$M\rightarrow E$. In the language of [BGH+18, Section 4], we have recovered a
choice $M_{\infty}$ of $p$-power roots in $E$ of the lattice $M$, and our
homomorphism is precisely the map $M_{\infty}\rightarrow E_{\infty}$ from
[BGH+18, Definition 4.2]. Continuing with the notation from [BGH+18], we
define $D_{\infty}:=M_{\infty}\times_{\mathbb{Z}}\mathbb{Z}_{p}$. Then by
[BGH+18, Theorem 4.6.5], we have
$A_{\infty}=D_{\infty}\times^{M_{\infty}}E_{\infty}$. By symmetry, the same
bilinear form also defines $M^{\vee}\rightarrow E^{\vee}_{\infty}$, from which
we can recover $A^{\vee}_{\infty}$. In summary, we have:

######  Proposition 5.4.6.

The spaces $E_{\infty}$, $E^{\vee}_{\infty}$, $A_{\infty}$, and
$A^{\vee}_{\infty}$ are determined by the top triangle of Diagram (5.4.2).

Tilting the top triangle of (5.4.2) and applying Proposition 5.4.3 to fill in
the bottom triangle, we get a diagram full of leading notation which will take
some unpacking

(5.4.3) ${P^{\prime\times}_{\infty}}$${M^{\prime}\times
M^{\prime\vee}}$${B^{\prime}_{\infty}\times
B^{\prime\vee}_{\infty}}$${P^{\times}_{B^{\prime}\times
B^{\prime\vee}}}$${M^{\prime}\times M^{\prime\vee}}$${B^{\prime}\times
B^{\prime\vee}.}$$\scriptstyle{\langle\cdot,\cdot\rangle^{\prime}_{\infty}}$$\scriptstyle{\phi^{\prime}_{\infty}\times\phi^{\prime\vee}_{\infty}}$$\scriptstyle{\langle\cdot,\cdot\rangle^{\prime}}$$\scriptstyle{\phi^{\prime}\times\phi^{\prime\vee}}$

Here we have applied the isomorphism $B_{\infty}^{\flat}\cong
B^{\prime}_{\infty}$ which comes because the special fibers of $B$ and
$B^{\prime}$ agree, the analogous isomorphism for $B^{\prime\vee}$, and the
isomorphism $P^{\times}_{\infty}\cong P^{\prime\times}_{\infty}$ from
Proposition 5.4.3, which also gives the commutativity of the right vertical
square. We are _defining_ $M^{\prime}\times M^{\prime\vee}$ to be the tilt of
$M\times M^{\vee}$, and the morphisms filling out the top triangle are defined
by the tilting correspondence. The morphisms in the bottom triangle are
defined by composition of the morphisms in the top triangle with projections
to the bottom. The map $\phi^{\prime}\times\phi^{\prime\vee}$ is the
composition of homomorphisms of adic groups, so it is itself a group
homomorphism.

######  Lemma 5.4.7.

The map $\langle\cdot,\cdot\rangle^{\prime}:M^{\prime}\times
M^{\prime\vee}\rightarrow P^{\times}_{B^{\prime}\times B^{\prime\vee}}$ is a
non-degenerate bilinear form.

###### Proof.

Bilinearity follows because tilting preserves the two partial group laws on
$P_{\infty}$. Non-degeneracy follows as in Lemma 4.2.1. ∎

Applying [Lüt16, Proposition 6.1.8] to this bilinear form, we get a pair of
abeloids $A^{\prime}:=E^{\prime}/M^{\prime}$ and
$A^{\prime\vee}:=E^{\prime\vee}/M^{\prime\vee}$. Our goal is to show that
$A^{\prime}_{\infty}\cong A_{\infty}^{\flat}$, which will imply by symmetry
that $A^{\prime\vee}_{\infty}\cong(A^{\prime\vee}_{\infty})^{\flat}$. By
Theorem 4.2.2, it is enough to show that $E^{\prime}_{\infty}\cong
E_{\infty}^{\flat}$ and that the map $M^{\prime}\rightarrow
E^{\prime}_{\infty}$ induced by $\langle\cdot,\cdot\rangle^{\prime}_{\infty}$
agrees with the tilt of the map $M\rightarrow E_{\infty}$ induced by
$\langle\cdot,\cdot\rangle^{\prime}_{\infty}$.

By the decomposition in Proposition 5.4.5, to see that
$E^{\prime}_{\infty}\cong E_{\infty}^{\flat}$ it is enough to see that

$(P^{\times}_{\phi^{\vee}_{\infty}(m^{\vee})})^{\flat}\cong
P^{\prime\times}_{\phi^{\prime\vee}_{\infty}(m^{\flat})},$

but this is precisely the statement of Corollary 5.4.4. The statement about
the bilinear forms follows because $h^{\prime}:M^{\prime}\rightarrow
E^{\prime}$ is determined by the $r$ maps

$\langle\cdot,\phi^{\prime\vee}(m^{\prime\vee}_{i})\rangle^{\prime}_{\infty}:M^{\prime}\rightarrow
P^{\prime\times}_{B^{\prime}_{\infty}\times\phi^{\prime\vee}_{\infty}(m_{i}^{\vee})},$

which are by definition the tilts of the $r$ analogous maps defining $h$ in
terms of $\langle\cdot,\cdot\rangle_{\infty}.$ We therefore get the desired
result:

######  Proposition 5.4.8.

Let $A$ be an abeloid over a perfectoid field $K$ with dual $A^{\vee}$, let
$A_{\infty}$ and $(A^{\vee})_{\infty}$ be the corresponding perfectoid covers.
Then there is an abeloid $A^{\prime}$ over $K^{\flat}$ with
$A^{\prime}_{\infty}\cong(A_{\infty})^{\flat}$ and
$(A^{\prime\vee})_{\infty}\cong(A^{\vee})_{\infty}^{\flat}$.

With this in hand, we tilt our line bundles.

######  Theorem 5.4.9.

Given a line bundle $L$ on an abeloid $A/K$, there is some model
$\mathfrak{A}_{\alpha}$ of $A$ such that $L$ extends to a formal line bundle
$\mathfrak{L}_{\alpha}$ on $\mathfrak{A}_{\alpha}$. If the degree of $L$ is
prime to $p$, then after replacing $K$ by a pro-$p$ extension, there is some
$A^{\prime}/K^{\flat}$ as in Theorem 4.2.2 and line bundle $L^{\prime}$
extending to a formal line bundle $\mathfrak{L}^{\prime}_{\alpha}$ such that
the special fibers of the formal line bundles agree. The line bundle $L$ is
ample if and only if $L^{\prime}$ is.

###### Proof.

By Proposition 5.1.2, $L$ is equivalent to a line bundle $N$ on $B$, group
homomorphism $\sigma:M\rightarrow M^{\vee}$, and trivialization
$\chi:M\rightarrow\phi^{*}N$ such that the diagram

(5.4.4)
${M}$${B}$${M^{\vee}}$${B^{\vee}.}$$\scriptstyle{\phi}$$\scriptstyle{\sigma}$$\scriptstyle{\varphi_{N}}$$\scriptstyle{\phi^{\vee}}$

commutes, and the equation

(5.4.5)
$\chi(m_{1}+m_{2})\otimes\chi(m_{1})^{-1}\otimes\chi(m_{2})^{-1}=\langle
m_{1},\sigma(m_{2})\rangle$

holds for all $m_{1},m_{2}\in M$.

