$\unicode[STIX]{x1D706}=(\unicode[STIX]{x1D706}_{1}\geqslant \cdots \geqslant \unicode[STIX]{x1D706}_{n}\geqslant 0)$
of nonnegative integers into at most
$n$
parts. Order these partitions lexicographically, and write
$|\unicode[STIX]{x1D706}|$
for
$\unicode[STIX]{x1D706}_{1}+\cdots +\unicode[STIX]{x1D706}_{n}$
. The irreducible representation of
$\text{GL}(\unicode[STIX]{x1D714}(W_{k}))$
with highest weight
$\unicode[STIX]{x1D706}$
may be identified with
$\unicode[STIX]{x1D714}(S^{\unicode[STIX]{x1D706}}W_{k})$
, where
$S^{\unicode[STIX]{x1D706}}$
is the Schur functor associated to
$\unicode[STIX]{x1D706}$
; see [Reference Fulton and HarrisFH91, ch. 15] for a concrete discussion. We may decompose
$\unicode[STIX]{x1D714}(S^{\unicode[STIX]{x1D706}}W_{k})$
into eigenspaces
$X_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}$
indexed by partitions
$\unicode[STIX]{x1D707}$
with
$|\unicode[STIX]{x1D707}|=|\unicode[STIX]{x1D706}|$
, with the dimension
$d_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}$
of
$X_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}$
being given by the Weyl character formula; the only fact we need here is that
$d_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}=1$
if
$\unicode[STIX]{x1D707}=\unicode[STIX]{x1D706}$
and
$d_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}=0$
if
$\unicode[STIX]{x1D707}>\unicode[STIX]{x1D706}$
. This implies the existence of an isomorphism
$$\begin{eqnarray}W_{k+1}\cong \mathop{\prod }_{|\unicode[STIX]{x1D706}|=p}(S^{\unicode[STIX]{x1D706}}W_{k})^{\otimes e_{\unicode[STIX]{x1D706}}}\end{eqnarray}$$
of virtual representations of
$G^{r-}(V)$
for some integers
$e_{\unicode[STIX]{x1D706}}$
with
$e_{(p,0,\ldots ,0)}=1$
.
Apply Corollary 2.1.6 to find a cyclic vector
$\mathbf{v}$
for
$W_{k}$
and then write
$D^{n}(\mathbf{v})=a_{0}\mathbf{v}+\cdots +a_{n-1}D^{n-1}(\mathbf{v})$
with
$a_{0},\ldots ,a_{n-1}\in E$
as in Proposition 2.2.6. Let
$\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n}$
be the multiset of roots of the polynomial
$Q(T)=T^{n}-a_{n-1}T^{n-1}-\cdots -a_{0}$
in some algebraic closure of
$E$
; we then have
$-\text{log}\,|\unicode[STIX]{x1D6FC}_{i}|=\log \mathit{IR}(W_{k})-\log \unicode[STIX]{x1D714}+\log \unicode[STIX]{x1D70C}$
for
$i=1,\ldots ,n$
. Using the quantitative form of Corollary 2.2.7 described above, we see that there exists
$c>1$
depending on
$n,p,r$
(but not
$k$
) such that for any partition
$\unicode[STIX]{x1D706}$
with
$|\unicode[STIX]{x1D706}|=p$
, the following two multisets coincide in their values not exceeding
$\log \max \{c\mathit{IR}(W_{k}),r\}-\log \unicode[STIX]{x1D714}+\log \unicode[STIX]{x1D70C}$
. (In principle, the choice of
$c$
depends also on
$\mathit{IR}(W_{k})$
, but since
$\log \mathit{IR}(W_{k})<r<\unicode[STIX]{x1D714}$
we may make the choice uniform in this parameter.)
- –
-
The multiset consisting of
$\log s-\log \unicode[STIX]{x1D714}+\log \unicode[STIX]{x1D70C}$
for
$s$
ranging over the intrinsic subsidiary radii of
$S^{\unicode[STIX]{x1D706}}W_{k}$
. - –
-
The multiset consisting of
$-\text{log}\,|\unicode[STIX]{x1D70B}_{1}\unicode[STIX]{x1D6FC}_{1}+\cdots +\unicode[STIX]{x1D70B}_{n}\unicode[STIX]{x1D6FC}_{n}|$
taken with multiplicity
$d_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}$
, for each partition
$\unicode[STIX]{x1D707}$
and each distinct permutation
$(\unicode[STIX]{x1D70B}_{1},\ldots ,\unicode[STIX]{x1D70B}_{n})$
of
$(\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{n})$
.
