# A trustless decentralized protocol for distributed consensus of public
quantum random numbers

Lac Nguyen  Center for Quantum Science and Engineering, Stevens Institute of
Technology, Hoboken, NJ 07030 U.S.A  Physics Department, Stevens Institute of
Technology, Hoboken, NJ 07030 U.S.A  QPhoton, Inc. 78 John Miller Way, Kearny,
NJ 07032  Jeevanandha Ramanathan  Center for Quantum Science and Engineering,
Stevens Institute of Technology, Hoboken, NJ 07030 U.S.A  Physics Department,
Stevens Institute of Technology, Hoboken, NJ 07030 U.S.A  QPhoton, Inc. 78
John Miller Way, Kearny, NJ 07032  Michelle Mei Wang  Center for Quantum
Science and Engineering, Stevens Institute of Technology, Hoboken, NJ 07030
U.S.A  Physics Department, Stevens Institute of Technology, Hoboken, NJ 07030
U.S.A  QPhoton, Inc. 78 John Miller Way, Kearny, NJ 07032  Yong Meng Sua
Center for Quantum Science and Engineering, Stevens Institute of Technology,
Hoboken, NJ 07030 U.S.A  Physics Department, Stevens Institute of Technology,
Hoboken, NJ 07030 U.S.A  QPhoton, Inc. 78 John Miller Way, Kearny, NJ 07032
Yuping Huang  Center for Quantum Science and Engineering, Stevens Institute of
Technology, Hoboken, NJ 07030 U.S.A  Physics Department, Stevens Institute of
Technology, Hoboken, NJ 07030 U.S.A  QPhoton, Inc. 78 John Miller Way, Kearny,
NJ 07032<EMAIL_ADDRESS>
###### Abstract

Quantum random number (QRNG) beacons distinguish themselves from classical
counterparts by providing intrinsic unpredictability originating from the
fundamental laws of quantum mechanics. Most demonstrations have focused on
certifiable randomness generators to guarantee the public that their
genuineness is independent from imperfect implementations. These efforts
however do not benefit applications where multiple distrusted users need a
common set of random numbers, as they must rely on the honesty of beacon
owners. In this paper, we formally introduce a design and proof-of-principle
experiment of the first consensus protocol producing QRNs in a decentralized
environment (dQRNG). Such protocol allows N number of participants contribute
in the generation process and publicly verify numbers they collect. Security
of the protocol is guaranteed given (N-1) dishonest participants. Our method
is thus suited for distribute systems that requires a bias-resistant, highly
secure, and public-verifiable random beacon.

††preprint: APS/123-QED

## Introduction

Quantum information science is fundamentally changing the way sensitive
information is distributed, shared, utilized, and perceived. While quantum key
distribution has been a global focus of research and development in this
field, there are many other areas where quantum physics principles can be
applied to solve previously prohibitive problems and create significant
impacts. Among them is the generation and distribution of shared resources
amongst distrusted parties over decentralized, unsecured networks.

Suppose Alice, Bob, Dave, and Charles, who are four validators in a
blockchain, need to choose one consensus leader among themselves to create the
next block in a secure and fair manner, without relying on any centralized
randomness source. Suppose a lottery service must prove to claimants that the
process of declaring winners is purely random and not controlled by any
central entity. The common ground of such scenarios is, in the absence of a
trusted third party, how a group of mutually distrusted participants can agree
on some genuine random choices for a public settlement among themselves [1].
This is the problem of constructing a random number generator in a
decentralized environment (dRNG) where RNs are not only unbiased,
unpredictable, tamper-resistant, but also publicly verifiable. Post-election
auditing, creating public parameters for cryptographic protocols, [2, 3],
gambling and lotteries services, and private preserved messages [4] are some
applications requiring dRNGs. In recent years, with the booming of blockchain
applications such as distributed computing platforms, smart contracts, and
Proof-of-Stake (PoS) based consensus algorithms [5, 6], dRNG has significantly
become more demanding. With the rapid rise in the field of quantum random
number generators (QRNGs), one might wonder why not exploit the fundamentally
unpredictable randomness from quantum process [7, 8, 9, 10] [11, 12], [13, 14]
to build a beacon for dRNG. In the past years, most QRNs research focused on
creating either a high speed or a self-tested, device-independent generator.
In the first scheme, QRNGs are built by fully characterized devices, offering
up to Gbits speeds, comparatively as high as classical pseudo-random number
generators (PRNGs), to accommodate numerical simulations or gaming service
applications [15, 16]. Meanwhile, the latter QRNGs type serves mostly
cryptography purposes by delivering rather slower speeds but self-tested
capability in which quantum randomness can be monitored and verified without
depending on any trusted physical implementations [17, 18, 19]. Nevertheless,
these demonstrations benefit only individual users that have physical access
to collect QRNs directly from their own devices. To solve the problem of
dRNGs, a , users must remotely receive numbers broadcasted from a centralized
source of quantum randomness. Because there is no way to verify if those
outcomes truly originated from quantum process or whether they are protected
in transit, the randomness of these numbers become subjective. Beacon users
have no choice but to trust the honesty of QRNG manufacturers, or rely on
verification from a third party and security of the datatransfer process.
These two considerable approach however do not solve the problem of dRNG. From
the point of view of dRNG users, a QRNG remains a centralized source of
randomness that they send request and receive digital RNs, therefore, it only
meets the technical trust of users but not the social trust. dRNGs users still
have to count on the honesty of some third party. For example, in 2018, NIST
built a QRNG beacon where randomness is certified by the impossibility of
superluminal signals. This brought RNG beacon to the next level where users no
longer worry of adversarial activities on measurement settings, measurement
devices, or randomness seeds. Nevertheless, the loophole Bell tests must be
isolated from hackers and customers [20], forcing public users to put trust in
the beacon owners. Clearly, a crucial element is missing in current QRNG
technology: not all or no RNs users participate in verifying the randomness
publicly. Building on these observations, we recognize an urgent need to
unlock more potential of quantum properties into solving these problems. In
this paper, we propose the first dQRNG protocol that, on-demand, produces one
or one set of random numbers used for decision-making on a matter involving N
parties. Our protocol is motivated from both quantum and classical
cryptography. We show that this novel decentralized quantum random number
generator (dQRNG) protocol is:
\- Genuine: no participant has control over the final outcomes.
\- Flexible in randomness distribution: without post-processing, QRNs follow
arbitrary probability distribution specified and agreed upon by all N parties.
\- Decentralized: anyone is able to participate in the QRN generation process.
\- Publicly verifiable: all N parties have the power to verify the genuineness
of the outcome RN(s).
\- Highly tolerant: outcomes are unpredictable if at least one party is honest
among N parties.
\- Quantum secured: immune to quantum computer attacks.

We provide a general framework of how to implement our protocol into current
classical and quantum network infrastructure. Along the way, we demonstrate
experimentally dQRNG with four participants and report the robustness of the
protocol by showing randomness testing results, probability distribution
verification, and fairness for users in QRNs generation process.

## Results

### Previous works

To our best knowledge, there is currently no quantum protocol for dRNG. Thus,
we would like to briefly explain how classical techniques generate RNs over a
decentralized network before presenting our quantum solution. Many classical
consensus protocols have been developed and implemented the past years [21,
22, 23]. A popular approach is producing RNs based on a commit-reveal scheme
[24]. In the commit phase, each participant submits the hash of their own
secret RN. In the reveal phase, all participants announce the RNs of their
choices so that others can compare with the encrypted RN they sent earlier.
The final RN is the outcome of a previously agreed upon function combining all
secret RNs from the entire pool. A challenge this method faces is that as each
participant publicizes their RNs, the last participant to reveal has an
advantage of knowing the resulting RN ahead of time. Therefore, the protocol
could fail because the last revealer either manipulates the output RN or
refuses to submit their secret RN, thus terminating the whole RN generation
process. To avoid this, RNs are fed into a verifiable delay function (VDF)
with the assumption that the malicious participant does not possess
specialized hardware capable of cracking the inherently slow hash computation
of the VDF [25]. Other directions for dRNG are built upon threshold digital
signature methods where RNs are obtained by amalgamating majority of random
signatures from participants [26]. These methods rely on public-private keys
scheme and require distributed keys generation process, hence, are vulnerable
against quantum computer attacks [27].

### Model

Figure 1: Decentralized quantum random number generator (dQRNG) architecture

All communication layers of dQRNG are implemented in parallel corresponding to
those in classical communication [28]. Figure 1 depicts quantum related layers
of dQRNG. Physical layers are identical to quantum network where there must be
an entanglement source, private quantum channels between communication
parties, and entanglement verification setup. Detailed schematic illustration
of dQRNG physical layers are described in figure 2a and 2b.

Figure 2: Details of necessary devices inside physical 1 (a) and 2 (b) layers
of dQRNG architecture. (c) explains the timing diagram of the synchronized
reference signal for all nodes. A detected photon carries an identification
(ID) corresponding to the closest reference pulse arriving before it. Figure
3: Diagram of the dQRNG protocol procedure involving four parties, Alice, Bob,
Charles, and Dave.

We apply a model that has been reported in many previous studies as a solution
for large-scale quantum networks, in which entangled photons are passed
through a beamsplitter of N, distributing to N nodes such that each party
shares photon pairs with every other party (total of
$P_{num}=\frac{N(N-1)}{2}$ number of possible channel pairs) [29]. Each
communication node must have local measurement and data processing devices
such as single photon detectors and classical information processors.
Transport layer in classical communication is co-used in this protocol for
transmission of QRNs between nodes. Unlike quantum network architecture, our
protocol can be built without the assumption of secured authentication.
Consensus algorithm of dQRNG is operated on classical application layer. All
parties in the communication pool are synchronized to a common reference
signal of period $T$ to keep track order of their detected photon state. Such
a reference can be supplied through the same optical communication channels to
distribute the entangled photons (see figure 2b and 2c).

### Construction

We first introduce the notations necessary for reading this protocol. Let $N$
be the number of participants, and $k$ be $(N-1)$. This pool would like to
agree on a list of random numbers $R$ of length $l$ within a bounded range
$B$. Depending on the application, parameter $\mu$, representing the
probability of being generated of a random numbers, is decided publicly in
advance [30, 31]. Parameter ${I}$ indicates reference identification number or
pulse index representing the time slot number of a detected photon.
$R\textsuperscript{c}$ is a list of classical true random numbers generated by
participants’ choices, $R\textsuperscript{q}$ is a list of QRNs generated by
quantum process, and $R\textsuperscript{qc}$ is a list representing RNs
produced from the dQRNG protocol. $F$ is a function, such as $XOR$ operation
or summation operation, agreed ahead of time by participants such that
$R=F(R\textsuperscript{qc})$. We assume the existence of a quantum network
infrastructure, as described in the previous section, connecting all
communication nodes. All participants publicly agree upon a common photon
measurement basis. We do not assume any necessary authentication between nodes
nor existence of any encryption and decryption, given quantum or classical
method, for data transferring during the protocol operation. A set of random
numbers $R$ is generated in five steps:

Reveal $\&$ Consensus 1. Each participant generates and publicizes their own
list of random values of length $m$ ($m>>l$) called
$R\textsuperscript{c}_{i}={\\{r\textsuperscript{c}_{i,m}\\}}$ and probability
weight $\mu_{i}$ ($i$ indicates participant such as Alice (A), Bob (B),
Charles(C), Dave(D), etc; this parameter is optional)
$r\textsuperscript{c}_{i,m}\in N$, $r\textsuperscript{c}_{i,m}\leq B$. List of
all $R\textsuperscript{c}_{i}$ are combined to create $R\textsuperscript{c}$ =
$R\textsuperscript{c}_{A}\cup R\textsuperscript{c}_{B}\cup
R\textsuperscript{c}_{C}\cup R\textsuperscript{c}_{D}...\cup
R\textsuperscript{c}_{N}=\\{r\textsuperscript{c}_{m_{c}}\\}$, where
$m_{c}=m\times N$. A probability density function (pdf) $C$ is constructed by
combining probability weight $\mu_{i}$.

Quantum measurements. Entangled photon pairs are sent to all participants.
Each participant then performs photon measurement privately to obtain their
collapsed photon states
$R\textsuperscript{q}_{i}=\\{\ket{r\textsuperscript{q}_{i,m\prime_{i}}}\\}$
with corresponding indices $I_{i}=\\{ind_{i,m\prime_{i}}\\}$ ($m\prime$ is the
number of detected photons. In practice, length $m\prime$ is different for
each node depending on the loss of the quantum channel and measurement noise,
$m\prime_{i}>>m>>l$).

Quantum results reveal. Measurement results of photon states with
corresponding indices are publicly revealed by each participant. Quantum
entanglement verification is performed by extracting detected photon results
previous step from $P_{num}$ number of channel pairs,
$R\textsuperscript{q}_{ij}=R\textsuperscript{q}_{i}\cap
R\textsuperscript{q}_{j}=\\{\ket{r\textsuperscript{q}_{i,m\prime_{i}}}\\}\cap\\{\ket{r\textsuperscript{q}_{j,m\prime_{j}}}\\}=\\{\ket{r\textsuperscript{q}_{ij,m\prime_{ij}}}\\}$
with corresponding list of indices $I_{ij}=I_{i}\cap I_{j}$. In the ideal
case, $m\prime_{ij}=m\prime_{i}=m\prime_{j}$, however,
$m\prime_{ij}<m\prime_{i}$ and $m\prime_{j}$ due to quantum channel loss and
measurement noise. For example, for pair Alice and Bob, correlated list of
$R\textsuperscript{q}_{AB}=R\textsuperscript{q}_{A}\cap
R\textsuperscript{q}_{B}=\\{\ket{r\textsuperscript{q}_{A,m\prime_{A}}}\\}\cap\\{\ket{r\textsuperscript{q}_{B,m\prime_{B}}}\\}$
with corresponding list of indices $I_{AB}=I_{A}\cap
I_{B}=\\{ind_{A,m\prime}\\}\cap\\{ind_{B,m\prime}\\}=\\{ind_{AB,m\prime_{AB}}\\}$.
Similarly, using the same measurement results, probability distribution
verification is performed to compare with pdf $C$.