By Proposition 5.2.3 and Proposition 5.2.5, $L$ extends to some formal model
$\mathfrak{A}_{\alpha}$, and is equivalent to a choice of formal model
$\mathfrak{N}^{\prime}$ of $N$ and a continuous function $f$ built from
“tropical data” associated to $\sigma$ and $\chi$. To construct the desired
$L^{\prime}$, we use the data for $L$ to construct analogous data
$N^{\prime},\sigma^{\prime},$ and $\chi^{\prime}$ for $A^{\prime}$.

As we saw while proving Proposition 5.3.4, there is some lift of
$\widetilde{B^{\prime}}$ to a formal abelian scheme $\overline{B^{\prime}}$
over $K^{\flat\circ}$ such that the line bundle $\overline{\mathfrak{N}}$
lifts to a line bundle $\mathfrak{N}^{\prime}$ with generic fiber
$N^{\prime}$. The line bundle $N^{\prime}$ is ample if and only if its special
fiber is. Using this $B^{\prime}$ when we construct $A^{\prime}$, we get the
desired $N^{\prime}$. We note that this step requires the condition on the
degree of $L$ (and is the only place where it is used).

Constructing $A^{\prime}$ and $A^{\prime\vee}$ simultaneously as in
Proposition 5.4.8, we get isomorphisms $\flat:M\cong M^{\prime}$ and
$\flat:M^{\prime}\cong M^{\prime\vee}$ which agree on the special fiber (that
is, their actions on the special fiber agree). We use these isomorphisms to
transfer $\sigma$ to a group homomorphism
$\sigma^{\prime}:M^{\prime}\rightarrow M^{\prime\vee}$ which agrees with
$\sigma$ on the special fiber. These isomorphisms therefore respect the
resulting bilinear forms (5.1.10), so one is positive definite precisely when
the other is. By the ampleness criteria in Proposition 5.1.2, we see that $L$
is ample if and only if $L^{\prime}$ is.

To tilt the trivialization $\chi:M\rightarrow\phi^{*}N\cong M\times K$, we let
$m^{\prime}_{1},\dots,m^{\prime}_{r}$ be the image in $M^{\prime}$ of the
basis $m_{1},\dots,m_{r}$ of $M$ under $\flat$. We define
$r^{\prime}(m^{\prime}_{i})$ to be $(m_{i}^{\prime},a_{i}^{\prime})$ for any
$a_{i}^{\prime}$ in $\sharp^{-1}(\chi(m_{i}))$ \- perhaps making a pro-$p$
extension of $K$ to ensure that such an element exists - then extend to all of
$M^{\prime}$ using the rule (5.4.5). ∎

The condition on the degree of $L$ is somewhat annoying, but we believe it is
unavoidable. For our perfectoid applications, Lemma 5.3.5 will let us get
around the issue.

As a first application, we can upgrade Theorem 4.2.2 to work for abelian
varieties, not just abeloids.

######  Theorem 5.4.10.

Let $A$ be an abelian variety over a perfectoid field $K$ with value group
contained in $\mathbb{Q}$. Let $A_{\infty}$ be the perfectoid cover
constructed in Theorem 3.2.5. After replacing $K$ by a pro-$p$ extension,
there is an abelian variety $A^{\prime}$ over $K^{\flat}$ such that
$(A_{\infty})^{\flat}\cong(A^{\prime})_{\infty}$.

###### Proof.

Proposition 5.4.9 proves this for abelian varieties with a polarization of
degree prime to $p$. Fixing any polarization $\varphi_{A}$ of $A$, we can
quotient $A$ by a $p$-Sylow subgroup of $\ker(\varphi_{A})$, giving us an
isogeny $A\rightarrow A/\ker(\varphi_{A})=:A_{1}$ of degree $p^{k}$ for some
positive integer $k$. Then $\varphi_{A}$ induces a polarization on $A_{1}$ of
degree prime to $p$. By Lemma 5.3.5 we can prove the theorem after replacing
$A$ by $A_{1}$. ∎

######  Corollary 5.4.11.

Given a line bundle $L$ on an abelian variety $A$ over $K$, after replacing
$K$ by a pro-$p$ extension, there is a $(\mathbb{G}_{m,K})_{\infty}$ torsor
$L^{\times}_{\infty}$ living over $q_{1}^{*}(L^{\times})$.

###### Proof.

Now that we can transfer line bundles from $A$ to $A^{\prime}$, the argument
is essentially the same as in Proposition 5.3.4. We can again use the Theorem
of the cube to construct $p$th power roots at finite levels of the tower, and
we can use Theorem 5.4.9 to find a system of line bundles over $A^{\prime}$
with isomorphic special fibers. On the tilt side, Lemma 5.3.2 gives a
$\mathbb{G}_{m,\infty}^{\flat}$-torsor. Untilting, we get a
$\mathbb{G}_{m,\infty}$ torsor with the correct special fiber, so it must be
the one we are looking for. ∎

######  Remark 5.4.12.

Ben Heuer tells us that he can prove the stronger result that
$\operatorname{Pic}(A_{\infty})\cong\operatorname{Pic}(A_{\infty}^{\flat})$,
using that $H^{i}(A_{\infty},\mathbb{Z}_{p})=0$ for $i>0$. In particular, this
gives an affirmative answer to [DH19, Open Problem 5.14] in this case. See
[Heu21a, Theorem 1.8] for a description of
$\operatorname{Pic}^{0}(A_{\infty})$.

##  6\. Weight-monodromy for complete intersections in abelian varieties

We are now ready to prove our main result: the weight-monodromy conjecture for
complete intersections in abelian varieties. In Section 6.1, we give a more
detailed overview of the setup of the weight-monodromy conjecture. We then
prove the theorem in Section 6.2.

###  6.1. Background on the weight-monodromy conjecture

Let $X$ be a proper smooth $d$-dimensional variety over a local field $k$ of
residue characteristic $p$, let $q$ be the cardinality of the residue field.
Let $\ell$ be a prime not equal to $p$, let $i$ be an integer between 0 and
$2d$. Then the $i$th étale cohomology group
$V:=H^{i}_{\text{\'{e}t}}(X_{\overline{k}},\overline{\mathbb{Q}}_{\ell})$ is a
finite-dimensional $\overline{\mathbb{Q}}_{\ell}$-vector space with an action
of the absolute Galois group $G_{k}:=\operatorname{Gal}(\overline{k}/k)$ of
$k$. This action defines a Galois representation
$\rho:G_{k}\rightarrow\operatorname{GL}(V)$.

In this section, we explain how to define two filtrations on $V$. The
monodromy filtration comes from the inertia subgroup $I_{k}$ of $G_{k}$, and
the weight filtration comes from the action of a lift of Frobenius from the
residue field. The weight-monodromy conjecture says that these filtrations are
the same.

The monodromy filtration comes out of the following result of Grothendieck.