This formally implies an analogous assertion with
$S^{\unicode[STIX]{x1D706}}W_{k}$
replaced by
$\prod _{\unicode[STIX]{x1D706}}(S^{\unicode[STIX]{x1D706}}W_{k})^{\otimes e_{\unicode[STIX]{x1D706}}}$
, or equivalently by
$W_{k+1}$
, and
$d_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}$
replaced by
$\sum _{\unicode[STIX]{x1D706}}e_{\unicode[STIX]{x1D706}}d_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}}$
, or equivalently by
$1$
if
$\unicode[STIX]{x1D707}=(p,0,\ldots ,0)$
and
$0$
otherwise. This implies at once that
$\mathit{IR}(W_{k+1})\geqslant \max \{c\mathit{IR}(W_{k}),r\}$
, as desired.◻
Using the lemma, we may apply Proposition 1.1.2 to see that
$G^{r-}(V)$
is finite for
$r<\unicode[STIX]{x1D714}^{p}$
. It follows that for
$r<\unicode[STIX]{x1D714}$
, the set
$(0,r]$
contains only finitely many distinct values which can occur as
$\mathit{IR}(W)$
as
$W$
runs over
$[V]$
. Using Proposition 2.3.5 and Lemma 2.3.6, we inductively deduce the same statement for
$r\leqslant \unicode[STIX]{x1D714}^{p^{-h}}$
for
$h=0,1,\ldots \,$
. We thus obtain condition (v) of Remark 1.1.3 for general
$r$
, and applying Proposition 1.1.2 again finishes the proof.
Definition 3.4.2. The notation
$\langle x\rangle$
is not defined; it is not the usage of Definition 3.1.4. Rather, for
$x\in \mathbb{Q}_{p}$
, one writes
$\langle x\rangle$
for the smallest positive rational number
$a$
such that one of
$x-a,x+a$
is in
$\mathbb{Z}_{p}$
.
Corollary 3.4.6. In the statement and the proof, ‘
$p$
-adic non-Liouville number’ should be ‘
$p$
-adic Liouville number’.
Corollary 3.4.7. The proof given is incorrect: given the correction to Corollary 3.4.6, we cannot rule out the possibility that
$a-b$
is a
$p$
-adic Liouville number. See [Reference KedlayaKed10, Lemma 13.4.3] for a correct argument.
Remark 3.4.14. We provide a more detailed derivation of part (a) of this statement in the case
$0\notin I$
. Let
$\mathbf{e}_{1},\ldots ,\mathbf{e}_{n}$
be a basis of
$M_{1,J}$
, and choose
$\mathbf{e}_{1}^{\prime },\ldots ,\mathbf{e}_{n^{\prime }}^{\prime }\in M_{J}$
lifting a basis of
$M_{2,J}$
. By [Reference KedlayaKed10, Lemma 13.5.4], there exists
$k>0$
such that for each
$\unicode[STIX]{x1D701}\in K^{\text{alg}}$
with
$\unicode[STIX]{x1D701}^{p^{m}}=1$
, the matrix of action of
$\unicode[STIX]{x1D701}^{\ast }$
on
$\mathbf{e}_{1},\ldots ,\mathbf{e}_{n},\mathbf{e}_{1}^{\prime },\ldots ,\mathbf{e}_{n^{\prime }}^{\prime }$
has operator norm at most
$p^{km}$
. Of course this remains true after enlarging
$k$
, which we will do several times hereafter.