Consensus 2. A list of QRNs is obtained by appending lists of correlated
photon measurement results $R\textsuperscript{q}_{ij}$ from all $P_{num}$
channel pairs, forming $R\textsuperscript{q}=R\textsuperscript{q}_{AB}\cup
R\textsuperscript{q}_{AC}\cup R\textsuperscript{q}_{AD}\cup
R\textsuperscript{q}_{BC}...=\\{\ket{r\textsuperscript{q}_{m\prime\prime}}\\}$,
for all $P_{num}$ and $I=I_{AB}\cup I_{AC}\cup I_{AD}\cup
I_{BC}...=\\{ind_{m\prime\prime}\\}$, for all $P_{num}$ where
$m^{\prime\prime}=\sum_{P_{num}}m\prime_{ij}$.
Quantum randomness is combined with users’ randomness by first publicly
perform $I\cap R\textsuperscript{c}$ to assemble a new list of indices called
$I_{final}=\\{ind_{m\prime\prime}\\}\cap\\{r\textsuperscript{c}_{m_{c}}\\}$.
Then, the final list of random numbers is filtered so that only values with
corresponding indices in $I_{final}$ remain, called
$R_{qc}=\\{r\textsuperscript{qc}_{m_{c}}\\}$. It can be seen that $R_{qc}$ are
selected as subset of QRNs harvested in the quantum process by consensus
happened before quantum measurements were operated, and therefore, still are
QRNs.

Computing output. To obtain the list of $R$ random numbers of length $l$ as
desired, an aggregation function $F$ (e.g modular addition) is used to
calculate publicly and can be confirmed by any participants if needed,
$R=F(\\{r\textsuperscript{qc}_{m_{c}}\\})$.

### Analysis

Figure 4: Experimental setup to demonstrate four parties performing random
number generation consensus

Correctness: For as long as at most $k$ participants are malicious, meaning at
least 1 participants follow the protocol honestly, the results will still be
unbiased and unpredictable, the protocol will be executed and all participants
will agree upon the final result when the protocol is completed.
Proof. Assume the case of four participants Alice, Bob, Charlie, and Dave
where they need to agree on one truly random number ($l=1$) in the range from
0 to 6, given each of number is equally likely to be selected. Alice is the
only one being honest ($k=3$). We proceed the dQRNG protocol to show that the
final result is still random and incalculable.
Reveal $\&$ Consensus 1. Each participant Alice (A), Bob (B), Charles(C),
Dave(D) generates and publicizes their own list of random values of length
$m$, let’s make $m=10$:

$\displaystyle
R\textsuperscript{c}_{A}={\\{r\textsuperscript{c}_{A,1\rightarrow{10}}\\}},R\textsuperscript{c}_{B}={\\{r\textsuperscript{c}_{B,1\rightarrow{10}}\\}},$
$\displaystyle
R\textsuperscript{c}_{C}={\\{r\textsuperscript{c}_{C,1\rightarrow{10}}\\}},R\textsuperscript{c}_{D}={\\{r\textsuperscript{c}_{D,1\rightarrow{10}}\\}},$
$\displaystyle R\textsuperscript{c}=R\textsuperscript{c}_{A}\cup
R\textsuperscript{c}_{B}\cup R\textsuperscript{c}_{C}\cup
R\textsuperscript{c}_{D}=\\{r\textsuperscript{c}_{1\rightarrow{40}}\\},m_{c}=40$
$\displaystyle C=1$

Quantum measurements.

$\displaystyle
R\textsuperscript{q}_{A}={\\{r\textsuperscript{q}_{A,m\prime_{A}}}\\},R\textsuperscript{q}_{B}={\\{r\textsuperscript{q}_{B,m\prime_{B}}}\\},$
$\displaystyle
R\textsuperscript{q}_{C}={\\{r\textsuperscript{q}_{C,m\prime_{C}}}\\},R\textsuperscript{q}_{D}={\\{r\textsuperscript{q}_{D,m\prime_{D}}}\\},$
$\displaystyle(m\prime_{i}>>10>>1).$

Quantum results reveal. Alice reveals her results honestly while Bob, Charles,
and Dave do not. The three of them might cheat without knowing there are other
cheaters in the pool. They could alter their results partially or entirely
with artificial data. This method, however, does not favor the cheaters in
anyway. Quantum entanglement verification easily displays them because their
announced results must be compared with each others and with Alice.
In another possible scenario, all three cheaters might consult each other and
assemble set of data such that they can pass the quantum entanglement
verification with each other in public. At the same time, they still embedded
some honest results to verify with Alice. In such situation,

$\displaystyle R\textsuperscript{q}_{AB}=R\textsuperscript{q}_{A}\cap
R\textsuperscript{q}_{B}(\text{non-deterministic}),$ $\displaystyle
R\textsuperscript{q}_{AC}=R\textsuperscript{q}_{A}\cap
R\textsuperscript{q}_{C}(\text{non-deterministic}),$ $\displaystyle
R\textsuperscript{q}_{AD}=R\textsuperscript{q}_{A}\cap
R\textsuperscript{q}_{D}(\text{non-deterministic}),$ $\displaystyle
R\textsuperscript{q}_{BC}=R\textsuperscript{q}_{B}\cap
R\textsuperscript{q}_{C}(\text{deterministic}),$ $\displaystyle
R\textsuperscript{q}_{BD}=R\textsuperscript{q}_{B}\cap
R\textsuperscript{q}_{D}(\text{deterministic}),$ $\displaystyle
R\textsuperscript{q}_{CD}=R\textsuperscript{q}_{C}\cap
R\textsuperscript{q}_{D}(\text{deterministic}),$

with corresponding indices behave in the same manner $I_{AB},I_{AC},I_{AD}$
are non-deterministic; $I_{BC},I_{BD},I_{CD}$ are deterministic.
Consensus 2. $R\textsuperscript{q}=R\textsuperscript{q}_{AB}\cup
R\textsuperscript{q}_{AC}\cup R\textsuperscript{q}_{AD}\cup
R\textsuperscript{q}_{BC}\cup R\textsuperscript{q}_{BD}\cup
R\textsuperscript{q}_{CD}$ and therefore, it’s a combination of random numbers
extracted from quantum process and falsified ones made up by Bob, Charles, and
Dave. Likewise, $I=I_{AB}\cup I_{AC}\cup I_{AD}\cup I_{BC}\cup I_{BD}\cup
I_{CD}$ is a combination of random indices and fabricate ones. Assume in the
first step,
$R\textsuperscript{c}_{B},R\textsuperscript{c}_{C},R\textsuperscript{c}_{D}$
are agreed and arranged by Bob, Charles, and Dave before they proclaim,
$R\textsuperscript{c}$ still carries private randomness comes from Alice.
Thus, the outcome indices of $I_{final}=I\cap R\textsuperscript{c}$ remains
unpredictable. By post-selecting only numbers with indices $I_{final}$ in
$R\textsuperscript{q}$ list, $R_{qc}=\\{r\textsuperscript{qc}\\}$ is composed
from mixture of quantum random numbers and arbitrary subset of predetermined
numbers.
Computing output. As a consequence, computing
$R=F(\\{r\textsuperscript{qc}\\})$ returns an unpredictable number.

Now let’s examine some common attacks on quantum networks and how dQRNG
protocols are immune against them.

Digital emulation attack. The quantum results reveal step occurs over the
classical network, it is vulnerable to digital emulation attack. Without the
access to the quantum channel, an adversary can pretend to be one of the
participant because the consensus happens in the application layer of the
classical channel. Observing others revealing their results first, he then
fabricates
$R\textsuperscript{q}_{i}=\\{\ket{r\textsuperscript{q}_{i,m\prime_{i}}}\\}$
with corresponding indices $I_{i}=\\{ind_{i,m\prime_{i}}\\}$ to emulate that
he has matching measurement results with every other participants. This
malicious action is rather meaningless because it neither helps him know the
final random numbers result before hand nor manipulate them.

Man-in-the-middle attack (MITM). Since we do not assume any secure quantum
authentication requirement, an outside adversary can perform MITM attack and
take the identity of that node. In this case, the adversary gains full access
to quantum measurement results and has the choice to publicize any type of
results he wants. His dishonest reveal, if he chooses to do so, does not
affect the unpredictability of the final results as proved previously.
Participants do not hide their quantum measurement results and corresponding
indices, therefore, knowing these information ahead of time does not give the
adversary any advantages either.

### Results

Case | 1 | 2
---|---|---
N | 4 | 4
R | 8-bit (binary) value | Number represents A, B, C or D
B | [1,256] | [0,3]
l | 1 | 1
$\mu_{i}$ | N/A | $\mu_{A}=\mu_{B}=\mu_{C}=\mu_{D}=0.25$
$R\textsuperscript{c}_{i}$ | generated by PRNG | generated by PRNG
m | 8388607 | 8388607
$R\textsuperscript{q}_{i}$ | collected from | collected from
| quantum measurements | quantum measurements
Table 1: For this typical example, users A, B, C, and D received
$m\prime_{A}=1029054$, $m\prime_{B}=964265$, $m\prime_{C}=952180$, and
$m\prime_{D}=1019553$ photons respectively. This parameter will vary each time
the protocol is run. Other parameters that will vary are user-input random
number list $R_{c}$, matching pairs $R^{q}_{ij}$ with indices $I_{ij}$ where
$i,j=A,B,C,D$ Figure 5: Case 1: dQRNG protocol is used when four parties must
together repeatedly generate an 8-bit random number after a certain time
interval for cryptography purposes. The protocol is run 100 times with various
amount of 8-bit random numbers produced each time. Typical results of the
probability that an 8-bit random number (total 256 possible values) is
generated by dQRNG protocol with four participants is recorded. Comparing to
the theoretical probability of $\frac{1}{256}=0.0039$, experimental results
fluctuate between 0.0032 and 0.0047 (this gives errors less than $20.5\%$).
Figure 6: Case 2: dQRNG protocol is used when four parties must together
select a winner among themselves for some voting protocol purposes. The
protocol is run 100 times with one party is chosen each time. Comparing to the
theoretical probability of 0.25 experimental results only fluctuate less than
$5.28\%$ showing the protocol is unbiased.

Here we realize the dQRNG protocol in setups purposing for different
applications. Table 1 records parameters used and measured in each experiment.
The first type of scenario occurs when a pool of parties repeatedly shares a
new random string for cryptography protocols such as private messages or
multi-party computation[32]. In particular, we create a case where four
parties must generate an 8-bit string periodically. We use pseudo-random
number generators (PRNG) to simulate choices of $R\textsuperscript{c}_{i}$
list from each party. The rest of the dQRNG steps are executed following the
raw data collected from the experiment whose details are explained in section
Methods. To examine the uniformity of the dQRNG, we run protocol 100 times
with multiple random numbers string are produced each time. Figure
LABEL:Fig:dQRNG_simulation_uniform_case1 indicates the average probability of
generating a certain 8-bit value calculated from 100 experiment runs comparing
to the theoretically probability. The second type of dQRNG use case happens in
scenario requiring a random fair voting mechanism. Specific examples could be
a winner must be decided in a lottery or a validator must be selected to
create the next block in a blockchain protocol. In our experimental
demonstration, a pool of four parties must decide who is chosen with the
condition that each party equally likely has chance of being chosen
($\mu_{i}=0.25$). Similar to case 1, PRNGs are used to generate list
$R\textsuperscript{c}_{i}$ in step 1 of dQRNG while quantum measurement
outcomes are used for rest of the steps. We run our protocol 100 times and
recorded the average probability results in Figure
LABEL:Fig:dQRNG_simulation_uniform_case2.

(a)

(b)

(c)

Figure 7: Consider the independent timing jitters of the SPD (33 ps) and the
TDC (20 ps) at each communication node, photon arrival times collected are
rounded to 250 ps per bin-width to compensate the overall timing jitter of all
nodes in the network. Joint photon arrival time plots which present correlated
QRNs of a typical channel pair are showed in (a), (b), and (c) following
uniform, gaussian, and rayleigh probability distributions.

Figure 8: Left figure: Temporal cross-correlation between photon arrival times
of each channel pair in different probability distributions. From left to
right are uniform, Gaussian, and Rayleigh distributions. Total coincidence
photon pairs are collected over 60s interval delaying within 500 ps around the
maximum peak. Right figure: Accidental coincidence counts with photon arrival
times rounded to 250 ps binwidth

## Discussion

The invention of distributed applications and blockchain have promised to
bring security, fairness, and transparency to digital technologies, business
solutions, as well as socioeconomic and political structure by enabling
consensus decision in a global scale. Nonetheless, this relies on dRNG and
heavily depends on keeping it conspiracy-resistant using current classical
communication techniques. It will be deleterious when a fast enough quantum
computer breaks the current dRNG and from there, vandalizes this
decentralized-everything idea completely. One might argue, a natural approach
to quantum proof dRNG could be applying quantum cryptography protocols such as
quantum keys distribution, quantum secret sharing, or quantum digital
signatures to protect data transfer or signatures encryption [33, 34].
However, in trying to make a scalable and applicable dRNG protocol, we create
and present a quantum version that is simple, lightweight. Our dQRNG avoids
the cumbersome procedures in quantum cryptography including keys distillation,
quantum error correction, or slow key rates, but still holds the quantum
advance in being provably random and information-theoretic secure. At the same
time, comparing to classical domain, our protocol operates without colluding
given only one honest party while requiring no encryption algorithm, and thus,
reduces the communication complexity between parties. Furthermore, with the
arbitrary probability distribution QRNs generation feature, our protocol
outperforms others by avoiding an extra step of complex mathematics and not
wasting random numbers when transforming QRNs from uniform distribution to
others.