######  Proposition 6.1.1 ([ST68, Proposition on page 515]).

Let $\rho:G_{k}\rightarrow\operatorname{GL}(V)$ be a finite-dimensional
$\overline{\mathbb{Q}}_{\ell}$-representation. Then there is an open subgroup
$I_{1}$ of the inertia subgroup $I_{k}$ such that for all $g$ in $I_{1}$, the
matrix $\rho(g)$ is unipotent.

We can now define the monodromy filtration using the nilpotent operator
$N:=\log(\rho(g))$ on $V$ constructed in the course of the previous proof
(after a Tate twist).

######  Defintion/Proposition 6.1.2.

Let $N:V\rightarrow V$ be a nilpotent operator on a finite-dimensional vector
space. Then there is a unique finite increasing filtration
$\operatorname{Fil}_{j}^{N}$ of $V$ such that $N\operatorname{Fil}_{j}^{N}$ is
contained in $\operatorname{Fil}_{j-2}^{N}$ and such that $N^{k}$ induces an
isomorphism $\operatorname{Gr}_{i+j}^{N}V\cong\operatorname{Gr}_{i-j}^{N}V.$
Explicitly, we have

$\operatorname{Fil}_{j}^{N}V=\displaystyle\sum_{j_{1}-j_{2}=j}\ker
N^{j_{1}+1}\cap\operatorname{Im}N^{j_{2}}.$

###### Proof.

[Del80, Proposition 1.6.1] ∎

The monodromy filtration therefore depends only on the pro-$\ell$ part of the
inertia in the Galois group. Taking pro-$p$ extensions of our base field will
therefore preserve the monodromy filtration.

To define the weight filtration, we fix an element
$\Phi\in\operatorname{Gal}_{k}$ which maps to geometric Frobenius on the
residue field. We use the matrix $\rho(\Phi)\in\operatorname{GL}(V)$ to
decompose $V$ into eigenspaces. Work of Rapoport-Zink, along with de Jong’s
alterations theorem, gives us information about the eigenvalues.

######  Proposition 6.1.3.

With notation as above, we have a decomposition

$V=\bigoplus_{j=0}^{2i}W^{\prime}_{j},$

where the eigenvalues of $\rho(\Phi)$ acting on $W^{\prime}_{j}$ are Weil
numbers of weight $j$. That is, they are algebraic numbers, and for any
embedding of $k$ into $\mathbb{C}$, they have absolute value $q^{j/2}$.

###### Sketch.

See [RZ82] for details, or the nice expositions in [Ill94, Section 3] and
[Sch11, Section 2]. We use de Jong’s alterations theorem to reduce to the case
where $X$ has a semistable model. The special fiber of this model consists of
finitely many smooth varieties, intersecting transversely. By the Riemann
hypothesis part of the Weil conjectures, the Frobenius eigenvalues on each
irreducible component of the special fiber are all Weil numbers of weight $i$.
If $j$ components have non-empty intersection, it is a smooth variety with
codimension $j$, so the Frobenius eigenvalues are Weil numbers of weight
$i-j$. The Rapoport-Zink spectral sequence builds the actual cohomology of $X$
out of these pieces, giving the desired decomposition. ∎

If we chose a different lift of Frobenius, we’d get a different decomposition
of $V$. However, the weights and dimensions remain the same, as does the
weight filtration:

######  Definition 6.1.4.

With notation as in Proposition 6.1.3, the _weight filtration_ on $V$ is
defined by letting the $m$th piece be

$W_{m}=\bigoplus_{j=0}^{m}W^{\prime}_{j}.$

For any lift $\Phi$ of geometric Frobenius, we have $N\Phi=q\Phi N$. This
implies that the monodromy operator interacts nicely with the weight
filtration: we have $NW_{m}\subset W_{m-2}$. The weight-monodromy conjecture
says more:

######  Conjecture 6.1.5 (Deligne, [Del71]).

For a proper smooth variety $X$ over a local field $k$, the weight and
monodromy filtrations on the $i$th $\ell$-adic étale cohomology group of $X$
are the same. Equivalently, the eigenvalues of any lift $\Phi$ of geometric
Frobenius on any graded piece $\operatorname{gr}_{j}^{N}V$ of the monodromy
filtration are Weil numbers of weight $i+j$.

The equivalence of these statements is in [Del80, Section 1.6]. Deligne gave
another interpretation of this conjecture in terms of the poles of
$L$-functions associated to sheaves. He used this interpretation to prove the
conjecture when $k$ has characteristic $p$ and $X$ is defined over a curve.
See [Del80, Section 1.8] for the proof, or [Sch12, Section 9] for a quick
overview of the idea.

###  6.2. Weight-monodromy for complete intersections in abelian varieties

Scholze’s approach to weight-monodromy [Sch12, Section 9] is summarized in the
following:

######  Proposition 6.2.1.

Let $Y$ be a $d$-dimensional, geometrically connected, proper smooth variety
over a local field $k$ of residue characteristic $p$, fix a prime $\ell\neq
p$. Let $K$ be a perfectoid pro-$p$ extension of $k$, let $G$ be the absolute
Galois group of $K$ and $K^{\flat}$. To prove the weight-monodromy conjecture
for $Y$, it is enough to construct a $d$-dimensional, geometrically connected,
proper smooth variety $Z$ over $K^{\flat}$ which can be defined over some
local field contained in $K^{\flat}$, along with maps

$f_{i}^{*}:H^{i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})\rightarrow
H^{i}(Z_{\mathbb{C}_{p}^{\flat},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$

for $i\in\\{0,\dots,2d\\}$ which are $G$-equivariant, compatible with cup
product, and such that $f^{*}_{2d}$ is an isomorphism.

###### Proof.

We first check that the maps $f_{i}^{*}$ are all injective. To see this, we
work with the diagram

${H^{i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})\times
H^{2d-i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})}$${H^{2d}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})}$${\overline{\mathbb{Q}}_{\ell}}$${H^{i}(Z_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})\times
H^{2d-i}(Z_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})}$${H^{2d}(Z_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})}$${\overline{\mathbb{Q}}_{\ell}.\par}$$\scriptstyle{f_{i}^{*}\times
f_{2d-i}^{*}}$$\scriptstyle{\cup}$$\scriptstyle{f_{2d}^{*}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\cup}$$\scriptstyle{\cong}$

By Poincaré duality, the rows give perfect pairings between $H^{i}$ and
$H^{2d-i}$ for every $i$. The left square commutes because the $f_{i}^{*}$ are
compatible with cup product. The right square is all isomorphisms because
$f_{2d}^{*}$ is an isomorphism and $Y$ and $Z$ are geometrically connected.

Let $c\in
H^{i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$ be a
class with $f_{i}^{*}(c)=0\in
H^{i}(Z_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell}).$
As every class in
$H^{2d-i}(Z_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$
pairs to 0 with 0, commutativity of the diagram tells us that every class in
$H^{2d-i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$ must
pair to 0 with $c$. As the top row is a perfect pairing, $c$ must therefore be
0, so $f_{i}^{*}$ is injective as desired.