By hypothesis, after enlarging
$k$
, for each
$m$
there exist elements
$\mathbf{v}_{m,A_{1},j}\in M_{1,J}$
such that
$\unicode[STIX]{x1D701}^{\ast }(\mathbf{v}_{m,A_{1},j})=\unicode[STIX]{x1D701}^{a_{1,j}}\mathbf{v}_{m,A_{1},j}$
for all
$\unicode[STIX]{x1D701}\in K^{\text{alg}}$
with
$\unicode[STIX]{x1D701}^{p^{m}}=1$
, and the matrix
$S_{m,A_{1}}$
defined by
$\mathbf{v}_{m,A_{1},j}=\sum _{i}(S_{m,A_{1}})_{ij}\mathbf{e}_{i}$
is invertible and satisfies
$|S_{m,A_{1}}|_{J},|S_{m,A_{1}}^{-1}|_{J}\leqslant p^{km}$
. After enlarging
$k$
again, there also exist elements
$\mathbf{v}_{m,A_{2},j}^{\prime }\in M_{J}$
in the span of
$\mathbf{e}_{1}^{\prime },\ldots ,\mathbf{e}_{n^{\prime }}^{\prime }$
such that
$\unicode[STIX]{x1D701}^{\ast }(\mathbf{v}_{m,A_{2},j}^{\prime })\equiv \unicode[STIX]{x1D701}^{a_{2,j}}\mathbf{v}_{m,A_{2},j}\hspace{0.6em}({\rm mod}\hspace{0.2em}M_{1,J})$
for all
$\unicode[STIX]{x1D701}\in K^{\text{alg}}$
with
$\unicode[STIX]{x1D701}^{p^{m}}=1$
, and the matrix
$S_{m,A_{2}}$
defined by
$\mathbf{v}_{m,A_{2},j}^{\prime }=\sum _{i}(S_{m,A_{2}})_{ij}\mathbf{e}_{i}^{\prime }$
is invertible and satisfies
$|S_{m,A_{2}}|_{J},|S_{m,A_{2}}^{-1}|_{J}\leqslant p^{km}$
. By replacing
$\mathbf{v}_{m,A_{2},j}^{\prime }$
with
$$\begin{eqnarray}\mathbf{v}_{m,A_{2},j}^{\prime \prime }=\frac{1}{p^{m}}\mathop{\sum }_{\unicode[STIX]{x1D701}^{p^{m}}=1}\unicode[STIX]{x1D701}^{-a_{2,j}}\unicode[STIX]{x1D701}^{\ast }(\mathbf{v}_{m,A_{2},j}^{\prime }),\end{eqnarray}$$
we ensure that
$\unicode[STIX]{x1D701}^{\ast }(\mathbf{v}_{m,A_{2},j}^{\prime \prime })=\unicode[STIX]{x1D701}^{a_{2,j}}\mathbf{v}_{m,A_{2},j}^{\prime \prime }$
. The change-of-basis matrix
$S_{m,A}$
from
$\mathbf{e}_{1},\ldots ,\mathbf{e}_{n},\mathbf{e}_{1}^{\prime },\ldots ,\mathbf{e}_{n^{\prime }}^{\prime }$
to
$\mathbf{v}_{m,A_{1},1},\ldots ,\mathbf{v}_{m,A_{1},n},\mathbf{v}_{m,A_{2},1}^{\prime \prime },\ldots ,\mathbf{v}_{m,A_{2},n^{\prime }}^{\prime }$
is an upper-triangular block matrix with diagonal blocks
$S_{m,A_{1}},S_{m,A_{2}}$
; by comparing to the original basis, we see that the off-diagonal block of
$S_{m,A}$
is bounded in norm by
$p^{4km}$
. We then have
$|S_{m,A}|_{J},|S_{m,A}^{-1}|_{J}\leqslant p^{6km}$
.