## Methods

We realize our dQRNG protocol by constructing a simple experimental model
comprised of four participants depicted in Figure 4. We first apply the same
technique of generating QRNs with arbitrary probability distribution as we
introduced in our previous work to create programmable QRN[35]. The difference
is that the dQRNG protocol requires broadcasting QRNs to all participants,
thus an entangled photons generation and distribution process must follow
afterwards. The procedure is described as follow: first, intensity of an
1559.67 nm continuous wave (CW) laser is modulated using an electro-optical
modulator (EOM) before coupled into a PPLN waveguide for second-harmonic
generation (SHG) at 779.8 nm ; next, the SHG light is coupled into another
identical PPLN waveguide for generating entangled photon pairs via
spontaneous-parametric-down-conversion (SPDC) process; and lastly, a system of
three $50/50$ fiber beamsplitters are used to randomly transmit photons into
four different nodes. Note that quantum entanglement verification is an
essential procedure for dQRNG protocol. SPDC is ubiquitous for many
entanglement-based quantum networks demonstrated thus far. At each node,
photons are passed through fiber polarization controller (FPC) before detected
by superconducting nanowire single photon detector (SNSPD). In this
experiment, using a time-to-digital converter (TDC), we record the photon
arrival times compared to a network synchronized 10 MHz reference signal thus
harvest the high dimensional temporal mode of each single photon detection.
Moreover, the dQRNG protocol is scalable to multiple degrees of freedom in
photonics states (ie. polarization, spectral, and spatial modes ) for photon
efficient generation of QRNs. Data are then transferred to a local machine
which is connected to others in the classical network to perform consensus
tasks in the application layer.

In the experimental demonstration of our dQRNG protocol, three different pump
shapes including CW, Gaussian, and Rayleigh pulses are sent into the SPDC
source, showing the versatility in probability distribution of our QRNs.
Correlated QRNs of each channel pair $R\textsuperscript{q}_{ij}$ must satisfy
the designated distribution to pass the quantum results reveal step. Figure 5
displays joint QRNs between a typical channel pair when photon arrival times
are rounded to 250 ps binwidth. Figure 6 (left) describes the time resolved
coincidence peaks of all available channel pairs AB, AC, AD, BC, BD, and CD
with corresponding CW, Rayleigh, and Gaussian SPDC pump pulses, respectively.
Discrepancies in coincidence counts between channel pairs are due to the
imperfection of beamsplitters as well as contributions by different detection
efficiency and timing jitters of each SPD channel. Using one SPDC source to
distribute photon pairs to the entire network reduces the coincidence count
rate or correlated QRNs rate. The dQRNG produces one or one small set of RNs
at a time, thus, even though the average $R\textsuperscript{q}_{ij}$ rate we
achieve is rather low, it does not affect the performance of the protocol.
With the same time bin’s width, coincidence to accidental coincidence ratios
(CARs) of all possible channel pairs is described in Figure 8b. Security
threshold of dQRNG protocol depends on CAR to decide when the SPDC process
should end. For example, assuming CAR threshold is 50 for gaussian pump shape,
photon pairs generation continue until all channel pairs achieve this value.
From our experiment data, this means Alice, Bob, Charles, and Dave each has
length $m\prime_{A}=,m\prime_{B}=,m\prime_{C}=,m\prime_{D}=$, respectively. It
can be seen that $m\prime_{i}>>l=1$ in this case. Lastly, following consensus
2 step, we append all $R\textsuperscript{q}_{ij}$ into one list
$R\textsuperscript{q}$ and examine this sequence using National Institute of
Standards and Technology (NIST) SP 800-22 Statistical Test Suite [36]. Results
of a typical QRNs sequence in uniform distribution is presented in figure 9,
showing our QRNs pass these test suites with high confidence level.

Figure 9: NIST results of typical 63182 correlated QRNs of uniform probability
distribution. QRNs are harvested from 250 ps binwidth photon arrival times and
then QRNs are converted into 8-bit representation of 505456 bits and inputted
into the NIST test suite. P-values greater than significance level 0.01
certifies this sequence passes the randomness threshold rigorously for
standard cryptography purposes.

## References

  * Wasserman [1996] D. Wasserman, Let them eat chances: Probability and distributive justice, Economics and Philosophy 12 (1996).
  * Lenstra and Wesolowski [2015] A. K. Lenstra and B. Wesolowski, A random zoo: sloth, unicorn, and trx, Cryptology ePrint Archive, Report 2015/366 (2015), https://eprint.iacr.org/2015/366.
  * Baignères  _et al._ [2015] T. Baignères, C. Delerablée, M. Finiasz, L. Goubin, T. Lepoint, and M. Rivain, Trap me if you can – million dollar curve, Cryptology ePrint Archive, Report 2015/1249 (2015), https://eprint.iacr.org/2015/1249.
  * van den Hooff  _et al._ [2015] J. van den Hooff, D. Lazar, N. Zeldovich, and M. A. Zaharia,  _Vuvuzela: scalable private messaging resistant to traffic analysis_ (Association for Computing Machinery (ACM), 2015) pp. 137–152.
  * Buterin [2014] V. Buterin,  _A next generation smart contract and decentralized application platform_, Tech. Rep. (2014).
  * Gazi1  _et al._ [2018] P. Gazi1, A. Kiayias, and D. Zindr,  _Proof-of-Stake Sidechains_, Tech. Rep. (2018).
  * Gabriel  _et al._ [2010] C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, A generator for unique quantum random numbers based on vacuum states, Nat Photon 4, 711 (2010).
  * Shi  _et al._ [2016] Y. Shi, B. Chng, and C. Kurtsiefer, Random numbers from vacuum fluctuations, Applied Physics Letters 109, 041101 (2016).
  * Samsonov  _et al._ [2020] E. O. Samsonov, B. E. Pervushin, A. E. Ivanova, A. A. Santev, V. I. Egorov, S. M. Kynev, and A. V. Gleim, Vacuum-based quantum random number generator using multi-mode coherent states (2020), 2004.06552 [quant-ph] .
  * Zhou  _et al._ [2019] Q. Zhou, R. Valivarthi, C. John, and W. Tittel, Practical quantum random-number generation based on sampling vacuum fluctuations, Quantum Engineering 1, e8 (2019), https://onlinelibrary.wiley.com/doi/pdf/10.1002/que2.8 .
  * Qi  _et al._ [2010] B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, High-speed quantum random number generation by measuring phase noise of a single-mode laser, Opt. Lett. 35, 312 (2010).
  * Guo  _et al._ [2010] H. Guo, W. Tang, Y. Liu, and W. Wei, Truly random number generation based on measurement of phase noise of a laser, Phys. Rev. E 81, 051137 (2010).
  * Nie  _et al._ [2014] Y.-Q. Nie, H.-F. Zhang, Z. Zhang, J. Wang, X. Ma, J. Zhang, and J.-W. Pan, Practical and fast quantum random number generation based on photon arrival time relative to external reference, Applied Physics Letters 104, 051110 (2014), http://dx.doi.org/10.1063/1.4863224 .
  * Xu  _et al._ [2016] F. Xu, J. H. Shapiro, and F. N. C. Wong, Experimental fast quantum random number generation using high-dimensional entanglement with entropy monitoring, Optica 3, 1266 (2016).
  * Z  _et al._ [2019] Z. Z, Z. Y, H. W, Y. S, and G. H, 6 gbps real-time optical quantum random number generator based on vacuum fluctuation, Rev Sci Instrum (2019).
  * Kim [2021] K. Kim, Massively parallel ultrafast random bit generation with a chip-scale laser, Science 371, https://doi.org/10.1126/science.abc2666 (2021).
  * Lunghi  _et al._ [2015] T. Lunghi, J. B. Brask, C. C. W. Lim, Q. Lavigne, J. Bowles, A. Martin, H. Zbinden, and N. Brunner, Self-testing quantum random number generator, Phys. Rev. Lett. 114, 150501 (2015).
  * Avesani  _et al._ [2018] M. Avesani, D. G. Marangon, G. Vallone, and P. Villoresi, Source-device-independent heterodyne-based quantum random number generator at 17 gbps, Nature Communications 10.1038/s41467-018-07585-0 (2018).
  * Y  _et al._ [2018] L. Y, Z. Q, L. M.H, G. J.Y, Z. Y, Bai.B, Zhang.W, Liu.W.Z, Wu.C, Yuan.X, L. H, Munro.W.J, W. Z, You.L, Zhang.J, Ma.X, Fan.J, Zhang.Q, and Pan.J.W, Device-independent quantum random-number generation, Nature 10.1038/s41586-018-0559-3 (2018).
  * Bierhorst  _et al._ [2018] P. Bierhorst, E. Knill, S. Glancy, Y. Zhang, A. Mink, S. Jordan, A. Rommal, Y.-K. Liu, B. Christensen, S. W. Nam, M. J. Stevens, and L. K. Shalm, Experimentally generated randomness certified by the impossibility of superluminal signals, Nature 556, 10.1038/s41586-018-0019-0 (2018).
  * Schindler  _et al._ [2020] P. Schindler, A. Judmayer, N. Stifter, and E. Weippl, Hydrand: Efficient continuous distributed randomness, in  _2020 IEEE Symposium on Security and Privacy (SP)_ (2020) pp. 73–89.
  * Nguyen Van  _et al._ [2019] T. Nguyen Van, T.-D. Le, T. Nguyen-Anh, M.-P. Nguyen, T. Nguyen-Van, M.-Q. Le-Tran, Q. Le, H. Pham, and K. Nguyen-An, A system for scalable decentralized random number generation (2019).
  * Syta  _et al._ [2016] E. Syta, P. Jovanovic, E. K. Kogias, N. Gailly, L. Gasser, I. Khoffi, M. J. Fischer, and B. Ford, Scalable bias-resistant distributed randomness, Cryptology ePrint Archive, Report 2016/1067 (2016), https://eprint.iacr.org/2016/1067.
  * RAN [2017] _Randao: Verifiable Random Number Generation_ , Tech. Rep. (2017).
  * Boneh  _et al._ [2019] D. Boneh, J. Bonneau, B. Bunz, and B. Fisch, Verifiable delay functions, Cryptology ePrint Archive, Report 2018/601 (2019), https://eprint.iacr.org/2018/601.pdf.
  * Skidanov [2019] A. Skidanov,  _Near Protocol Randomness Beacon_, Tech. Rep. (2019).
  * Libert  _et al._ [2014] B. Libert, M. Joye, and M. Yung, Born and raised distributively: Fully distributed non-interactive adaptively-secure threshold signatures with short shares, Proceedings of the Annual ACM Symposium on Principles of Distributed Computing 645 (2014).
  * Cerf and Cain [1983] V. Cerf and E. Cain, The dod internet architecture model, Comput. Networks 7, 307 (1983).
  * Liu  _et al._ [2020] X. Liu, X. Yao, R. Xue, H. Wang, H. Li, Z. Wang, L. You, X. Feng, F. Liu, K. Cui, Y. Huang, and W. Zhang, An entanglement-based quantum network based on symmetric dispersive optics quantum key distribution, APL Photonics 5, 076104 (2020), https://doi.org/10.1063/5.0002595 .
  * Nguyen  _et al._ [2019] C. T. Nguyen, D. T. Hoang, D. Nguyen, D. Niyato, H. Nguyen, and E. Dutkiewicz, Proof-of-stake consensus mechanisms for future blockchain networks: Fundamentals, applications and opportunities, IEEE Access 7, 85727 (2019).
  * Saleh [2020] F. Saleh, Blockchain without Waste: Proof-of-Stake, The Review of Financial Studies 34, 1156 (2020), https://academic.oup.com/rfs/article-pdf/34/3/1156/36264599/hhaa075_supplementary_data.pdf .
  * Baum  _et al._ [2014] C. Baum, I. Damgard, and C. Orlandi, Conference on Security and Cryptography for Networks 8642, 175 (2014).
  * Hillery  _et al._ [1999] M. Hillery, V. Bužek, and A. Berthiaume, Quantum secret sharing, Physical Review A 59, 1829–1834 (1999).
  * Collins  _et al._ [2017] R. Collins, R. Amiri, M. Fujiwara, and et al., Experimental demonstration of quantum digital signatures over 43 db channel loss using differential phase shift quantum key distribution, Scientific Report 7, https://doi.org/10.1038/s41598-017-03401-9 (2017).
  * Nguyen  _et al._ [2018] L. Nguyen, P. Rehain, Y. M. Sua, and Y.-P. Huang, Programmable quantum random number generator without postprocessing, Opt. Lett. 43, 631 (2018).
  * Rukhin  _et al._ [2001] A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, M. Levenson, M. Vangel, D. Banks, N. Heckert, J. Dray, and S. Vo, A statistical test suite for random and pseudorandom number generators for cryptographic applications (2001).
  * Coates  _et al._ [1988] R. F. W. Coates, G. J. Janacek, and K. V. Lever, Monte carlo simulation and random number generation, IEEE Journal on Selected Areas in Communications 6, 58 (1988).
  * James and Moneta [2020] F. James and L. Moneta, Review of high-quality random number generators, Computing and Software for Big Science 4, 10.1007/s41781-019-0034-3 (2020).
  * Rabin [1983] M. O. Rabin, Transaction protection by beacons, Journal of Computer and System Sciences 27, 256 (1983).
  * [40] Anu quantum random numbers.
  * [41] Random.org true random numbers service.