By $G$-equivariance, the map $f_{i}^{*}$ respects the monodromy filtrations on
$H^{i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$ and
$H^{i}(Z_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$.
By injectivity, the $j$th graded piece of
$H^{i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$ is a
sub-vector space of the $j$th graded piece of
$H^{i}(Z_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell}).$
Every Frobenius eigenvalue for $Y$ is therefore a Frobenius eigenvalue for
$Z$. As $Z$ is proper, smooth, and can be defined over a characteristic $p$
local field, Deligne proved weight-monodromy for $Z$. That is, every Frobenius
eigenvalue of $Z$ is a Weil number of degree $i+j$. The same is therefore true
for $Y$ and weight-monodromy holds. ∎

Specializing to our case of interest, let $Y$ be a complete intersection in an
abelian variety $A$ which satisfies the properties of Proposition 6.2.1. We
must construct a variety $Z\subset A^{\prime}$ and maps $f^{*}_{i}$ which also
satisfy the properties of Proposition 6.2.1. To construct $Z$, we use the
following approximation result: our analogue of Scholze’s approximation result
[Sch12, Proposition 8.7].

######  Proposition 6.2.2.

Let $Y\subset A$ be a hypersurface, let $\tilde{Y}$ be a small open
neighborhood of $Y$, where we are considering $Y$ and $A$ as adic spaces. Let
$q_{1}:|(A_{\infty})^{\flat}|\rightarrow|A|$ be the map of topological spaces
induced by tilting to $A_{\infty}$, then projecting to the bottom of the
inverse system. Let
$q_{1}^{\prime}:|(A_{\infty})^{\flat}|\rightarrow|A^{\prime}|$ be the map
induced by projection. Then there exists a hypersurface $Z^{\prime}\subset
A^{\prime}$ such that $q_{1}^{\prime-1}(Z^{\prime})\subset
q_{1}^{-1}(\tilde{Y})$. Equivalently, there exists some open
$\tilde{Y^{\prime}}\subset A^{\prime}$ with
$q_{1}^{\prime-1}(\tilde{Y^{\prime}})=q_{1}^{-1}(\tilde{Y})$ and a
hypersurface $Z^{\prime}\subset\tilde{Y^{\prime}}$.

###### Proof.

There is some invertible sheaf $\mathcal{L}_{1}$ on $A$ and an element $f\in
H^{0}(A,\mathcal{L}_{1})$ with zero locus $Y$. We will inductively define an
inverse system of line bundles $L_{1}\rightarrow L_{2}\rightarrow\cdots$ so
that the induced direct system of rings of global functions $R_{1}\leftarrow
R_{2}\leftarrow\cdots$ gives a perfectoid ring $R_{\infty}=(\varinjlim
R_{i})^{\wedge}$. The tilt $R_{\infty}^{\flat}$ will be given by an analogous
limit over $A^{\prime}$. This will allow us to approximate $q_{1}^{*}f$ by a
function pulled back from $A^{\prime}$.

Given an invertible sheaf $\mathcal{L}_{n}$ on $A$, the associated line bundle
(in the category of schemes, not adic or rigid spaces) is

$L_{n}:=\underline{\operatorname{Spec}}\big{(}\bigoplus_{m\in\mathbb{N}}\mathcal{L}_{n}^{\otimes
m}\big{)}.$

We write

$R_{n}\cong\bigoplus_{m\in\mathbb{N}}H^{0}(A,\mathcal{L}_{n}^{\otimes m})$

for the ring of global functions of this line bundle. Note that if we worked
in the category of rigid or adic spaces and analytified this line bundle (as
we did in the previous chapter), we would get many more global functions: the
rigid affine line has many more global functions than just $K[T]$. However, if
we start with a formal model $\mathfrak{L}_{n}$ in the sense of 5.2.1, we have
$H^{0}(\mathfrak{A}_{n},\mathfrak{L}_{n})[\frac{1}{p}]\cong
H^{0}(A^{an},\mathcal{L}^{an}_{n})\cong H^{0}(A,\mathcal{L}_{n})$, where the
last isomorphism is rigid GAGA [Köp74, §5], [Con06, Example 3.2.6]. These
isomorphisms compatibly extend to the tensor powers $\mathcal{L}_{n}^{\otimes
m}$, so we can use them define a $K^{\circ}$-submodule of $R_{n}$. We
topologize $R_{n}$ by letting this submodule be a ring of definition. This
allows us to make sense of the completion in the definition of $R_{\infty}$.

When $p$ is odd, we define

$L_{n+1}=L_{n}^{\otimes\frac{p+1}{2}}\otimes[-1]^{*}L_{n}^{\frac{p-1}{2}}.$

By the Theorem of the cube, we have $L_{n+1}^{\otimes p}\cong[p]^{*}L_{n}$. If
$p=2$, we instead let $L_{n+1}=L_{n}^{\otimes 5}\otimes[-1]^{*}L_{n}^{\otimes
3}$, which satisfies $L_{n+1}^{\otimes 2}\cong[4]^{*}L_{n}$, then argue
similarly. As in the discussion after Definition 5.3.1, we have a morphism
$L_{n+1}\rightarrow L_{n+1}^{\otimes p}$ induced by the inclusion
$\bigoplus_{m\in\mathbb{N}}\mathcal{L}_{n+1}^{\otimes
pm}\hookrightarrow\bigoplus_{m\in\mathbb{N}}\mathcal{L}_{n+1}^{\otimes m}$.
Now we can define the morphism $L_{n+1}\rightarrow L_{n}$ as the composition

$L_{n+1}\rightarrow L_{n+1}^{\otimes p}\cong[p]^{*}{L_{n}}\rightarrow L_{n}.$

This fits into the diagram

(6.2.1)
${L_{n+1}}$${A}$${L_{n}}$${A.}$$\scriptstyle{f_{n}}$$\scriptstyle{[p]}$

By Proposition 5.2.3, there is some formal model $\mathfrak{A}_{\alpha}$ of
$A$ and formal line bundle $\mathfrak{L}_{1}$ on $\mathfrak{A}_{\alpha}$ which
is a model of $L_{1}$. Inductively applying Corollary 5.2.6, there is then a
formal line bundle $\mathfrak{L}_{n}$ on $\mathfrak{A}_{\alpha/p^{n}}$ which
is a model of $L_{n}$. We therefore get a formal model of the above diagram

(6.2.2)
${\mathfrak{L}_{n+1}}$${\mathfrak{A}_{\alpha/p^{n+1}}}$${\mathfrak{L}_{n}}$${\mathfrak{A}_{\alpha/p^{n}}.}$$\scriptstyle{[p]}$

######  Lemma 6.2.3.

The mod $\varpi$ special fiber of the map
$\mathfrak{L}_{n+1}\rightarrow\mathfrak{L}_{n}$ in diagram 6.2.2 factors
through relative Frobenius.

###### Proof.

In the proof of Theorem 3.2.4, we constructed formal analytic covers
$\mathcal{U}_{n+1}=\\{U_{i,n+1}\\}$ and $\mathcal{U}_{n}=\\{U_{j,n}\\}$ of $A$
leading to the models $\mathfrak{A}_{\alpha/p^{n+1}}$ and
$\mathfrak{A}_{\alpha/p^{n}}$. These covers were constructed so that the map
$[p]:A\rightarrow A$ sends each $U_{i,n+1}$ to some $U_{j,n}$, and the special
fibers of the corresponding map of formal affine opens factors through
relative Frobenius. We have chosen $\alpha$ so that the line bundles $L_{n+1}$
and $L_{n}$ are trivial over the covers $\mathcal{U}_{n+1}$ and
$\mathcal{U}_{n}$ respectively. So locally on the source, $f_{n}$ restricts to
the map $\mathbb{A}^{1}_{K}\times U_{i,n+1}\rightarrow\mathbb{A}^{1}_{K}\times
U_{j,n}$ which is the product of the map
$\mathbb{A}^{1}_{K}\rightarrow\mathbb{A}^{1}_{K}$ sending the coordinate $T$
to $T^{p}$ with the restriction of $[p]$ to $U_{i,n+1}$. Both of these factor
through relative Frobenius on the special fiber, so their product does as
well. ∎

######  Corollary 6.2.4.

The ring $R_{\infty}$ is perfectoid.