Theorem 3.4.20. The given argument alone only implies that the
$S_{m_{0},A}U$
is the change-of-basis matrix to a basis
$\mathbf{e}_{1},\ldots ,\mathbf{e}_{n}$
of
$M_{[\unicode[STIX]{x1D6FC}^{\prime },\unicode[STIX]{x1D6FD}^{\prime }]}$
which is fixed by parallel transport by all
$p$
-power roots of unity. To deduce that the matrix of action
$N$
of
$D$
on this basis has entries in
$K$
, note that for any
$\unicode[STIX]{x1D706}\in K^{\times }$
, the operations
$\unicode[STIX]{x1D706}^{\ast }$
and
$D$
commute; consequently, for
$\unicode[STIX]{x1D701}$
a
$p$
-power root of unity,
$$\begin{eqnarray}\mathop{\sum }_{i}N_{ij}\mathbf{e}_{i}=D(\mathbf{e}_{i})=(D\circ \unicode[STIX]{x1D701}^{\ast })(\mathbf{e}_{i})=(\unicode[STIX]{x1D701}^{\ast }\circ D)(\mathbf{e}_{i})=\mathop{\sum }_{i}\unicode[STIX]{x1D701}^{\ast }(N_{ij})\mathbf{e}_{i}.\end{eqnarray}$$
This implies that each
$N_{ij}$
belongs to
$K$
. To see that
$N$
is nilpotent, let
$\unicode[STIX]{x1D706}$
be any eigenvalue; by [Reference KedlayaKed10, Example 9.5.2], we must have
$\unicode[STIX]{x1D706}\in \mathbb{Z}_{p}$
. We can then find a nonzero eigenvector
$\mathbf{v}$
of
$N$
in the
$K$
-span of
$\mathbf{e}_{1},\ldots ,\mathbf{e}_{n}$
, which is fixed by
$\unicode[STIX]{x1D701}^{\ast }$
; however, by computing
$\unicode[STIX]{x1D701}^{\ast }$
in terms of
$N$
we obtain
$\unicode[STIX]{x1D701}^{\ast }(\mathbf{v})=\unicode[STIX]{x1D701}^{\unicode[STIX]{x1D706}}\mathbf{v}$
. This forces
$\unicode[STIX]{x1D706}=0$
, so
$N$
is nilpotent as claimed. (Compare the errata for [Reference KedlayaKed10, Theorem 13.6.1].)
Theorem 3.4.22. The given argument implies that the
$\unicode[STIX]{x1D6F1}_{m}$
converge to a projector
$\unicode[STIX]{x1D6F1}$
stable under parallel transport by all
$p$
-power roots of unity. However, it is not immediately apparent either that the limiting projector is horizontal, or that the summands have the correct exponents.
To remedy this, it is convenient to work in a somewhat larger category. As in the given argument, we assume that
$0\notin I$
. Let
$\unicode[STIX]{x1D707}_{p^{\infty }}$
be the group of
$p$
-power roots of unity in an algebraic closure of
$K$
. Let
${\mathcal{C}}$
be the category in which an object is a finite projective
$R_{I}$
-module
$M$
together with a semilinear action of
$\unicode[STIX]{x1D707}_{p^{\infty }}$
on
$M\otimes _{K}K(\unicode[STIX]{x1D707}_{p^{\infty }})$
satisfying the following conditions.
- –
-
The action of
$\unicode[STIX]{x1D701}\in \unicode[STIX]{x1D707}_{p^{\infty }}$
is semilinear with respect to the substitution
$t\mapsto \unicode[STIX]{x1D701}t$
on
$R_{I}$
. - –
-
The action of
$\unicode[STIX]{x1D707}_{p^{\infty }}$
is equivariant with respect to the action of
$\text{Gal}(K(\unicode[STIX]{x1D707}_{p^{\infty }})/K)$
on both
$\unicode[STIX]{x1D707}_{p^{\infty }}$
and
$M\otimes _{K}K(\unicode[STIX]{x1D707}_{p^{\infty }})$
. In particular, for each positive integer
$n$
, the action of
$\unicode[STIX]{x1D707}_{p^{n}}$
is induced by an action on
$M\otimes _{K}K(\unicode[STIX]{x1D707}_{p^{n}})$
. - –
-
For each closed subinterval
$J$
of
$I$
, for some (and hence any) basis
$\mathbf{e}_{1},\ldots ,\mathbf{e}_{n}$
of
$M_{J}$
, there exists
$k>0$
such that for each positive integer
$m$
, for each
$\unicode[STIX]{x1D701}\in \unicode[STIX]{x1D707}^{p^{m}}$
, the matrix
$E(\unicode[STIX]{x1D701})$
defined by
$\unicode[STIX]{x1D701}^{\ast }(\mathbf{e}_{j})=\sum _{i}E(\unicode[STIX]{x1D701})_{ij}\mathbf{e}_{i}$
satisfies
$|E(\unicode[STIX]{x1D701})|\leqslant p^{km}$
.
We use the following facts about
${\mathcal{C}}$
.