$\widetilde{\mathcal{F}}$, and $c$ are as in Corollary 6.5.

Let $c_{r}^{\alpha,\alpha}$ and $c_{r}^{\alpha,\beta}$ be appropriate
modifications of $\widetilde{c}_{r}^{\alpha,\alpha}$ and
$\widetilde{c}_{r}^{\alpha,\beta}$ as in [LLZ14, §7.3] so that they satisfy
the Euler system norm relations for ${\mathrm{Ad}}(V_{f})(\psi+j)$, i.e. for
$r\ell\in\mathcal{R}_{\psi}$ we have

$\textup{cor}_{{\mathbf{Q}}(r\ell)/{\mathbf{Q}}(r)}\left(c_{r\ell}^{\alpha,\mu}\right)=\widetilde{P}_{\ell}(\ell^{j-1}{\rm
Frob}_{\ell}^{-1})\cdot d_{r}^{\alpha,\mu},$

where $\widetilde{P}_{\ell}(X)$ is the Euler factor of
${\mathrm{Ad}}(V_{f}^{*})(\psi^{-1})$.

Now we need to modify $c_{r}^{\alpha,\mu}$ so that they satisfy the Euler
system norm relations for ${{\rm Ad}^{0}}(V_{f})(\psi+j)$. Let
$m=r\ell\in\mathcal{R}_{\psi}$. Note that the Euler factor
$1-\ell^{j-1}\psi^{-1}(\ell)\sigma_{\ell}^{-1}$ for $\psi^{-1}+1-j$ is
invertible in
${{\mathbf{Z}}_{p}}[[\mathrm{Gal}({\mathbf{Q}}(m){\mathbf{Q}}_{\rm
cyc}/{\mathbf{Q}})]]$ since $\ell\equiv 1\ ({\rm mod}\ p)$, where
$\sigma_{\ell}\in\mathrm{Gal}({\mathbf{Q}}(m){\mathbf{Q}}_{\rm
cyc}/{\mathbf{Q}})$ is the unique element acting trivially on
${\mathbf{Q}}(\ell)$ and maps to ${\rm
Frob}_{\ell}\in\mathrm{Gal}({\mathbf{Q}}(r){\mathbf{Q}}_{\rm
cyc}/{\mathbf{Q}})$. Hence for any $m\in\mathcal{R}_{\psi}$ let

$d_{m}^{\alpha,\mu}\mathrel{\mathop{\mathchar 58\relax}}=\prod_{\ell\mid
m}(1-\ell^{1-j}\psi^{-1}(\ell)\sigma_{\ell}^{-1})^{-1}c_{m}^{\alpha,\mu}.$

This proves (ii), and (i) holds true due to [LZ16, Theorem 8.1.4-(ii)].

Note that $d_{1}^{\alpha,\mu}\neq 0$ iff $\widetilde{c}_{1}^{\alpha,\mu}\neq
0$, which is true thanks to Corollary 6.5. Hence by [LZ16, Theorem 8.1.4-(iv)
and (v)], the images of $d_{r}^{\alpha,\alpha}$ and $d_{r}^{\alpha,\beta}$ in
$H^{1}_{(p)}({\mathbf{Q}}(r),V_{{\mathrm{Ad}}})$ are linearly independent. On
the other hand, using the exact triangles in Theorem 5.10 for $p$-relaxed
Selmer complexes (and noticing that the local conditions split w.r.t. the
splitting of the representations), we obtain

$H^{1}_{(p)}({\mathbf{Q}}(r),V_{{\mathrm{Ad}}})=H^{1}_{(p)}({\mathbf{Q}}(r),V_{{{\rm
Ad}^{0}}})\oplus H^{1}_{(p)}({\mathbf{Q}}(r),E(\psi+j)),$

and $H^{1}_{(p)}({\mathbf{Q}},E(\psi+j))$ vanishes since $\psi+j$ is even and
the validity of Leopoldt conjecture for abelian number fields holds true (see
[BLV21, Proposition 3.2.3]). Thus we get

$H^{1}_{(p)}({\mathbf{Q}}(r),V_{{\mathrm{Ad}}})=H^{1}_{(p)}({\mathbf{Q}}(r),V_{{{\rm
Ad}^{0}}}),$

which concludes (iii). ∎

######  Corollary 6.7.

Suppose that $\psi(-1)=(-1)^{j}$ and $j\in(-1-k,0]$. If

$L_{p}^{{\rm
imp}}({\mathrm{Sym}}^{2}(f),\varepsilon_{f}^{-1}\psi^{-1},-j+k_{f}+2)L_{p}(\psi,1-j)\neq
0,$

then the Bloch-Kato Selmer groups

$H^{1}_{f}({\mathbf{Q}},{{\rm Ad}^{0}}(V_{f})(\psi+j))$

and

$H^{1}_{f}({\mathbf{Q}},{{\rm Ad}^{0}}(V_{f})(\psi^{-1}+1-j))$

vanish.

###### Proof.

First of all, we notice that Lemma 6.6 gives us a (horizontal) Euler system
(see [Rub00, Chapter 2]), hence we have

$\dim_{E}H^{1}_{(p)}({\mathbf{Q}},{{\rm Ad}^{0}}(V_{f})(\psi+j))=2\textnormal{
and }\dim_{E}H^{1}_{{\rm str}}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi^{-1}+1-j))=0,$

where the latter is thanks to [Rub00, Theorem 2.2(iii)] combined with the big
image hypothesis, and the former is due to the Euler-Poincaré characteristic
formula applied to $\mathbf{R\Gamma}_{(p)}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))$ and the duality theorem (see Proposition 5.16)
between this complex and $\mathbf{R\Gamma}_{\rm str}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi^{-1}+1-j))$.

We will follow the arguments of [LZ16, Theorem 8.2.1] to conclude that
$H^{1}_{f}({\mathbf{Q}},{{\rm Ad}^{0}}(V_{f})(\psi+j))=0$. We have the
following exact sequence:

$\displaystyle 0\longrightarrow H^{1}_{f}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))\longrightarrow H^{1}_{(p)}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))$ $\displaystyle\xrightarrow{{\rm
loc}_{p}}H^{1}({{\mathbf{Q}}_{p}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))/H^{1}_{f}({{\mathbf{Q}}_{p}},{{\rm
Ad}^{0}}(V_{f})(\psi+j)).$

By Lemma 6.6 we know that the images of the elements $d_{1}^{\alpha,\alpha}$
and $d_{1}^{\alpha,\beta}$ are linearly independent. Since

$\dim_{E}(H^{1}({{\mathbf{Q}}_{p}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))/H^{1}_{f}({{\mathbf{Q}}_{p}},{{\rm
Ad}^{0}}(V_{f})(\psi+j)))=2,$

we conclude that the map ${\rm loc}_{p}$ above is surjective, hence the above
exact sequence can be completed to a short exact sequence:

$\displaystyle 0\longrightarrow H^{1}_{f}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))\longrightarrow H^{1}_{(p)}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))$ $\displaystyle\xrightarrow{{\rm
loc}_{p}}H^{1}({{\mathbf{Q}}_{p}},V_{{{\rm
Ad}^{0}}})/H^{1}_{f}({{\mathbf{Q}}_{p}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))\longrightarrow 0.$

This gives us that $\dim_{E}(H^{1}_{f}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi+j)))=0$, i.e. $H^{1}_{f}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))=0$. The duality arguments in the proof of [LZ16,
Corollary 8.3.2] gives $H^{1}_{f}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi^{-1}+1-j))=0$. ∎

##  7\. Algebraic Factorization

This section is where our main theorem is stated and proved. First we will
present the hypotheses that our objects of interest need to satisfy, then
state our main result, and prove it using the theorems in previous sections.
In the last subsection, we will relate the algebraic $p$-adic $L$-function
(recall Corollary 5.32 for the definition) attached to certain twists of
${\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})$ to the algebraic $p$-adic
$L$-function attached to certain twists of
$\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}}^{*}$.

###  7.1. Setup and Assumptions

Let $f\in S_{k_{f}+2}(\Gamma_{1}(N_{f}),\varepsilon_{f})$ be a $p$-non-
ordinary normalized newform such that $(N_{f},p)=1$ with roots
$\alpha\mathrel{\mathop{\mathchar 58\relax}}=\alpha_{f}$ and
$\beta\mathrel{\mathop{\mathchar 58\relax}}=\beta_{f}$ of the Hecke polynomial
$X^{2}-a_{p}(f)X+\varepsilon_{f}(p)p^{k_{f}+1}$ at $p$. Let $\psi$ be a finite
order Dirichlet character with conductor $N_{\psi}$ coprime to $pN_{f}$. Let
$\mathcal{F}$ be the unique Coleman family over the affinoid space
$U=\mathrm{Spm}(A)\subseteq\mathcal{W}$ passing through $f_{\alpha}$, and
$\mathbf{V}_{\mathcal{F}}$ be the big Galois representation attached to
$\mathcal{F}$ as in Theorem 4.6.

######  Remark 7.1.

Throughout this section we will assume that $\psi+j$ is an even character,
i.e. $\psi(-1)=(-1)^{j}$. When $\psi+j$ is odd, the character
$\psi^{-1}+(1-j)$ is an even character, thus we can replace the roles of
$\psi$ by $\psi^{-1}$ and $j$ by $1-j$ in our assumptions below. Indeed we
will present our main result so that these two cases can be switch by duality.

Recall that in the previous section we assumed the following hypotheses on our
objects:

  1. (i)

$\alpha\neq\beta$,

  2. (ii)

$-k_{f}-1<j\leq 0$ if $\psi(-1)=(-1)^{j}$,

  3. (iii)

$L_{p}^{{\rm
imp}}({\mathrm{Sym}}^{2}(f),\varepsilon_{f}^{-1}\psi^{-1},-j+k_{f}+2)L_{p}(\psi,1-j)\neq
0$,

  4. (iv)

$\psi(p)\neq p^{j}$, $\psi(p)\frac{\beta}{\alpha}\neq p^{j}$,
$\frac{\alpha}{\beta}\psi(p)\neq p^{j-1}$,

  5. (v)

${\rm Im}(G_{\mathbf{Q}}\rightarrow{\rm Aut}(V_{f}))$ contains a conjugate of
${\rm SL}_{2}({{\mathbf{Z}}_{p}})$,

  6. (vi)

The prime above $p$ in the Hecke field of $f$ has degree $1$, i.e.
$E\cong{{\mathbf{Q}}_{p}}$,

  7. (vii)

There exists $u\in({\mathbf{Z}}/N_{f}N_{\psi})^{\times}$ such that
$\psi(u)\neq\pm 1\ ({\rm mod\ \mathfrak{P}})$ and
$\varepsilon_{f}^{-1}\psi(u)$ is a square modulo $\mathfrak{P}$, where
$\mathfrak{P}$ is the prime of $E$,

  8. (viii)

$p\geq 7$.

Let us briefly recall the hypotheses above. i is the $p$-regularity condition
which is conjectured to be always true. This is required to be able to
decompose $\mathbf{D}_{{\rm cris}}(V_{f}^{*})$ into rank $1$ distinct
eigenspaces. ii is needed so that $j$ is in the critical range of ${{\rm
Ad}^{0}}(V_{f})(\psi+j)$ (in the sense of Deligne, see Definition 3.7 and
Definition 3.6). iii is imposed in the corollary of the explicit reciprocity
law (Corollary 6.5) so that the specializations of Beilinson-Flach elements
are nonzero. iv is the no local zero hypothesis of [LZ16, §8.1] adapted to our
setting (see also the remark below). The hypotheses v \- viii are for
verifying the big image hypothesis, which is crucial for being able to apply
the Euler system arguments (see [Rub00, Remark 2.4] for the big image
hypothesis, and see [LZ19, §5.2] and [BLV21] for the verification of this
hypothesis via those assumptions). We would like to remark that in (vii), we
replaced $\psi$ by $\varepsilon_{f}^{-1}\psi$ in Loeffler and Zerbes’
assumption in [LZ19, 5.1].

######  Remark 7.2.

Note that the whole point of the assumptions above is to prove that the Bloch-
Kato Selmer groups $H^{1}_{f}({\mathbf{Q}},{{\rm Ad}^{0}}(V_{f})(\psi+j))$ and
$H^{1}_{f}({\mathbf{Q}},V_{{\rm Ad}^{0}}(\psi^{-1}+1-j))$ vanish. If one can
show this vanishing result by relaxing these conditions, our factorization
theorem will be true without requiring those hypotheses (if we also assume the
hypotheses below) except (i) and (v) above. Those two assumptions are required
so that we can relate the Selmer complex attached to ${{\rm
Ad}^{0}}(V_{f})(\psi+j)$ to the Selmer group $H^{1}_{f}({\mathbf{Q}},{{\rm
Ad}^{0}}(V_{f})(\psi+j))$ via Proposition 5.19.

In addition, we will use the following hypotheses to prove the factorization
theorems:

  1. (a)

For each prime ideal $\mathfrak{p}$ of
$\Lambda_{A}[1/p]\mathrel{\mathop{\mathchar
58\relax}}=\Lambda[1/p]\widehat{\otimes}A$, we have

$({{\rm
Ad}^{0}}(\mathbf{V}_{\mathcal{F}})(\psi+j)\otimes_{A}(\Lambda_{A}[1/p]^{\iota})/\mathfrak{p})^{H_{{\mathbf{Q}},S}}=0,$

and

$({{\rm
Ad}^{0}}(\mathbf{V}_{\mathcal{F}})(\psi^{-1}+1-j)\otimes_{A}(\Lambda_{A}[1/p]^{\iota})/\mathfrak{p})^{H_{{\mathbf{Q}},S}}=0,$

where $H_{{\mathbf{Q}},S}\mathrel{\mathop{\mathchar
58\relax}}=\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}}_{\infty}).$

  2. (b)

$H^{1}(I_{\ell},\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}}^{*}(\psi+j))$
and
$H^{1}(I_{\ell},\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}}^{*}(\psi^{-1}+1-j))$
are projective $A\widehat{\otimes}_{E}A$-modules.