###### Proof.

As the morphism
$\overline{\mathfrak{L}_{n+1}}\rightarrow\overline{\mathfrak{L}_{n}}$ factors
through relative Frobenius, the induced map $R^{\circ}_{n}/\varpi\rightarrow
R^{\circ}_{n+1}/\varpi$ also factors through relative Frobenius. Indeed, by
the Kunneth formula (see [Kem80, Theorem 14] for a version with the
appropriate generality), we have

$\Gamma(\overline{\mathfrak{L}_{n+1}}^{(p)})=\Gamma(\overline{\mathfrak{L}_{n+1}}\times_{\operatorname{Spec}(K^{\circ}/\varpi),\operatorname{Frob}}\operatorname{Spec}(K^{\circ}/\varpi))\cong\Gamma(\overline{\mathfrak{L}_{n+1}})\otimes_{K^{\circ}/\varpi,\operatorname{Frob}}K^{\circ}/\varpi\cong$
$R_{n+1}^{\circ}/\varpi\otimes_{K^{\circ}/\varpi,\operatorname{Frob}}K^{\circ}/\varpi\cong(R_{n+1}^{\circ}/\varpi)^{(p)},$

so the correct universal property is satisfied on global functions. Now the
local argument of Proposition 2.4.2 applies. ∎

Now by Theorem 5.4.9, we have a line bundle $L^{\prime}_{1}$ over an abelian
variety $A^{\prime}/K^{\flat}$ with a formal model $\mathfrak{L}^{\prime}_{1}$
defined over $\mathfrak{A}^{\prime}_{\alpha}$ such that the mod
$\varpi^{\flat}$ special fiber agrees with the mod $\varpi$ special fiber of
$\mathfrak{L}_{1}\rightarrow\mathfrak{A}_{\alpha}$. Tracing through the above
construction with $\mathfrak{L}^{\prime}_{1}$ in place of $\mathfrak{L}_{1}$
gives $R_{\infty}^{\flat}=(\varprojlim R^{\prime}_{i})^{\wedge}$.

We can let $\tilde{Y}=\\{x\in A:|f(x)|\leq\epsilon\\}$ for some $\epsilon$,
where the absolute value comes from the integral model for $L$ defined in
Proposition 5.2.3. Pulling back $\tilde{Y}$ and $f$ along $q_{1}$, we get an
open set in $A_{\infty}$ defined as an $\epsilon$-neighborhood of
$q_{1}^{*}(f)$, which we can view as an element of $R_{\infty}$. Using [Bha17,
Proposition 9.2.7], which builds off of [Sch12, Lemma 6.5], we can approximate
the vanishing locus of $q_{1}^{*}(f)$ by that of an element $g^{\sharp}$ in
the image of $\sharp:R_{\infty}^{\flat}\rightarrow R_{\infty}.$ We can tweak
$g$ so that it is the pullback of some $g\in R^{\prime}_{i}$ from some finite
level of the tower, and so that it is defined over $k$. Then
$q_{1}^{-1}(\tilde{Y})$ is an $\epsilon$-neighborhood of the vanishing locus
of $g$. Taking a big enough power of $g$, we can assume that it comes from the
bottom copy of $A^{\prime}$ on the tower. The vanishing locus $Z^{\prime}$ of
$g$ is the desired hypersurface. ∎

As in [Sch12, Corollary 8.8], we can intersect hypersurfaces to get the
analogous result when $Y$ is a complete intersection in $A$. We get an open
$\tilde{Y}^{\prime}\subset A^{\prime}$ such that
$q_{1}^{\prime-1}(\tilde{Y}^{\prime})=q_{1}^{-1}(\tilde{Y})$ as topological
spaces in $A^{\prime}_{\infty}$, and a closed variety $Z^{\prime}\subset
A^{\prime}$ of dimension $d$. We can choose $Z^{\prime}$ to be defined over a
local field in $K^{\flat}$, and take a component to ensure that it is
geometrically connected. We emphasize that even though we used the theory of
adic spaces to construct $Z^{\prime}$, it is a variety over a local field.

As in [Sch12, Section 9], we now make some reductions. By [Hub98, Theorem
3.6a], the relevant cohomology groups can be computed on the associated adic
spaces. We therefore analytify everything from now on, working strictly with
adic spaces. By [Hub96, Theorem 3.8.1], there is an open neighborhood
$\tilde{Y}$ of $Y$ in $A$ such that $Y$ and $Y^{\prime}$ have the same
cohomology. By de Jong’s alterations theorem [dJ96], we can choose a
projective smooth alteration $Z\rightarrow Z^{\prime}$. The induced maps on
cohomology are injective, giving
$H^{i}(Z^{\prime}_{\mathbb{C}_{p}^{\flat},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$
as a direct summand of
$H^{i}(Z_{\mathbb{C}_{p}^{\flat},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$.
It is therefore enough to construct maps from the cohomology of $Y$ to the
cohomology of $Z$.

The final difficulty, compared with Scholze’s proof for toric varieties, is
that the map $[p]:A^{\prime}\rightarrow A^{\prime}$ does not induce an
isomorphism of étale topoi. We therefore have no analogue of Scholze’s
projection map $\pi:\mathbb{P}^{n}_{K^{\flat}}\rightarrow\mathbb{P}^{n}_{K}$
(defined on étale topoi and topological spaces, and more generally defined for
toric varieties). To get around this, our cohomology computations must run
through the perfectoid covers more explicitly.

The tilting equivalence along with [Sch12, Theorem 7.17] gives isomorphisms of
étale topoi

$\varprojlim_{[p]}(A^{\prime}_{K^{\flat}})^{\sim}_{\text{\'{e}t}}\cong(A_{\infty}^{\flat})^{\sim}_{\text{\'{e}t}}\cong\varprojlim_{[p]}(A_{K})^{\sim}_{\text{\'{e}t}}.$

By [Sch12, Theorem 7.16], these inverse limits commute with taking fiber
products along étale maps to $A_{K}$ or $A^{\prime}_{K^{\flat}}$. We therefore
get the following diagram of étale topoi

${(A_{\infty}^{\flat})^{\sim}_{\text{\'{e}t}}}$${(A^{\prime}_{\mathbb{C}_{p}^{\flat}})^{\sim}_{\text{\'{e}t}}}$${(q_{1}^{-1}(\tilde{Y})_{\mathbb{C}_{p}^{\flat}})^{\sim}_{\text{\'{e}t}}}$${(A_{\mathbb{C}_{p}})^{\sim}_{\text{\'{e}t}}}$${(\tilde{Y}^{\prime}_{\mathbb{C}_{p}^{\flat}})^{\sim}_{\text{\'{e}t}}}$${(\tilde{Y}_{\mathbb{C}_{p}})^{\sim}_{\text{\'{e}t}}}$${(Z_{\mathbb{C}_{p}^{\flat}})^{\sim}_{\text{\'{e}t}}}$${(Y_{\mathbb{C}_{p}})^{\sim}_{\text{\'{e}t}}.\par}$$\scriptstyle{q_{1}}$$\scriptstyle{q_{1}^{\prime}}$$\scriptstyle{\cong}$

Here the parallelograms are fiber diagrams, and the diagonal maps are
projection maps to the bottom of the tilde-limits constructed by iterating the
maps $[p]$ for $A^{\prime}$ and $A$.