- –
-
Every
$\unicode[STIX]{x1D707}_{p^{\infty }}$
-stable submodule of an object of
${\mathcal{C}}$
is saturated. This follows from the fact that every finitely generated
$\unicode[STIX]{x1D707}_{p^{\infty }}$
-stable ideal of
$R_{I}\otimes _{K}K(\unicode[STIX]{x1D707}_{p^{\infty }})$
is either zero or trivial (the support of such an ideal cannot be both finite and nonempty). - –
-
The condition on
$|E(\unicode[STIX]{x1D701})|$
is preserved under passage to
$\unicode[STIX]{x1D707}_{p^{\infty }}$
-stable subquotients. Using the previous point, it follows that
${\mathcal{C}}$
is an abelian category. Beware that the condition on
$|E(\unicode[STIX]{x1D701})|$
is not preserved under formation of extensions; however, for an extension which does belong to
${\mathcal{C}}$
, the discussion of Remark 3.4.14 given above carries over. - –
-
The category
${\mathcal{C}}$
contains the differential modules over
$R_{I}$
satisfying the Robba condition as a nonfull subcategory. This is a consequence of [Reference KedlayaKed10, Lemma 13.5.4]. - –
-
Neither the final part of the proof of [Reference KedlayaKed10, Theorem 13.5.5] (starting from ‘We next choose
$A$
…’), nor any of the proof of [Reference KedlayaKed10, Theorem 13.5.6], makes any direct reference to differentiation, only to parallel transport. Consequently, both parts of Theorem 3.4.16 remain true for objects of
${\mathcal{C}}$
. - –
-
The given argument for Theorem 3.4.22 applies without change to an object
$M$
of
${\mathcal{C}}$
. That is, if
$M$
has an exponent admitting a nontrivial Liouville partition
$A_{1},A_{2}$
, then
$M$
splits nontrivially in
${\mathcal{C}}$
into summands
$M_{1},M_{2}$
. However, it is not apparent from the construction that
$M_{i}$
is an exponent of
$A_{i}$
.
In order to proceed, we must formally upgrade this last point to a stronger statement. Namely, let
$M$
be an object of
${\mathcal{C}}$
with an exponent admitting a Liouville partition
$A_{1},A_{2}$
. We will show that
$M$
admits a unique splitting
$M_{1}\oplus M_{2}$
in
${\mathcal{C}}$
for which
$M_{i}$
admits an exponent weakly equivalent to
$A_{i}$
. (Note that this does not guarantee that
$A_{i}$
itself occurs as an exponent for
$M_{i}$
.)
We first verify uniqueness. Suppose that
$M$
splits as
$M_{1}^{\prime }\oplus M_{2}^{\prime }$
,
$M_{1}^{\prime \prime }\oplus M_{2}^{\prime \prime }$
with
$M_{i}^{\prime },M_{i}^{\prime \prime }$
having exponents
$A_{i}^{\prime },A_{i}^{\prime \prime }$
weakly equivalent to
$A_{i}$
(and hence to each other). Let
$f$
be the composition of the inclusion
$M_{1}^{\prime }\rightarrow M$
with the projection
$M\rightarrow M_{2}^{\prime \prime }$
. Since
${\mathcal{C}}$
is an abelian category, we may form the exact sequences
$$\begin{eqnarray}0\rightarrow \ker (f)\rightarrow M_{1}^{\prime }\rightarrow \text{image}(f)\rightarrow 0,\quad 0\rightarrow \text{image}(f)\rightarrow M_{2}^{\prime \prime }\rightarrow \text{coker}(f)\rightarrow 0\end{eqnarray}$$
and choose exponents
$E_{1},E_{2},E_{3}$
for
$\ker (f),\text{image}(f),\text{coker}(f)$
. By Remark 3.4.14 and Theorem 3.4.16(b),
$A_{1}^{\prime }$
is weakly equivalent to
$E_{1}\cup E_{2}$
and
$A_{2}^{\prime \prime }$
is weakly equivalent to
$E_{2}\cup E_{3}$
. By Proposition 3.4.5(a),
$A_{1}^{\prime },A_{2}^{\prime \prime }$
constitute a Liouville partition, so we must have
$E_{2}=\emptyset$
, which is to say
$f=0$
. Similarly, the composition
$M_{2}^{\prime }\rightarrow M\rightarrow M_{1}^{\prime \prime }$
equals zero, so we have
$M_{1}^{\prime }=M_{1}^{\prime \prime }$
,
$M_{2}^{\prime }=M_{2}^{\prime \prime }$
as desired.