  3. (c)

For each prime ideal $\mathfrak{p}$ of
$\Lambda_{A\widehat{\otimes}_{E}A}[1/p]\mathrel{\mathop{\mathchar
58\relax}}=\Lambda[1/p]\widehat{\otimes}_{E}{A\widehat{\otimes}_{E}A}$, we
have

$(\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}}^{*}(\psi+j)\otimes_{A\widehat{\otimes}_{E}A}(\Lambda_{A\widehat{\otimes}_{E}A}[1/p]^{\iota})/\mathfrak{p})^{H_{{\mathbf{Q}},S}}=0,$

and

$(\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}}^{*}(\psi^{-1}+1-j)\otimes_{A\widehat{\otimes}_{E}A}(\Lambda_{A\widehat{\otimes}_{E}A}[1/p]^{\iota})/\mathfrak{p})^{H_{{\mathbf{Q}},S}}=0.$

######  Remark 7.3.

Note that conditions (a) and (c) imply conditions i and ii of Theorem 5.21 for
the representations considered in these conditions (we will explain it in more
detail later). Also note that if for each specialization $x\in
U=\mathrm{Spm}(A)$ the representation ${{\rm Ad}^{0}}(V_{\mathcal{F}_{x}})$ is
absolutely irreducible as a
$\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}}_{\infty})$ representation (which
we know for all classical specializations if $U$ is sufficiently small, see
[LZ19, Note 3.2.2] and [HTU97, §1]), the condition (a) is automatically
satisfied. Condition (b) is required for the perfectness of the Selmer
complexes attached to the twists of
$\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}_{\mathcal{F}}^{*}$.

###  7.2. The “Correct” Local Conditions

Before defining the local conditions at $p$, for the sake of notational
simplicity we set

$\displaystyle\mathbf{V}_{{\mathrm{Ad}}}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})(\psi+j),$
$\displaystyle\mathbf{V}_{{{\rm Ad}^{0}}}$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}={{\rm
Ad}^{0}}(\mathbf{V}_{\mathcal{F}})(\psi+j),$ $\displaystyle\mathbf{V}_{1}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=A(\psi+j)\cong\mathcal{R}_{A}(\psi+j),$

and denote the specializations of these representations at $x_{0}\in U$ by
$V_{{\mathrm{Ad}}}$, $V_{{{\rm Ad}^{0}}}$ and $V_{1}$, i.e.

$\displaystyle V_{{\mathrm{Ad}}}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Ad}}(V_{f})(\psi+j),$ $\displaystyle V_{{{\rm Ad}^{0}}}$
$\displaystyle\mathrel{\mathop{\mathchar 58\relax}}={{\rm
Ad}^{0}}(V_{f})(\psi+j),$ $\displaystyle V_{1}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=E(\psi+j)\cong\mathcal{R}_{A}(\psi+j).$

Recall that we assumed (without loss of generality) that $\psi+j$ is even.
Then we have the following exact sequence of $G_{\mathbf{Q}}$-representations
which splits via ${\rm Tr}^{*}/2\otimes{\rm Id}$:

(7.1) $\displaystyle 0\longrightarrow{{\rm
Ad}^{0}}(V_{f})(\psi+j)\longrightarrow{\mathrm{Ad}}(V_{f})(\psi+j)\xrightarrow{\rm{Tr}\otimes{\rm
Id}}E(\psi+j)\longrightarrow 0.$

Dually, the following exact sequence splits which splits via ${\rm
Tr}/2\otimes{\rm Id}$:

(7.2) $\displaystyle 0\longrightarrow
V_{1}^{*}(1)\xrightarrow{\rm{Tr^{*}}\otimes{\rm
Id}}V_{\mathrm{Ad}}^{*}(1)\longrightarrow V_{{\rm
Ad}^{0}}^{*}(1)\longrightarrow 0.$

Recall also from Section 4.3 that there exists a $(\varphi,\Gamma)$-submodule
$\mathbf{D}\subseteq\mathbf{D}_{\rm{rig},A}^{\dagger}(\mathbf{V}_{\mathcal{F}})$
of rank $1$ over the Robba ring $\mathcal{R}_{A}$ (see Section 3.1), which
satisfies the properties in Theorem 4.7. Let $D\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{D}_{x_{0}}\subseteq\mathbf{D}_{\rm rig}^{\dagger}(V_{f})$
denote its specialization at $x_{0}\in U$, which is a rank $1$
$(\varphi,\Gamma)$-module over the Robba ring $\mathcal{R}_{E}$ that is
saturated in $\mathbf{D}_{\rm rig}^{\dagger}(V_{f})$ (as $f_{\alpha}$ is
non-$\theta$-critical).

We set

$D_{{\mathrm{Ad}}}\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}_{E}(\mathbf{D}_{\rm
rig}^{\dagger}(V_{f}),D)(\psi+j).$

Then $D_{{\mathrm{Ad}}}\subseteq\mathbf{D}_{\rm
rig}^{\dagger}(V_{\mathrm{Ad}})$ is a rank $2$ $(\varphi,\Gamma)$-submodule
over $\mathcal{R}_{E}$. This is the most natural choice for the local
condition at $p$ attached to $V_{\mathrm{Ad}}$. Indeed ${\rm
rank}_{\mathcal{R}_{E}}D_{\mathrm{Ad}}=2=d^{+}(V_{\mathrm{Ad}})$, hence
$(V_{\mathrm{Ad}},D_{\mathrm{Ad}})$ satisfies the weakly Panchishkin condition
of [BCS23, Definition 4.19] (in [Pot13, Pot12], the local conditions
satisfying this condition is called (strict) ordinary), even though
$V_{\mathrm{Ad}}$ is not Panchishkin (or critical in the sense of Definition
3.6).

This choice of local condition $D_{\mathrm{Ad}}\subseteq\mathbf{D}_{\rm
rig}^{\dagger}(V_{\mathrm{Ad}})$ at $p$ induces the local conditions $D_{{\rm
Ad}^{0}}\subseteq\mathbf{D}_{\rm rig}^{\dagger}(V_{\mathrm{Ad}})$ and
$D_{1}\subseteq\mathbf{D}_{\rm rig}^{\dagger}(V_{\mathrm{Ad}})$ at $p$ via the
maps in the exact sequence (7.1). More precisely we have

$\displaystyle D_{{{\rm Ad}^{0}}}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}(\mathbf{D}_{\rm rig}^{\dagger}(V_{f})/D,D)(\psi+j),$
$\displaystyle D_{1}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}{\mathrm{Hom}}(D,D)(\psi+j)\cong\mathcal{R}_{E}(\psi+j).$

In this case $(V_{{\rm Ad}^{0}},D_{{\rm Ad}^{0}})$ and $(V_{1},D_{1})$ also
satisfy the weakly Panchishkin condition, i.e.

${\rm rank}_{\mathcal{R}_{E}}D_{{\rm Ad}^{0}}=1=d^{+}(V_{{\rm Ad}^{0}})$

and

${\rm rank}_{\mathcal{R}_{E}}D_{1}=1=d^{+}(V_{1}),$

even though only $V_{{\rm Ad}^{0}}$ is critical (as we assumed $\psi+j$ is
even and $-1-k_{f}<j\leq 0$) and $V_{1}$ is not.

We also define the local conditions attached to $\mathbf{V}_{\mathrm{Ad}}$,
$\mathbf{V}_{{\rm Ad}^{0}}$, and $\mathbf{V}_{1}$ in a similar manner, i.e.

$\displaystyle\mathbf{D}_{{\mathrm{Ad}}}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}(\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V}_{\mathcal{F}}),\mathbf{D})(\psi+j),$
$\displaystyle\mathbf{D}_{{{\rm Ad}^{0}}}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}(\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V}_{\mathcal{F}})/\mathbf{D},\mathbf{D})(\psi+j),$
$\displaystyle\mathbf{D}_{1}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}={\mathrm{Hom}}(\mathbf{D},\mathbf{D})(\psi+j)\cong\mathcal{R}_{A}(\psi+j).$

In this case we have
$D_{{\mathrm{Ad}}}\cong(\mathbf{D}_{{\mathrm{Ad}}})_{x_{0}}$, $D_{{{\rm
Ad}^{0}}}\cong(\mathbf{D}_{{{\rm Ad}^{0}}})_{x_{0}}$, and
$D_{1}\cong(\mathbf{D}_{1})_{x_{0}}$.

Let $\mathbf{D}_{\mathrm{Ad}}^{\perp}$, $\mathbf{D}_{{\rm Ad}^{0}}^{\perp}$,
$\mathbf{D}_{1}^{\perp}$ be the (dual) local conditions (see Proposition 5.16
for the definitions) attached to $\mathbf{V}_{\mathrm{Ad}}^{*}(1)$,
$\mathbf{V}_{{\rm Ad}^{0}}^{*}(1)$, $\mathbf{V}_{1}^{*}(1)$, respectively.
Also let
$D_{\mathrm{Ad}}^{\perp}\cong(\mathbf{D}_{\mathrm{Ad}}^{\perp})_{x_{0}}$,
$D_{{\rm Ad}^{0}}^{\perp}\cong(\mathbf{D}_{{\rm Ad}^{0}}^{\perp})_{x_{0}}$,
and $D_{1}^{\perp}\cong(\mathbf{D}_{1}^{\perp})_{x_{0}}$ denote the local
conditions attached to $V_{\mathrm{Ad}}^{*}(1)$, $V_{{\rm Ad}^{0}}^{*}(1)$,
$V_{1}^{*}(1)$, respectively. Note that the local conditions $D_{{\rm
Ad}^{0}}^{\perp}$ and $D_{1}^{\perp}$ are induced by $D_{\mathrm{Ad}}^{\perp}$
via the maps in the exact sequence (7.2).

Note that $(V_{\mathrm{Ad}}^{*}(1),D_{\mathrm{Ad}}^{\perp})$,
$(V_{1}^{*}(1),D_{1}^{\perp})$, and $(V_{1}^{*}(1),D_{1}^{\perp})$ are weakly
Panchishkin, since we have

$\displaystyle{\rm rank}_{\mathcal{R}_{E}}D_{\mathrm{Ad}}^{\perp}$
$\displaystyle=2=d^{+}(V_{\mathrm{Ad}}^{*}(1)),$ $\displaystyle{\rm
rank}_{\mathcal{R}_{E}}D_{{\rm Ad}^{0}}^{\perp}$
$\displaystyle=2=d^{+}(V_{{\rm Ad}^{0}}^{*}(1)),$ $\displaystyle{\rm
rank}_{\mathcal{R}_{E}}D_{1}^{\perp}$ $\displaystyle=0=d^{+}(V_{1}^{*}(1)).$

By Theorem 5.10, we have exact triangles of the following (Pottharst style)
Selmer Complexes:

(7.3) $\displaystyle\mathbf{R\Gamma}_{?}(\mathbf{V}_{{{\rm
Ad}^{0}}},\mathbf{D}_{{{\rm Ad}^{0}}})\rightarrow$
$\displaystyle\mathbf{R\Gamma}_{?}(\mathbf{V}_{{\mathrm{Ad}}},\mathbf{D}_{{\mathrm{Ad}}})\rightarrow\mathbf{R\Gamma}_{?}(\mathbf{V}_{1},\mathbf{D}_{1}),$
(7.4) $\displaystyle\mathbf{R\Gamma}_{?}(V_{{{\rm Ad}^{0}}},\mathbf{D}_{{{\rm
Ad}^{0}}})\rightarrow$
$\displaystyle\mathbf{R\Gamma}_{?}(V_{{\mathrm{Ad}}},\mathbf{D}_{{\mathrm{Ad}}})\rightarrow\mathbf{R\Gamma}_{?}(V_{1},\mathbf{D}_{1}),$
(7.5) $\displaystyle\mathbf{R\Gamma}_{?}(\mathbf{V}_{{{\rm
Ad}^{0}}}^{*}(1),\mathbf{D}_{{{\rm Ad}^{0}}}^{\perp})\rightarrow$
$\displaystyle\mathbf{R\Gamma}_{?}(\mathbf{V}_{{\mathrm{Ad}}}^{*}(1),\mathbf{D}_{{\mathrm{Ad}}}^{\perp})\rightarrow\mathbf{R\Gamma}_{?}(\mathbf{V}_{1}^{*}(1),\mathbf{D}_{1}^{\perp}),$
(7.6) $\displaystyle\mathbf{R\Gamma}_{?}(V_{{{\rm
Ad}^{0}}}^{*}(1),\mathbf{D}_{{{\rm Ad}^{0}}}^{\perp})\rightarrow$
$\displaystyle\mathbf{R\Gamma}_{?}(V_{{\mathrm{Ad}}}^{*}(1),\mathbf{D}_{{\mathrm{Ad}}}^{\perp})\rightarrow\mathbf{R\Gamma}_{?}(V_{1}^{*}(1),\mathbf{D}_{1}^{\perp}),$

where $?\in\\{\emptyset,{\rm Iw}\\}$.

######  Proposition 7.4.

Euler-Poincaré characteristics of the Selmer Complexes

$\mathbf{R\Gamma}(V_{\mathrm{Ad}},D_{\mathrm{Ad}}),\ \mathbf{R\Gamma}(V_{{\rm
Ad}^{0}},D_{{\rm Ad}^{0}}),\ \mathbf{R\Gamma}(V_{1},D_{1}),$

and

$\mathbf{R\Gamma}(V_{\mathrm{Ad}}^{*}(1),D_{\mathrm{Ad}}^{\perp}),\
\mathbf{R\Gamma}(V_{{\rm Ad}^{0}}^{*}(1),D_{{\rm Ad}^{0}}^{\perp}),\
\mathbf{R\Gamma}(V_{1}^{*}(1),D_{1}^{\perp})$

are all $0$.