We first construct the maps $f^{*}_{i}$ from this diagram, then check that it
satisfies the desired properties. In the above diagram, pullback induces
isomorphisms

$H^{i}(\tilde{Y}_{\mathbb{C}_{p},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow
H^{i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z})$

for all $i$ and $m$, and maps

$H^{i}(\tilde{Y}_{\mathbb{C}_{p},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow
H^{i}(q_{1}^{-1}(\tilde{Y})_{\mathbb{C}_{p}^{\flat},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z}),$
$H^{i}(\tilde{Y}^{\prime}_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow
H^{i}(Z_{\mathbb{C}_{p}^{\flat},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z}),$

It remains to construct a map

$H^{i}(q_{1}^{-1}(\tilde{Y})_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow
H^{i}(\tilde{Y}^{\prime}_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z})$

To do this, we factor the morphism $[p]:A^{\prime}\rightarrow A^{\prime}$ as

$A^{\prime}\xrightarrow{\varphi}A^{\prime}/A^{\prime}[p]^{0}\xrightarrow{\psi}A^{\prime}.$

Here $A^{\prime}[p]^{0}$ is the identity component of the finite group scheme
$A^{\prime}[p]$. The morphism $\varphi$ is radicial and induces a
homeomorphism of topological spaces, so it induces an isomorphism of étale
topoi as in [Sch12, Corollary 7.19]. The morphism $\psi$ is finite étale.

Restricting these morphisms to

$[p]^{-1}(\tilde{Y}^{\prime}_{\mathbb{C}_{p}^{\flat}})\xrightarrow{\varphi}\psi^{-1}(\tilde{Y}^{\prime}_{\mathbb{C}_{p}^{\flat}})\xrightarrow{\psi}\tilde{Y}^{\prime}_{\mathbb{C}_{p}^{\flat}},$

we again have a radicial homeomorphism followed by a finite étale map. As the
radicial map induces an isomorphism of étale topoi, we get isomorphisms

$H^{i}([p]^{-1}(\tilde{Y})_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z})\xrightarrow{\cong}H^{i}(\psi^{-1}(\tilde{Y}_{\mathbb{C}_{p}^{\flat}}),\mathbb{Z}/\ell^{m}\mathbb{Z})$

for all $i$ and $m$. As $\psi$ is finite étale, the trace map [Sta20, Tag
03SH] gives maps

$H^{i}(\psi^{-1}(\tilde{Y}_{\mathbb{C}_{p}^{\flat}}),\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow
H^{i}(\tilde{Y}_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z})$

for all $i$ and $m$. These maps are induced by the adjunction maps
$\psi_{*}\psi^{*}(\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow\mathbb{Z}/\ell^{m}\mathbb{Z}$
of sheaves on $\tilde{Y}_{\mathbb{C}^{\flat}_{p}}$, and are defined as the
composition

$H^{i}(\psi^{-1}(\tilde{Y}_{\mathbb{C}_{p}^{\flat}}),\mathbb{Z}/\ell^{m}\mathbb{Z})\xrightarrow{\cong}H^{i}(\tilde{Y}_{\mathbb{C}_{p}^{\flat}},\psi_{*}\psi^{*}(\mathbb{Z}/\ell^{m}\mathbb{Z}))\rightarrow
H^{i}(\tilde{Y}_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z}).$

Composing these, we get a map

$H^{i}([p]^{-1}(\tilde{Y})_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow
H^{i}(\tilde{Y}_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z}).$

Iterating this process for the inverse system defining
$q_{1}^{-1}(\tilde{Y})_{\mathbb{C}_{p}^{\flat}}\sim\varprojlim_{n}[p^{n}]^{-1}(\tilde{Y}^{\prime}_{\mathbb{C}_{p}}),$
we get the desired map

$H^{i}(q_{1}^{-1}(\tilde{Y})_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow
H^{i}(\tilde{Y}^{\prime}_{\mathbb{C}^{\flat}_{p},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z}).$

All of the maps are induced by a combination of pullback along morphisms of
adic spaces and (in the case of trace) homomorphisms of étale sheaves. By
functoriality of these constructions, the compositions
$H^{i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow
H^{i}(Z_{\mathbb{C}_{p}^{\flat},\text{\'{e}t}},\mathbb{Z}/\ell^{m}\mathbb{Z})$
are $G$-equivariant and compatible with cup product. Taking the inverse limit
and tensoring with $\overline{\mathbb{Q}}_{\ell}$, the same is true of the
maps
$f^{*}_{i}:H^{i}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})\rightarrow
H^{i}(Z_{\mathbb{C}_{p}^{\flat},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell}).$
There is only one condition left to check for Proposition 6.2.1 to apply.

######  Lemma 6.2.5.

The map
$f_{2d}^{*}:H^{2d}(Y_{\mathbb{C}_{p},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})\rightarrow
H^{2d}(Z_{\mathbb{C}_{p}^{\flat},\text{\'{e}t}},\overline{\mathbb{Q}}_{\ell})$
is an isomorphism.

###### Proof.

Both cohomology groups are isomorphic to $\overline{\mathbb{Q}}_{\ell}$, so it
is enough to show that the map is non-zero. It is enough to check this when
the coefficients are $\mathbb{Z}/\ell^{m}\mathbb{Z}$, then pass to the inverse
limit. In this situation, the above constructions give the following
commutative diagram.

${H^{2d}(A_{\infty}^{\flat},\mathbb{Z}/\ell^{m}\mathbb{Z})}$${H^{2d}(A^{\prime}_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z})}$${H^{2d}(q_{1}^{-1}(\tilde{Y})_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z})}$${H^{2d}(A_{\mathbb{C}_{p}},\mathbb{Z}/\ell^{m}\mathbb{Z})}$${H^{2d}(\tilde{Y}^{\prime}_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z})}$${H^{2d}(\tilde{Y}_{\mathbb{C}_{p}},\mathbb{Z}/\ell^{m}\mathbb{Z})}$${H^{2d}(Z_{\mathbb{C}_{p}^{\flat}},\mathbb{Z}/\ell^{m}\mathbb{Z})}$${H^{2d}(Y_{\mathbb{C}_{p}},\mathbb{Z}/\ell^{m}\mathbb{Z})\par}$$\scriptstyle{tr}$$\scriptstyle{tr}$$\scriptstyle{\cong}$

The argument is now essentially the same as [Sch12, Lemma 9.8], with a
slightly bigger diagram. As multiplication by $p$ induces an isomorphism on
all $\ell$-adic cohomology groups of an abelian variety, the two diagonal maps
at the top of the diagram are isomorphisms. It therefore suffices to show that
the map
$H^{2d}(A^{\prime}_{\mathbb{C}^{\flat}_{p}},\mathbb{Z}/\ell^{m}\mathbb{Z})\rightarrow
H^{2d}(Z_{\mathbb{C}^{\flat}_{p}},\mathbb{Z}/\ell^{m}\mathbb{Z})$ is non-zero.
The $d$th power of the first Chern class of an ample line bundle on
$A^{\prime}$ will have non-zero image, so we are done.