We next verify existence, proceeding by induction on the rank of
$M$
. There is nothing to check unless
$A_{1},A_{2}$
are both nonempty; in this case, we know that there exists a nontrivial splitting
$M\cong M_{1}^{\prime }\oplus M_{2}^{\prime }$
in
${\mathcal{C}}$
. Choose exponents
$A_{1}^{\prime },A_{2}^{\prime }$
for
$M_{1}^{\prime },M_{2}^{\prime }$
; then
$A_{1}^{\prime }\cup A_{2}^{\prime }$
is an exponent for
$M$
. By Theorem 3.4.16(b),
$A_{1}^{\prime }\cup A_{2}^{\prime }$
is weakly equivalent to
$A_{1}\cup A_{2}$
, so, by Proposition 3.4.5(b),
$A_{1}^{\prime }\cup A_{2}^{\prime }$
admits a Liouville partition
$A_{1}^{\prime \prime },A_{2}^{\prime \prime }$
with
$A_{i}^{\prime \prime }$
being weakly equivalent to
$A_{i}$
. In particular,
$A_{1}^{\prime \prime },A_{2}^{\prime \prime }$
are disjoint.
Identify finite multisubsets of
$\mathbb{Z}_{p}$
with finitely supported functions
$\mathbb{Z}_{p}\rightarrow \mathbb{Z}_{{\geqslant}0}$
, and put
$$\begin{eqnarray}A_{ij}^{\prime \prime }(x)=\min \{A_{i}^{\prime }(x),A_{j}^{\prime \prime }(x)\}\quad (x\in \mathbb{Z}_{p}).\end{eqnarray}$$
Since
$A_{1}^{\prime \prime },A_{2}^{\prime \prime }$
are disjoint, we have
$$\begin{eqnarray}(A_{1}^{\prime \prime }(x),A_{2}^{\prime \prime }(x))\in \{(A_{1}^{\prime }(x)+A_{2}^{\prime }(x),0),(0,A_{1}^{\prime }(x)+A_{2}^{\prime }(x))\}.\end{eqnarray}$$
We thus have
$$\begin{eqnarray}A_{i1}^{\prime \prime }(x)+A_{i2}^{\prime \prime }(x)=\min \{A_{i}^{\prime }(x),A_{1}^{\prime \prime }(x)\}+\min \{A_{i}^{\prime }(x),A_{2}^{\prime \prime }(x)\}=A_{i}^{\prime }(x)\end{eqnarray}$$
and hence
$A_{i1}^{\prime \prime },A_{i2}^{\prime \prime }$
form a Liouville partition of
$A_{i}^{\prime }$
. On the other hand,
$$\begin{eqnarray}A_{1i}^{\prime \prime }(x)+A_{2i}^{\prime \prime }(x)=\min \{A_{1}^{\prime }(x),A_{i}^{\prime \prime }(x)\}+\min \{A_{2}^{\prime }(x),A_{i}^{\prime \prime }(x)\}=A_{i}^{\prime \prime }(x).\end{eqnarray}$$
By the induction hypothesis, we may split
$M_{i}^{\prime }$
as
$M_{i1}^{\prime \prime \prime }\oplus M_{i2}^{\prime \prime \prime }$
with
$M_{ij}^{\prime \prime \prime }$
admitting an exponent
$A_{ij}^{\prime \prime \prime }$
weakly equivalent to
$A_{ij}^{\prime \prime }$
. Now define
$M_{i}=M_{1i}^{\prime \prime \prime }\oplus M_{2i}^{\prime \prime \prime }$
, so that
$M\cong M_{1}\oplus M_{2}$
. By construction,
$M_{i}$
admits the exponent
$A_{1i}^{\prime \prime \prime }\cup A_{2i}^{\prime \prime \prime }$
which is weakly equivalent to
$A_{i}^{\prime \prime }$
and hence to
$A_{i}$
, as desired.