###### Proof.

Due to the computations of ranks above, all of those representations and local
conditions appearing are weakly Panchishkin. By Corollary 5.17 the result
follows. ∎

###  7.3. Proof of the Main Results

In this section we state and prove our factorization theorems. First we will
present our factorization result for the Selmer complexes attached to the
representations $V_{\mathrm{Ad}}$, $V_{{\rm Ad}^{0}}$, and $V_{1}$. After
proving that result, we will use control theorems and the structures of
coadmissible modules to generalize it to Selmer complexes attached to
$\mathbf{V}_{\mathrm{Ad}}$, $\mathbf{V}_{{\rm Ad}^{0}}$, and $\mathbf{V}_{1}$.

######  Theorem 7.5 (Factorization for single newform).

Under the hypotheses i-viii we have:

  1. (1)

For the data

$(V,D_{V})\in\\{(V_{{\mathrm{Ad}}},D_{{\mathrm{Ad}}}),(V_{{{\rm
Ad}^{0}}},D_{{{\rm Ad}^{0}}}),(V_{1},D_{1})\\},$

the Selmer complexes $\mathbf{R\Gamma}_{\rm Iw}(V,D_{V})$ and
$\mathbf{R\Gamma}_{\rm Iw}(V^{*}(1),D_{V}^{\perp})$ lie in the category
$\mathcal{D}_{\rm parf}^{[1,2]}(\mathcal{H})$.

  2. (2)

The cohomologies $H^{i}_{\rm Iw}(V,D_{V})$ and $H^{i}_{\rm
Iw}(V^{*}(1),D_{V}^{\perp})$ vanish except $i\neq 2$. Therefore the exact
triangles (7.4) and (7.6) degenerate to the following short exact sequences:

(7.7) $\displaystyle 0\rightarrow H^{2}_{\rm Iw}(V_{{{\rm Ad}^{0}}},D_{{{\rm
Ad}^{0}}})$ $\displaystyle\rightarrow H^{2}_{\rm
Iw}(V_{{\mathrm{Ad}}},D_{{\mathrm{Ad}}})\rightarrow H^{2}_{\rm
Iw}(V_{1},D_{1})\rightarrow 0,$ (7.8) $\displaystyle 0\rightarrow H^{2}_{\rm
Iw}(V_{1}^{*}(1),D_{1}^{\perp})\rightarrow$ $\displaystyle H^{2}_{\rm
Iw}(V_{{\mathrm{Ad}}}^{*}(1),D_{{\mathrm{Ad}}}^{\perp})\rightarrow H^{2}_{\rm
Iw}(V_{{{\rm Ad}^{0}}}^{*}(1),D_{{{\rm Ad}^{0}}}^{\perp})\rightarrow 0.$

  3. (3)

For the pairs $(V,D_{V})$ as above, $H^{2}_{\rm Iw}(V,D_{V})$ and $H^{2}_{\rm
Iw}(V^{*}(1),D_{V}^{\perp})$ are torsion, and we have

$\displaystyle\det(\mathbf{R\Gamma}_{\rm Iw}(V,D_{V}))$ $\displaystyle={\rm
char}_{\mathcal{H}}H^{2}_{\rm Iw}(V,D_{V}),$
$\displaystyle\det(\mathbf{R\Gamma}_{\rm Iw}(V^{*}(1),D_{V}^{\perp}))$
$\displaystyle={\rm char}_{\mathcal{H}}H^{2}_{\rm
Iw}(V^{*}(1),D_{V}^{\perp}),$

and we obtain the following factorizations:

(7.9) $\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(V_{{\mathrm{Ad}}},D_{{\mathrm{Ad}}}))$
$\displaystyle=\det(\mathbf{R\Gamma}_{\rm Iw}(V_{{{\rm Ad}^{0}}},D_{{{\rm
Ad}^{0}}}))\widehat{\otimes}_{\mathcal{H}}\det(\mathbf{R\Gamma}_{\rm
Iw}(V_{1},D_{1})),$ (7.10) $\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(V_{{\mathrm{Ad}}}^{*}(1),D_{{\mathrm{Ad}}}^{\perp}))$
$\displaystyle=\det(\mathbf{R\Gamma}_{\rm Iw}(V_{{{\rm
Ad}^{0}}}^{*}(1),D_{{{\rm
Ad}^{0}}}^{\perp}))\widehat{\otimes}_{\mathcal{H}}\det(\mathbf{R\Gamma}_{\rm
Iw}(V_{1}^{*}(1),D_{1}^{\perp})).$

######  Lemma 7.6 (Vanishing of $H^{3}$ for specialization).

For $(V,D_{V})$ as in Theorem 7.5 \- 1 we have

$H^{3}_{\rm Iw}(V,D_{V})=0=H^{3}_{\rm Iw}(V^{*}(1),D_{V}^{\perp}).$

###### Proof.

First note that ${{\rm Ad}^{0}}(V_{f})$ is irreducible, see [LZ19, Note 3.2.2]
and [HTU97, §1]. Hence $V_{{\rm Ad}^{0}}$ and $V_{{\rm Ad}^{0}}^{*}(1)$ are
also irreducible as
$\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}}_{\infty})$-representations, where
$S$ is the finite set of primes dividing $pN_{\psi}N_{f}$. This and
$\dim_{E}V_{{\rm Ad}^{0}}$ being bigger than 1 imply that

$(V_{{\rm
Ad}^{0}}^{*}(1))^{\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}}_{\infty})}=0,$

and

$V_{{\rm Ad}^{0}}^{\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}}_{\infty})}=0.$

By Corollary 5.18 we obtain

$H^{3}_{\rm Iw}(V_{{\rm Ad}^{0}},D_{{\rm Ad}^{0}})=0=H^{3}_{\rm Iw}(V_{{\rm
Ad}^{0}}^{*}(1),D_{{\rm Ad}^{0}}^{\perp}).$

On the other hand, $V_{1}=E(\psi+j)$ and $V_{1}^{*}(1)=E(\psi^{-1}+1-j)$ are
one dimensional (hence irreducible) and not characters of
$\Gamma_{0}\cong\mathrm{Gal}({\mathbf{Q}}_{\infty}/{\mathbf{Q}})$ (since
$\psi$ is nontrivial with conductor coprime to $p$), hence

$(V_{1}^{*}(1))^{\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}}_{\infty})}=0,$

and

$V_{1}^{\mathrm{Gal}({\mathbf{Q}}_{S}/{\mathbf{Q}}_{\infty})}=0.$

Again by Corollary 5.18 we obtain

$H^{3}_{\rm Iw}(V_{1},D_{1})=0=H^{3}_{\rm Iw}(V_{1}^{*}(1),D_{1}^{\perp}).$

The exact triangles (7.4) and (7.6) yields exact sequences

$\displaystyle H^{3}_{\rm Iw}(V_{{\rm Ad}^{0}},D_{{\rm Ad}^{0}})\rightarrow$
$\displaystyle H^{3}_{\rm Iw}(V_{\mathrm{Ad}},D_{\mathrm{Ad}})\rightarrow
H^{3}_{\rm Iw}(V_{1},D_{1}),$ $\displaystyle H^{3}_{\rm
Iw}(V_{1}^{*}(1),D_{1}^{\perp})\rightarrow$ $\displaystyle H^{3}_{\rm
Iw}(V_{\mathrm{Ad}}^{*}(1),D_{\mathrm{Ad}}^{\perp})\rightarrow H^{3}_{\rm
Iw}(V_{{\rm Ad}^{0}}^{*}(1),D_{{\rm Ad}^{0}}^{\perp}),$

where we computed the leftmost and rightmost terms as $0$. Thus we obtain

$H^{3}_{\rm Iw}(V_{\mathrm{Ad}},D_{\mathrm{Ad}})=0=H^{3}_{\rm
Iw}(V_{\mathrm{Ad}}^{*}(1),D_{\mathrm{Ad}}^{\perp}).$

∎

###### Proof of Theorem 7.5.

Let $(V,D_{V})$ be as in Theorem 7.5 \- 1. Recall that in Lemma 7.6 we
calculated

$H^{3}_{\rm Iw}(V,D_{V})=0=H^{3}_{\rm Iw}(V^{*}(1),D_{V}^{\perp}).$

By Remark 5.31, we obtain 1.

The exact sequence in Proposition 5.15 (applied to the case $i=3$) gives an
isomorphism between torsion part of $H^{1}_{\rm Iw}(V,D_{V})$ and $H^{3}_{\rm
Iw}(V^{*}(1),D^{\perp})^{\iota}$, which we calculated as $0$ in Lemma 7.6.
Thus we see that $H^{1}_{\rm Iw}(V,D_{V})$ and $H^{1}_{\rm
Iw}(V^{*}(1),D_{V}^{\perp})$ must be free $\mathcal{H}$-modules, and of finite
rank due to the structure theorem of coadmissible $\mathcal{H}$-modules (see
Theorem 5.13).

When $V=V_{1}$, we know that

$H^{1}_{\rm Iw}({\mathbf{Q}},V_{1})=0$

when $\psi+j$ is an even character (see [BLV21, Corollary 3.2.4, Remark
3.2.5]). Since

$H^{1}_{\rm Iw}(V_{1},D_{1})\subseteq H^{1}_{\rm Iw}({\mathbf{Q}},V_{1}),$

we conclude that

$H^{1}_{\rm Iw}(V_{1},D_{1})=0.$

Now consider the case $V=V_{{\rm Ad}^{0}}$. By Proposition 5.11 (The control
theorem (CTRL-1) applied to $A=E$) gives

$H^{1}_{\rm Iw}({V}_{{\rm Ad}^{0}},{D}_{{\rm Ad}^{0}})_{\Gamma_{0}}\subseteq
H^{1}(V_{{\rm Ad}^{0}},D_{{\rm Ad}^{0}}),$

and by Proposition 5.19 we have

$H^{1}(V_{{\rm Ad}^{0}},D_{{\rm Ad}^{0}})\cong H^{1}_{f}({\mathbf{Q}},V_{{\rm
Ad}^{0}})\cong 0,$

where the latter isomorphism is due to Corollary 6.7. Thus we obtain

$H^{1}_{\rm Iw}({V}_{{\rm Ad}^{0}},{D}_{{\rm Ad}^{0}})_{\Gamma_{0}}=0.$

Since $H^{1}_{\rm Iw}({V}_{{\rm Ad}^{0}},{D}_{{\rm Ad}^{0}})$ is free
$\mathcal{H}$-module of with finite rank, it is isomorphic to
$\mathcal{H}^{r}$ for some $r\in{\mathbf{N}}$, hence

$0\cong H^{1}_{\rm Iw}({V}_{{\rm Ad}^{0}},{D}_{{\rm
Ad}^{0}})_{\Gamma_{0}}\cong E^{r}.$

Thus we obtain $r=0$, i.e.

$H^{1}_{\rm Iw}({V}_{{\rm Ad}^{0}},{D}_{{\rm Ad}^{0}})=0.$

By Corollary 5.17, we have

${\rm rank}_{\mathcal{H}}H^{2}_{\rm Iw}(V_{{\rm Ad}^{0}},D_{{\rm
Ad}^{0}})=0={\rm rank}_{\mathcal{H}}H^{2}_{\rm Iw}(V_{1},D_{1}).$

This proves the torsionness of these modules. Thus, the short exact sequence
in Proposition 5.15 (applied to the case $i=1$) gives

$H^{1}_{\rm Iw}(V_{{\rm Ad}^{0}}^{*}(1),D_{{\rm Ad}^{0}}^{\perp})=0=H^{1}_{\rm
Iw}(V_{1}^{*}(1),D_{1}^{\perp}).$

Hence $H^{2}_{\rm Iw}(V_{{\rm Ad}^{0}}^{*}(1),D_{{\rm Ad}^{0}}^{\perp})$ and
$H^{2}_{\rm Iw}(V_{1},D_{1})$ are also torsion.

The exact triangles (7.4) and (7.6) yields exact sequences

$\displaystyle H^{1}_{\rm Iw}(V_{{\rm Ad}^{0}},D_{{\rm Ad}^{0}})\rightarrow$
$\displaystyle H^{1}_{\rm Iw}(V_{\mathrm{Ad}},D_{\mathrm{Ad}})\rightarrow
H^{1}_{\rm Iw}(V_{1},D_{1}),$ $\displaystyle H^{1}_{\rm
Iw}(V_{1}^{*}(1),D_{1}^{\perp})\rightarrow$ $\displaystyle H^{1}_{\rm
Iw}(V_{\mathrm{Ad}}^{*}(1),D_{\mathrm{Ad}}^{\perp})\rightarrow H^{1}_{\rm
Iw}(V_{{\rm Ad}^{0}}^{*}(1),D_{{\rm Ad}^{0}}^{\perp}),$

and by previous computations leftmost and rightmost terms in these exact
sequences are $0$. Thus

$H^{1}_{\rm Iw}(V_{\mathrm{Ad}},D_{\mathrm{Ad}})=0=H^{1}_{\rm
Iw}(V_{\mathrm{Ad}}^{*}(1),D_{\mathrm{Ad}}^{\perp}),$

and we obtain 2. We also conclude that $H^{2}_{\rm
Iw}(V_{\mathrm{Ad}},D_{\mathrm{Ad}})$ and $H^{2}_{\rm
Iw}(V_{\mathrm{Ad}}^{*}(1),D_{\mathrm{Ad}}^{*}(1))$ are torsion, thus by
Corollary 5.32 we get

$\displaystyle\det(\mathbf{R\Gamma}_{\rm Iw}(V,D_{V}))$ $\displaystyle={\rm
char}_{\mathcal{H}}H^{2}_{\rm Iw}(V,D_{V}),$
$\displaystyle\det(\mathbf{R\Gamma}_{\rm Iw}(V^{*}(1),D_{V}^{\perp}))$
$\displaystyle={\rm char}_{\mathcal{H}}H^{2}_{\rm Iw}(V^{*}(1),D_{V}^{\perp})$

for $(V,D_{V})$ as before. Finally, the factorization of determinants in 3
follows formally from applying the determinant functor to the exact triangles
(7.4) and (7.6) (see [GKZ08, Appendix A, Proposition 31]). ∎

Before stating our factorization result for Coleman families, we will need a
lemma, which will make use of coadmissible modules and some ideas from
algebraic geometry.