∎

This concludes the proof of Theorem 1.1.2.

######  Remark 6.2.6.

The words “set-theoretic” in the statement of both Scholze’s theorem and our
version make it difficult to know what varieties the theorems apply to. For
most varieties, it is unknown if it is possible to write them in this fashion.
Ignoring the words set-theoretic then, most proper smooth complete
intersections $Y$ in abelian varieties are not also proper smooth complete
intersections in toric varieties. We learned the following argument from the
MathOverflow answer [Pol]. When $\dim(Y)\geq 3$, the Lefschetz hyperplane
theorem says that $\pi_{1}(Y_{\mathbb{C}_{p}})=\pi_{1}(A_{\mathbb{C}_{p}})$.
If $Y$ were also a complete intersection in a proper smooth toric variety $X$,
we would also have $\pi_{1}(Y_{\mathbb{C}_{p}})=\pi_{1}(X_{\mathbb{C}_{p}})$.
But $\pi_{1}(X_{\mathbb{C}_{p}})$ is trivial and $\pi_{1}(A_{\mathbb{C}_{p}})$
is not, giving a contradiction.

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# Spectrum Occupancy Detection Supported by Federated Learning ††thanks:
Copyright © 2023 IEEE. Personal use is permitted. For any other purposes,
permission must be obtained from the IEEE by emailing
<EMAIL_ADDRESS>This is the author’s version of an article that has
been published in the proceedings of 2023 International Conference on
Software, Telecommunications and Computer Networks (SoftCOM) and published by
IEEE. Changes were made to this version by the publisher prior to publication,
the final version of record is available at:
https://ieeexplore.ieee.org/document/10271668. To cite the paper use: Ł.
Kułacz, ”Spectrum Occupancy Detection Supported by Federated Learning,” 2023
International Conference on Software, Telecommunications and Computer Networks
(SoftCOM), Split, Croatia, 2023, pp. 1-3, doi:
10.23919/SoftCOM58365.2023.10271668. The presented work has been funded by the
National Science Centre in Poland within the project (no. 2021/41/N/ST7/01298)
of the PRELUDIUM program.

Ł. Kułacz  Institude of Radiocommunication
Poznan University of Technology
Poznan, Poland
<EMAIL_ADDRESS>
###### Abstract

Dynamic spectrum access is essential for radiocommunication and its limited
spectrum resources. The key element of dynamic spectrum access systems is
effective spectrum occupancy detection. In many cases, machine learning
algorithms improve detection effectiveness. Because of the recent trend of
using federated learning, a federated learning algorithm is presented in the
context of distributed spectrum occupancy detection. The results of the work
presented in the paper are based on actual signal samples collected in the
laboratory. The proposed algorithm is effective, especially in the context of
a set of sensors with faulty sensors.

###### Index Terms:

federated learning, machine learning, spectrum occupancy detection

##  I Introduction

Given the enormous pace of development of wireless technologies and the
problem of insufficient spectrum resources, dynamic spectrum access is a very
promising solution to meet the requirements for radiocommunication systems.
The possibility of using the spectrum for purposes other than those for which
it was originally intended (reserved) seems to be the easiest conceptual
solution. Unfortunately, for many reasons, this solution is not so
straightforward. First, a Dynamic Spectrum Access (DSA) system is required,
which will be able, using appropriate algorithms and data, to ensure access to
the spectrum for secondary (additional) users [1]. At the same time, care
should be taken to ensure that the primary user’s transmission is provided
with the best possible quality of service, whether through appropriate
separation of resources between users, control of the transmission power of
system users, or even temporary disabling the possibility of transmission for
unlicensed users. The critical component of the indicated system is access to
information about the presence of primary users’ transmissions. Once the
system has this information, it can appropriately protect the primary user’s
transmission. The primary source of information on spectrum occupancy is
spectrum sensing, a huge and separate part of the radiocommunication research
area. Many papers deal with spectrum occupancy detection algorithms in various
scenarios [2]. Machine learning algorithms are also successfully used to
improve the detection quality, which supports the detection process by
extracting essential, but not always noticeable, details of the transmission
[3]. An example of such details may be the periodicity of transmission,
different traffic volumes depending on the time of day or year, or the
characteristics of the transmission system used. It is worth noting one of the
significant features of wireless systems related to the use of machine
learning to detect spectrum occupancy - a very rare situation is the ability
to obtain verified information about whether a signal is transmitted in a
given place and time. This is the most critical aspect of supervised machine
learning and the collection of training data necessary for its operation. In
addition, it was noted that knowledge from a single sensor could often be
distorted due to the sensor’s specific location or its unreliability. To
eliminate this problem, many cooperative spectrum sensing algorithms have been
considered. They allow us to make better (more reliable) decisions in a
specific area; however, using machine learning algorithms is more complicated
due to the massive amount of data necessary to be transferred between the
sensors and the collecting node. This reduces the time available for the
actual transmission of user data and thus reduces the system’s efficiency,
ultimately aiming to send as much user data as possible. It should be
remembered that sensors should also be straightforward devices, which is an
additional challenge. Another critical problem is the transparency of the
transmitted data, i.e., the transmission of the collected measurement data by
sensors can be relatively easily intercepted and analyzed and, worse, replaced
[4]. One of the promising ideas for solving this problem, which has been
gaining popularity recently and is widely considered in many aspects and
applications, is federated learning [5].

##  II Federated Learning

Federated learning assumes the operation of individual sensors in a specific
area of the network, which collects data in their environment while training
their machine learning model. Then, the model coefficients of individual
sensors are exchanged, as opposed to the samples collected in cooperative
spectrum sensing, and an aggregated model is created (for example, in the
simplest Federated Averaging (FedAvg) scenario - created by averaging all
coefficients received from the sensors) [6]. Further, the collective model,
and its coefficients, are transferred to all sensors. In the simplest
scenario, the sensors use a collective model. Then, based on the newly
collected samples, they slightly adjust its coefficients or create a different
model resulting from the appropriate merge of the collective model and their
own model. The procedure described above is repeated many times. Consequently,
individual sensors have access to a collective model, which is at least
indirectly based on many more samples than those collected by a single sensor.
In addition, the amount of information transferred between the sensors is
limited to only the model coefficients, which depends on the complexity of the
machine learning model [7]. One benefit of using federated learning is the
ability to train a single-sensor model much faster in case of its replacement
or simply adding a new one. Therefore, the speed of adaptation of such a
system should be its advantage - especially in the context of the
aforementioned general challenge related to spectrum occupancy detection.
Similarly, the ability to detect a faulty sensor from ”significantly”
different reported model coefficients alone should be a valuable safeguard for
the system - increasing its reliability. Federated learning used in this way
cannot offer higher detection efficiency than cooperative spectrum occupancy
detection due to limited access to training data (only indirectly). However,
at the expense of slightly lower detection efficiency, it is possible to
significantly reduce the amount of sent control data and improve system
security [8].