We finally return to the original situation, in which
$M$
is a differential module over
$R_{I}$
satisfying the Robba condition, with an exponent admitting the Liouville partition
$A_{1},A_{2}$
. By the previous arguments, in
${\mathcal{C}}$
there is a unique splitting
$M\cong M_{1}\oplus M_{2}$
in which
$M_{i}$
admits an exponent weakly equivalent to
$A_{i}$
. For
$\unicode[STIX]{x1D706}\in K^{\times }$
with
$|1-\unicode[STIX]{x1D706}|<1$
, the substitutions
$t\mapsto \unicode[STIX]{x1D701}t$
and
$t\mapsto \unicode[STIX]{x1D706}t$
commute: consequently, the pullbacks
$M_{1,\unicode[STIX]{x1D706}}$
and
$M_{2,\unicode[STIX]{x1D706}}$
of
$M_{1},M_{2}$
along
$\unicode[STIX]{x1D706}^{\ast }$
are again objects of
${\mathcal{C}}$
with the same exponents as
$M_{1},M_{2}$
. Consequently, the splittings
$M_{1}\oplus M_{2}$
,
$M_{1,\unicode[STIX]{x1D706}}\oplus M_{2,\unicode[STIX]{x1D706}}$
of
$M$
must coincide, which is to say that
$M_{1}$
and
$M_{2}$
are preserved by
$\unicode[STIX]{x1D706}^{\ast }$
. By taking a sequence of values of
$\unicode[STIX]{x1D706}$
converging to 1, we deduce that
$M_{1}$
and
$M_{2}$
are differential submodules, as desired.
Definition 3.7.9. The displayed equation should read
$$\begin{eqnarray}D(\mathbf{v}_{1})=\mathbf{v}_{2},\ldots ,D(\mathbf{v}_{e-1})=\mathbf{v}_{e},\quad D(\mathbf{v}_{e})=\unicode[STIX]{x1D706}t^{h/m}\mathbf{v}_{1}.\end{eqnarray}$$
Lemma 3.7.11. In the fifth line of the proof, the inequality should read
$|\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D706}^{1/e}t^{h/(em)}|_{\unicode[STIX]{x1D70C}}<|\unicode[STIX]{x1D6FC}|_{\unicode[STIX]{x1D70C}}$
. We give some further justification for why this inequality can be achieved.
There is no issue unless
$\unicode[STIX]{x1D70C}$
belongs to the divisible closure of
$|F^{\times }|$
; we may thus assume without loss of generality that
$K$
is algebraically closed and
$\unicode[STIX]{x1D70C}=1$
. Because of the hypothesis that
$M$
is pure, in the notation of Proposition 3.6.3 we must have
$s_{\overline{\unicode[STIX]{x1D707}},i}(M)=0$
for all
$i$
and all nonzero
$\overline{\unicode[STIX]{x1D707}}$
. This means that for each index
$i$
such that
$a_{i}$
corresponds to a point (not necessarily a vertex) on the Newton polygon of the polynomial
$P(T)$
, we can find
$\unicode[STIX]{x1D706}_{i}\in K$
,
$m_{i}\in \mathbb{Z}$
such that
$|a_{i}-\unicode[STIX]{x1D706}_{i}t^{m_{i}}|_{\unicode[STIX]{x1D70C}}<|a_{i}|$
; moreover,
$\log |\unicode[STIX]{x1D706}_{i}|$
and
$m_{i}$
are determined by the affine function
$F_{n-i}(M,r)$
near
$r=0$
. From this, we may infer the desired approximation.
Theorem 3.8.16. In light of the modifications to Proposition 1.1.2 and Remark 1.1.3 described above, we must also verify the conditions (iv) and (v) added to Remark 1.1.3.
- –
-
To check (iv), we may enlarge
$K$
to contain all roots of unity; we may then reduce to checking that if
$M$
is of rank 1,
$M$
satisfies the Robba condition, and
$M^{\otimes \ell }$
is trivial for some prime
$\ell$
, then
$M$
becomes trivial after a finite tamely ramified extension (which does not change
$G^{0+}(M)$
). Apply Theorem 3.4.16 to construct an exponent
$A\in \mathbb{Z}_{p}$
of
$M$
; then
$\ell A$
is weakly equivalent to 0, so Corollary 3.4.7 implies that
$\ell A\in \mathbb{Z}$
. If
$\ell =p$
, then it follows that
$A\in \mathbb{Z}$
, so Theorem 3.4.20 implies that
$M$
is trivial; otherwise
$M$
becomes trivial after adjoining
$t^{1/\ell }$
. - –
-
To check (v), we may apply the lemma stated in the discussion of Theorem 2.3.17; it implies that for any
$M$
and any
$r>0$
, the filtration of
$G^{0+}(M)$
has only finitely many breaks in the interval
$[r,pr]$
.