######  Lemma 7.7.

Let $M$ be a coadmissible $\mathcal{H}_{A}$-module, and for each $x\in
U=\mathrm{Spm}(A)$, let $\mathcal{H}_{x}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{H}_{E_{x}}$, where $E_{x}\mathrel{\mathop{\mathchar
58\relax}}=A/\mathfrak{m}_{x}$ is the residue field of $x$. Suppose that
$M\widehat{\otimes}_{{\rm sp}_{x,{\rm Iw}}}\mathcal{H}_{x}=0$ for each $x\in
U$, where ${\rm sp}_{x,{\rm Iw}}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{A}\rightarrow\mathcal{H}_{x}$ is the specialization
map. Then $M=0$.

###### Proof.

Since $M$ is a coadmissible $\mathcal{H}_{A}$-module, $M=\varprojlim_{n}M_{n}$
for finitely generated $\mathcal{H}_{n}\widehat{\otimes}_{E}A$-modules
$M_{n}$. We will show that $M_{n}=0$ for each $n\in{\mathbf{N}}$.

For an arbitrary $x\in U$, let

(7.11) $\displaystyle\mathfrak{p}_{x,n}\mathrel{\mathop{\mathchar
58\relax}}=\ker((\mathcal{H}_{A})_{n}\rightarrow(\mathcal{H}_{x})_{n}).$

Recall that

$M_{n}\cong M\widehat{\otimes}_{\mathcal{H}_{A}}(\mathcal{H}_{A})_{n},$

thus

$M_{n}\widehat{\otimes}_{(\mathcal{H}_{A})_{n}}(\mathcal{H}_{x})_{n}\cong(M\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{x})\widehat{\otimes}_{\mathcal{H}_{A}}(\mathcal{H}_{A})_{n}.$

By our running assumption, we have

$M_{n}\widehat{\otimes}_{(\mathcal{H}_{A})_{n}}(\mathcal{H}_{x})_{n}\cong
M_{n}\widehat{\otimes}_{(\mathcal{H}_{A})_{n}}((\mathcal{H}_{A})_{n})/\mathfrak{p}_{x,n}=0.$

Let $n\in{\mathbf{N}}$ and a maximal ideal $\mathfrak{m}$ of
$(\mathcal{H}_{A})_{n}=\mathcal{H}_{n}\widehat{\otimes}A$ be arbitrary but
fixed. We claim that there exists an $x\in U$ such that
$\mathfrak{m}\supseteq\mathfrak{p}_{x,n}$. Set $Y\mathrel{\mathop{\mathchar
58\relax}}=W_{n}\times_{E}U=\mathrm{Spm}(\mathcal{H}_{n}\widehat{\otimes}_{E}A)$.
Denote by $\pi\mathrel{\mathop{\mathchar 58\relax}}Y\twoheadrightarrow U$ the
natural projection, which corresponds to the morphism
$A\hookrightarrow\mathcal{H}_{n}\widehat{\otimes}A$ of affinoid algebras
sending $a\in A$ to $1\otimes a$. Set $x\mathrel{\mathop{\mathchar
58\relax}}=\pi(\mathfrak{m})$ (where $\mathfrak{m}\in
Y=\mathrm{Spm}(\mathcal{H}_{n}\widehat{\otimes}_{E}A)$).

We claim that the fiber $Y_{x}\mathrel{\mathop{\mathchar
58\relax}}=W_{n}\times_{E}E_{x}\cong\pi^{-1}(x)$ of $x$ is the vanishing locus
$V(\mathfrak{p}_{x,n})$ of the prime ideal $\mathfrak{p}_{x,n}$. Note that
$Y_{x}=\mathrm{Spm}(\mathcal{H}_{n}\widehat{\otimes}_{E}E_{x})$ and
$\mathcal{H}_{n}\widehat{\otimes}_{E}E_{x}\cong({\mathcal{H}_{x}})_{n}$. Hence
(7.11) immediately implies our claim.

By construction $\mathfrak{m}\in\pi^{-1}(x)=Y_{x}=V(\mathfrak{p}_{x,n})$,
hence $\mathfrak{m}\supseteq\mathfrak{p}_{x,n}$.

Thus for all $\mathfrak{m}\in Y$ there is a point $x\in U$ with
$\mathfrak{m}\supseteq\mathfrak{p}_{x,n}$, and recall that we already obtained

$M_{n}\widehat{\otimes}_{(\mathcal{H}_{A})_{n}}((\mathcal{H}_{A})_{n})/\mathfrak{p}_{x,n}=0.$

These two results imply

$M_{n}\widehat{\otimes}_{(\mathcal{H}_{A})_{n}}((\mathcal{H}_{A})_{n})/\mathfrak{m}=0.$

By Nakayama lemma (as $M_{n}$ is a finitely generated
$(\mathcal{H}_{A})_{n}$-module) $(M_{n})_{\mathfrak{m}}=0$. Since
$\mathfrak{m}$ was arbitrary we get $M_{n}=0$, thus $M=0$. ∎

######  Theorem 7.8 (Factorization for Coleman families).

Under our hypotheses in Section 7.1, we have:

  1. (1)

For the data

$(\mathbf{V},\mathbf{D}_{\mathbf{V}})\in\\{(\mathbf{V}_{{\mathrm{Ad}}},\mathbf{D}_{{\mathrm{Ad}}}),(\mathbf{V}_{{{\rm
Ad}^{0}}},\mathbf{D}_{{{\rm Ad}^{0}}}),(\mathbf{V}_{1},\mathbf{D}_{1})\\},$

the Selmer complexes $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})$ and $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp})$ lie in the category
$\mathcal{D}_{\rm parf}^{[1,2]}(\mathcal{H}_{A})$.

  2. (2)

The cohomologies $H^{i}_{\rm Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})$ and
$H^{i}_{\rm Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp})$ vanish
except $i\neq 2$. Therefore the exact triangles (7.3) and (7.5) degenerate to
the following short exact sequences:

(7.12) $\displaystyle 0\rightarrow H^{2}_{\rm Iw}(\mathbf{V}_{{{\rm
Ad}^{0}}},\mathbf{D}_{{{\rm Ad}^{0}}})$ $\displaystyle\rightarrow H^{2}_{\rm
Iw}(\mathbf{V}_{{\mathrm{Ad}}},\mathbf{D}_{{\mathrm{Ad}}})\rightarrow
H^{2}_{\rm Iw}(\mathbf{V}_{1},\mathbf{D}_{1})\rightarrow 0,$ (7.13)
$\displaystyle 0\rightarrow H^{2}_{\rm
Iw}(\mathbf{V}_{1}^{*}(1),\mathbf{D}_{1}^{\perp})\rightarrow$ $\displaystyle
H^{2}_{\rm
Iw}(\mathbf{V}_{{\mathrm{Ad}}}^{*}(1),\mathbf{D}_{{\mathrm{Ad}}}^{\perp})\rightarrow
H^{2}_{\rm Iw}(\mathbf{V}_{{{\rm Ad}^{0}}}^{*}(1),\mathbf{D}_{{{\rm
Ad}^{0}}}^{\perp})\rightarrow 0.$

  3. (3)

For $(V,D_{V})$ as in Theorem 7.5 \- 1 with $\mathbf{V}_{x_{0}}=V$ and
$(\mathbf{D}_{\mathbf{V}})_{x_{0}}=D_{V}$, we have

$\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}}))\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/{\mathfrak{p}_{x_{0}}}$
$\displaystyle\cong{\rm char}_{\mathcal{H}}H^{2}_{\rm Iw}(V,D_{V}),$
$\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp}))\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/{\mathfrak{p}_{x_{0}}}$
$\displaystyle\cong{\rm char}_{\mathcal{H}}H^{2}_{\rm
Iw}(V^{*}(1),D_{V}^{\perp}),$

where $\mathfrak{p}_{x}$ is the kernel of the map ${{\rm sp}_{{x},{\rm
Iw}}}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{A}\rightarrow\mathcal{H}$ induced by the specialization
at $x\in U=\mathrm{Spm}(A)$. We have the following factorizations:

$\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{{\mathrm{Ad}}},\mathbf{D}_{{\mathrm{Ad}}}))$
$\displaystyle=\det(\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V}_{{{\rm
Ad}^{0}}},\mathbf{D}_{{{\rm
Ad}^{0}}}))\widehat{\otimes}_{\mathcal{H}_{A}}\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{1},\mathbf{D}_{1})),$ $\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{{\mathrm{Ad}}}^{*}(1),\mathbf{D}_{{\mathrm{Ad}}}^{\perp}))$
$\displaystyle=\det(\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V}_{{{\rm
Ad}^{0}}}^{*}(1),\mathbf{D}_{{{\rm
Ad}^{0}}}^{\perp}))\widehat{\otimes}_{\mathcal{H}_{A}}\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{1}^{*}(1),\mathbf{D}_{1}^{\perp})).$

###### Proof.

First note that for any $\mathbf{V}$ as above, $H^{1}(I_{\ell},\mathbf{V})$ is
projective $A$-module since $U$ is nice affinoid (hence $A$ is an Euclidean
domain). This verifies the condition iii in Theorem 5.21. Let us show that

(7.14) $\displaystyle H^{3}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})=0=H^{3}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp}).$

By the control theorem (CTRL-2), for any $x\in U$ we get

$H^{3}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x}\cong
H^{3}_{\rm Iw}(\mathbf{V}_{x},(\mathbf{D}_{\mathbf{V}})_{x})$

and

$H^{3}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x}\cong
H^{3}_{\rm Iw}(\mathbf{V}_{x}^{*}(1),(\mathbf{D}_{\mathbf{V}})_{x}^{\perp}).$

When $\mathbf{V}=\mathbf{V}_{{\rm Ad}^{0}}$, using our running assumption (a)
in Section 7.1 and applying the arguments of Lemma 7.6 we get

$H^{3}_{\rm Iw}(\mathbf{V}_{x},(\mathbf{D}_{\mathbf{V}})_{x})=0=H^{3}_{\rm
Iw}(\mathbf{V}_{x}^{*}(1),(\mathbf{D}_{\mathbf{V}})_{x}^{\perp}).$

Since this holds for all $x\in U$, by Lemma 7.7 we get

$H^{3}_{\rm Iw}(\mathbf{V}_{{\rm Ad}^{0}},\mathbf{D}_{{\rm
Ad}^{0}})=0=H^{3}_{\rm Iw}(\mathbf{V}_{{\rm Ad}^{0}}^{*}(1),\mathbf{D}_{{\rm
Ad}^{0}}^{\perp}).$

When $\mathbf{V}=\mathbf{V}_{1}$, for any $x\in U$ we obtain
$\mathbf{V}_{x}\cong E_{x}(\psi+j)$, and we can apply Lemma 7.6 directly to
conclude (7.14). When $\mathbf{V}=\mathbf{V}_{\mathrm{Ad}}$, (7.14) follows
from applying $H^{3}$ to the exact triangles (7.3) and (7.5). These
computations verify i in Theorem 5.21.

Our running assumption (a) in 7.1 immediately implies the condition ii of
Theorem 5.21 as $H_{{\mathbf{Q}},S}\subseteq G_{{\mathbf{Q}},S}$. Hence we can
apply Theorem 5.21 to conclude 1.

Now we will show that

(7.15) $\displaystyle H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})=0=H^{1}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp}).$

By control theorem (CTRL-2) for any $x\in U$ we have

$H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x}\subseteq
H^{1}_{\rm Iw}(\mathbf{V}_{x},(\mathbf{D}_{\mathbf{V}})_{x}).$

When $\mathbf{V}=\mathbf{V}_{1}$ (or $\mathbf{V}_{1}^{*}(1)$) the latter
(hence the former) vanishes thanks to Theorem 7.5. Thus by Lemma 7.7 we can
conclude (7.15). When $\mathbf{V}=\mathbf{V}_{{\rm Ad}^{0}}$ (or
$\mathbf{V}=\mathbf{V}_{{\rm Ad}^{0}}^{*}(1)$), we should use the control
theorem (CTRL-1) instead. We obtain

$H^{1}(\mathbf{V},\mathbf{D}_{\mathbf{V}})_{x_{0}}\subseteq
H^{1}(\mathbf{V}_{x_{0}},(\mathbf{D}_{\mathbf{V}})_{x_{0}})=0,$

where the latter is due to Theorem 7.5. Note that
$\mathbf{R\Gamma}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\in\mathscr{D}_{\rm
parf}^{[1,2]}(A)$, hence it is quasi isomorphic to a complex consisting of
finite projective $A$-modules concentrated at degrees $[1,2]$, which implies

$\mathbf{R\Gamma}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\simeq\left[A^{r}\rightarrow
A^{r}\right],$

since every projective module over a PID is free. Then
$H^{1}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\subseteq A^{r}$ is also free of
some finite rank, and $H^{1}(\mathbf{V},\mathbf{D}_{\mathbf{V}})_{x_{0}}=0$.
Thus we conclude that the rank must be zero, i.e.
$H^{1}(\mathbf{V},\mathbf{D}_{\mathbf{V}})=0$. Thus we obtain

$H^{1}_{\rm Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})_{\Gamma_{0}}\subseteq
H^{1}(\mathbf{V},\mathbf{D}_{\mathbf{V}})=0.$

Now let $x\in U$ be arbitrary. By the control theorem (CTRL-2) we get

$H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x}\subseteq
H^{1}_{\rm Iw}(\mathbf{V}_{x},(\mathbf{D}_{\mathbf{V}})_{x}),$

and note that $H^{1}_{\rm Iw}(\mathbf{V}_{x},(\mathbf{D}_{\mathbf{V}})_{x})$
is a free $\mathcal{H}$-module (by the structure theorem for coadmissible
$\mathcal{H}$-modules and the fact that $H^{3}_{\rm
Iw}(\mathbf{V}_{x}^{*}(1),(\mathbf{D}_{\mathbf{V}})_{x}^{\perp})=0$).
Therefore $H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x}$
is also a free $\mathcal{H}$-module with some finite rank $r\geq 0$, i.e.