##  III Data Collection

The following measurements were carried out to demonstrate the operation of
the spectrum occupancy detection system using federated learning. Two sets of
devices were used, each consisting of a Linux computer with GNU Radio software
(version 3.8) and a Universal Software Radio Peripheral (USRP) B210 device,
where one set acts as a transmitter and the other as a receiver of the test
radio signal. The transmitted signal was a random set of values modulated with
Gaussian Minimum Shift Keying (GMSK). The receiver recorded a specified number
of signal samples (IQ) for successively higher transmit signal gain settings.
The receiver collected 100 million IQ samples for noise and 10 million IQ
samples per transmit power level used by the transmitter. The center frequency
was set to 2.1 GHz and the bandwidth to 40 MHz. In addition, the receiver
collected reference data as noise samples when the transmitter was turned off.
The received signal was divided into every 10’000 samples from which Fast
Fourier Transform (FFT) was calculated. Then the average power received in
this channel was calculated for each channel and the kurtosis and skewness of
the autocorrelation function. In addition, a label has been adapted to each
set of these values depending on whether the transmitter is turned on during a
given measurement. The actual data collected in the described manner was then
used in a spectrum occupancy detection simulation using federated learning.
For simplicity, the current simulation results use data for only one frequency
channel (also, the tested signal was transmitted only in one of the analyzed
channels). The data presented in this study are openly available [9].

##  IV Simulation Setup and Reference Scenario

The first step in preparing data for machine learning purposes was to balance
the number of measurements for each label. Because after the measurements,
most of the data was collected for the signal’s turned-on transmission- some
measurements with significant transmit power were omitted. In this way, the
same amount of noise and signal data (with a transmit power close to the noise
power) were used to train the machine learning model. Since the experiment is
about federated learning, all the collected data was shuffled and divided into
five equal subsets - simulating that each of the five sensors contained some
stored information. This is a deliberate simplification to focus primarily on
analyzing the federated learning algorithm. The effectiveness of determining
the spectrum occupancy was calculated (for each sensor separately) based on
the classic energy detection algorithm (assuming a false alarm probability of
1%) to have a reference point for all obtained measurements. The average
efficiency of this algorithm is 93.18%, whereby efficiency means the
percentage of correctly determined spectrum occupancy values in the set of
test values (all collected data). The spectrum occupancy detection system’s
efficiency is measured by the accuracy of the ML model used in this process
(efficiency and accuracy are used alternately). Then, a simple machine
learning algorithm, logistic regression, was tested. In this case, the average
efficiency of spectrum occupancy determination was 98.83%. In addition, a
neural network was tested (with an input layer with four nodes, a hidden layer
with four nodes, and an output layer with one node), achieving an average of
99.04% efficiency. Data from each sensor was split into training (80%) and
testing (20%) datasets. Each of the mentioned algorithms has been tested for
many different parameters and validated using the stratified K-Fold cross-
validation algorithm; however, the effectiveness base on the entire dataset
(from the perspective of spectrum occupancy detection) has been shown to
standardize the results.

##  V Simulation Results

The conducted simulation of the operation of the federated learning algorithm
consisted in dividing the training data into equal batches and iteratively:
training single models (for each sensor separately - using only training data
for a single sensor), then creating a collective model based on the average
value of model coefficients and setting the calculated coefficients in the
models of each sensor. In parallel, copies of the models for each sensor were
retained (omitted from the coefficient exchange process) to reference the
scenario without using federated learning. First, federated learning using a
logistic regression model was tested. The average efficiency of the presented
federated learning algorithm was 94.51%, whereas the average efficiency of the
sensors (without federated learning) was 92.74%. It is worth explaining the
differences between the obtained values and the aforementioned efficiency of
the logistic regression algorithm itself (98.83%). This difference is since in
federated learning, training data enters the model gradually, and after
obtaining only a small part of the data, we try to estimate the effectiveness
of the model. In the reference scenario, however, all data is transferred to
the model and subjected to many iterations of the learning process. More
interesting results can be seen when one of the analyzed sensors is faulty and
reports random decisions about spectrum occupancy. In this situation, the
average efficiency of the presented federated learning algorithm was 94.51%,
whereas the average efficiency of the sensors (without federated learning) was
85.39%. With two faulty (in the same way) sensors, the average efficiency of
the presented federated learning algorithm was 94.17%, whereas the average
efficiency of the sensors (excluding federated learning) was 78.24%. It can be
noticed that using even such a simple algorithm as averaging the model
coefficients improves the system’s reliability when simple and unreliable
sensors are used. Fig. 1 shows the average accuracy of predictions on the
analyzed dataset in the scenario with Logistic Regression used in the FL
process.

Figure 1: Accuracy score for Logistic Regression machine learning model
utilizing all collected data

Then, federated learning using a neural network was tested. The average
efficiency of the presented federated learning algorithm was 96.46%, whereas
the average efficiency of the sensors (excluding federated learning) was
95.63%. As before, the difference with reference results is that training data
enters the model gradually in federated learning. After obtaining only a small
part of the data, we try to estimate the model’s effectiveness. In the
reference scenario, however, all data is transferred to the model and
subjected to many iterations of the learning process. Here, the scenario where
one of the analyzed sensors is faulty, and reports random decisions about
spectrum occupancy was checked again. In this situation, the average
efficiency of the presented federated learning algorithm was 96.25%, whereas
the average efficiency of the sensors (excluding federated learning) was
90.16%. With two broken (in the same way) sensors, the average efficiency of
the presented federated learning algorithm was 96.34%, whereas the average
efficiency of the sensors (excluding federated learning) was 77.79%. It can be
noticed that using even such a simple algorithm as averaging model
coefficients, also in the case of neural networks, improves system reliability
when simple and unreliable sensors are used. Fig. 2 shows the average accuracy
of predictions on the analyzed data set in the scenario with Logistic
Regression used in the Federated Learning process. However, using neural
networks in such a simple scenario requires exchanging a much larger number of
coefficients (41 instead of 4), and the model-learning process is
computationally more complex than logistic regression. It is also worth
emphasizing that it is possible to detect an incorrect model in both of the
analyzed situations based on how much its coefficients differ from those of
other models.

Figure 2: Accuracy score for Neural Network machine learning model utilizing
all collected data

##  VI Conclusions

The algorithms presented in the work show potential for using federated
learning in spectrum occupancy detection systems. In specific scenarios, they
provide increased reliability of detection and, more importantly, ensure
greater efficiency of decisions made compared to classic algorithms (such as
energy detection) and comparable efficiency compared to algorithms using
machine learning (compared to a model trained on a larger number of samples -
such as in cooperative spectrum occupancy detection). Further work on this
topic is required, mainly to investigate the influence of the environment
change on the created models, especially when received signal power in the
sensors will be close to the noise level. Moreover, simplification made in
this work, i.e., collection of samples only by one USRP device, should be
extended - as hardware imperfections, even minor differences in received
signal level, could be critical in federated learning.

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