$H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x}\cong\mathcal{H}^{r}.$

On the other hand,

$(H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x})_{\Gamma_{0}}\cong(H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})_{\Gamma_{0}})/\widehat{\otimes}_{A}A/\mathfrak{m}_{x},$

where $\mathfrak{m}_{x}\subseteq A$ is the maximal ideal corresponding to
$x\in U$. We know from the previous paragraph that the latter vanishes, hence

$(H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x})_{\Gamma_{0}}=0.$

This gives us that the rank of the (free) $\mathcal{H}$-module $H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x}$
must be zero, hence

$H^{1}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/\mathfrak{p}_{x}=0.$

Since $x\in U$ was arbitrary, it follows from Lemma 7.7 that

$H^{1}_{\rm Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})=0.$

For $\mathbf{V}=\mathbf{V}_{\mathrm{Ad}}$ or
$\mathbf{V}=\mathbf{V}_{\mathrm{Ad}}^{*}(1)$, we obtain

$H^{1}_{\rm Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})=0$

by applying $H^{1}$ to the exact triangles (7.3) and (7.5).

These computations show that the cohomologies of $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})$ and $\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp})$ vanishes except $i=2$,
thus the long exact sequence induced by the same exact triangles degenerate to
the short exact sequences in 2.

By the control theorem (CTRL-2) we get

$\displaystyle\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/{\mathfrak{p}_{x_{0}}}$
$\displaystyle\cong\mathbf{R\Gamma}_{\rm Iw}(V,D_{V}),$
$\displaystyle\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp})\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/{\mathfrak{p}_{x_{0}}}$
$\displaystyle\cong\mathbf{R\Gamma}_{\rm Iw}(V^{*}(1),D_{V}^{\perp}),$

hence applying the determinant functor and the fact that

$\det(\mathbf{R\Gamma}_{\rm Iw}(V,D_{V}))={\rm char}_{\mathcal{H}}(H^{2}_{\rm
Iw}(V,D_{V}))$

and

$\det(\mathbf{R\Gamma}_{\rm Iw}(V^{*}(1),D_{V}^{\perp}))={\rm
char}_{\mathcal{H}}(H^{2}_{\rm Iw}(V^{*}(1),D_{V}^{\perp}))$

(see the last paragraph of the proof of Theorem 7.5), we conclude

$\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{\mathbf{V}}))\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/{\mathfrak{p}_{x_{0}}}$
$\displaystyle\cong{\rm char}_{\mathcal{H}}H^{2}_{\rm Iw}(V,D_{V}),$
$\displaystyle\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}^{*}(1),\mathbf{D}_{\mathbf{V}}^{\perp}))\widehat{\otimes}_{\mathcal{H}_{A}}\mathcal{H}_{A}/{\mathfrak{p}_{x_{0}}}$
$\displaystyle\cong{\rm char}_{\mathcal{H}}H^{2}_{\rm
Iw}(V^{*}(1),D_{V}^{\perp}).$

Finally, the factorizations follow from applying the determinant functor to
the exact triangles (7.3) and (7.5). ∎

###  7.4. Comparison of $3$-variable and $2$-variable Algebraic $p$-adic
$L$-functions

In this subsection, we will give a partial result on the comparison of the
algebraic $p$-adic $L$-functions attached to (certain twists of)
$\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}^{*}$ and
${\mathrm{Ad}}(\mathbf{V}_{\mathcal{F}})$ (see Corollary 5.32). For the sake
of notational simplicity we will denote

$\mathbf{V}_{3}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{V}_{\mathcal{F}}\widehat{\otimes}_{E}\mathbf{V}^{*}_{\mathcal{F}}(\psi+j),$

and

$\mathbf{D}_{3}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{D}\widehat{\otimes}_{\mathcal{R}_{E}}\mathbf{D}_{{\rm
rig},A}^{\dagger}(\mathbf{V}_{\mathcal{F}})^{*}.$

The notation chosen refers to the fact that it should be the algebraic
counterpart of the $3$-variable $p$-adic $L$-function
$L_{p}(\mathcal{F},\mathcal{F}^{c},\psi+j+\mathbf{k}_{f}+1)$.

Let $B\mathrel{\mathop{\mathchar 58\relax}}=A\widehat{\otimes}_{E}A$. By our
running assumptions (c) and (b), we conclude by Theorem 5.21 that

$\mathbf{R\Gamma}_{\rm Iw}(\mathbf{V}_{3},\mathbf{D}_{3}),\
\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{3}^{*}(1),\mathbf{D}_{3}^{\perp})\in\mathscr{D}_{\rm
parf}^{[1,2]}(\mathcal{H}_{B}).$

To be more precise, the assumptions iii and ii of Theorem 5.21 are
automatically satisfied, meanwhile for the condition i, we can argue as in the
proof of Theorem 7.8 and use Lemma 7.7 to conclude that

$H^{3}_{\rm Iw}(\mathbf{V}_{3},\mathbf{D}_{3})=0=H^{3}_{\rm
Iw}(\mathbf{V}_{3}^{*}(1),\mathbf{D}_{3}^{\perp}).$

There exists a map $\pi\mathrel{\mathop{\mathchar
58\relax}}=\Delta^{*}\mathrel{\mathop{\mathchar 58\relax}}B\rightarrow A$
which induces $\widetilde{\pi}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{B}\rightarrow\mathcal{H}_{A}$. Notice that
$\mathbf{V}_{3}\otimes_{\pi}A\cong\mathbf{V}_{\mathrm{Ad}}$ and
$\mathbf{D}_{3}\otimes_{\mathcal{R}_{B}}\mathcal{R}_{A}\cong\mathbf{D}_{\mathrm{Ad}}$.

We can use the control theorem of [Pot13, Theorem 1.12 - (3)] for Selmer
complexes to deduce that

$\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V},\mathbf{D}_{3})\otimes_{\widetilde{\pi}}^{\mathbf{L}}\mathcal{H}_{A}\simeq\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{\mathrm{Ad}},\mathbf{D}_{\mathrm{Ad}}),$

hence

$\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3}))\otimes_{\widetilde{\pi}}^{\mathbf{L}}\mathcal{H}_{A}\simeq\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{\mathrm{Ad}},\mathbf{D}_{\mathrm{Ad}})).$

Similarly we get

$\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{3}^{*}(1),\mathbf{D}_{3}^{\perp}))\otimes_{\widetilde{\pi}}^{\mathbf{L}}\mathcal{H}_{A}\simeq\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{\mathrm{Ad}}^{*}(1),\mathbf{D}_{\mathrm{Ad}})^{\perp}).$

Let $\mathfrak{p}\mathrel{\mathop{\mathchar 58\relax}}=\ker(\pi)$ and
$\widetilde{\mathfrak{p}}\mathrel{\mathop{\mathchar
58\relax}}=\ker(\widetilde{\pi})$. Note that $\mathfrak{p}$ and
$\widetilde{\mathfrak{p}}$ are height-$1$ prime ideals of $B$ and
$\mathcal{H}_{B}$ respectively.

######  Theorem 7.9.

Suppose that $H^{1}_{\rm Iw}(\mathbf{V}_{3},\mathbf{D}_{3})$ is a finitely
generated $\mathcal{H}_{B}$-module. There exists an $f\in\mathcal{H}_{B}$ with
$f\equiv 1\ ({\rm mod}\ \widetilde{\mathfrak{p}})$ such that

$\displaystyle H^{1}(\mathbf{V}_{3}^{\prime},\mathbf{D}_{3}^{\prime})=0,$
$\displaystyle
H^{1}((\mathbf{V}_{3}^{\prime})^{*}(1),(\mathbf{D}_{3}^{\prime})^{\perp})=0,$

where $\mathbf{V}_{3}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{V}_{3}\widehat{\otimes}_{B}(\mathcal{H}_{B})_{f}^{\iota}$
and $\mathbf{D}_{3}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}=\mathbf{D}_{3}\widehat{\otimes}_{\mathcal{R}_{B}}{(\mathcal{H}_{B})}_{f}^{\iota}$.
Moreover, we have the following relationships between the algebraic $p$-adic
$L$-functions:

$\displaystyle\det(\mathbf{R\Gamma}(\mathbf{V}_{3}^{\prime},\mathbf{D}_{3}^{\prime}))\otimes_{(\mathcal{H}_{B})_{f}}\mathcal{H}_{A}$
$\displaystyle\cong\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{\mathrm{Ad}},\mathbf{D}_{\mathrm{Ad}})),$
$\displaystyle\det(\mathbf{R\Gamma}((\mathbf{V}_{3}^{\prime})^{*}(1),(\mathbf{D}_{3}^{\prime})^{\perp}))\otimes_{(\mathcal{H}_{B})_{f}}\mathcal{H}_{A}$
$\displaystyle\cong\det(\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{\mathrm{Ad}}^{*}(1),\mathbf{D}_{\mathrm{Ad}}^{\perp})).$

###### Proof.

From now on we will prove the theorem for $\mathbf{V}_{3}$ and
$\mathbf{V}_{\mathrm{Ad}}$, the exact same arguments will prove the results
for $(\mathbf{V}_{3})^{*}(1)$ and $\mathbf{V}_{\mathrm{Ad}}^{*}(1)$. By
control theorem [Pot13, Theorem 1.12] we have

$\mathbf{R\Gamma}(\mathbf{V}_{3},\mathbf{D}_{3})\otimes^{\mathbf{L}}_{\widetilde{\pi}}\mathcal{H}_{A}\cong\mathbf{R\Gamma}(\mathbf{V}_{\mathrm{Ad}},\mathbf{D}_{\mathrm{Ad}}).$

Since $\widetilde{\mathfrak{p}}$ is a height $1$ prime, the higher Tor terms
${\rm Tor}_{i}^{\mathcal{H}_{B}}({-},\mathcal{H}_{A})$ vanishes for $i\geq 2$,
and the corresponding spectral sequence degenerates into the following
information:

$\displaystyle H^{1}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3})[\widetilde{\mathfrak{p}}]\mathrel{\mathop{\mathchar
58\relax}}=0,$ $\displaystyle 0\rightarrow H^{1}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3})\widehat{\otimes}_{\mathcal{H}_{B}}\mathcal{H}_{B}/\widetilde{\mathfrak{p}}\rightarrow$
$\displaystyle H^{1}_{\rm
Iw}(\mathbf{V}_{\mathrm{Ad}},\mathbf{D}_{\mathrm{Ad}})\rightarrow H^{2}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3})[\widetilde{\mathfrak{p}}]\rightarrow 0,$
$\displaystyle H^{2}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3})\widehat{\otimes}_{\mathcal{H}_{B}}\mathcal{H}_{B}/\widetilde{\mathfrak{p}}$
$\displaystyle\cong H^{2}_{\rm
Iw}(\mathbf{V}_{\mathrm{Ad}},\mathbf{D}_{\mathrm{Ad}}).$

Thus we have $H^{1}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3})\widehat{\otimes}_{\mathcal{H}_{B}}\mathcal{H}_{B}/\widetilde{\mathfrak{p}}=0$.
By our finiteness assumption we can use Nakayama lemma, thus there exists an
$f\in\mathcal{H}_{B}$, $f\in 1+\widetilde{\mathfrak{p}}$ such that
$fH^{1}_{\rm Iw}(\mathbf{V}_{3},\mathbf{D}_{3})=0$. This implies that

$H^{1}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3})\widehat{\otimes}_{\mathcal{H}_{B}}(\mathcal{H}_{B})_{f}=0.$

Since $(\mathcal{H}_{B})_{f}$ is flat over $\mathcal{H}_{B}$, we have

$0=H^{1}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3})\widehat{\otimes}_{\mathcal{H}_{B}}(\mathcal{H}_{B})_{f}\cong
H^{1}(\mathbf{V}_{3}^{\prime},\mathbf{D}_{3}^{\prime}).$

Note that there exists a map $(\mathcal{H}_{B})_{f}\rightarrow\mathcal{H}_{A}$
which makes the following diagram commutative:

${\mathcal{H}_{B}}$${(\mathcal{H}_{B})_{f}}$${\mathcal{H}_{A}}$$\scriptstyle{\widetilde{\pi}}$

This follows from the fact that the localization functor is exact, thus we get
an exact sequence

$0\rightarrow\widetilde{\mathfrak{p}}_{f}\rightarrow(\mathcal{H}_{B})_{f}
\xrightarrow{{\widetilde{\pi}}_{f}}(\mathcal{H}_{A})_{\overline{f}}\rightarrow
0,$

where $\overline{f}\mathrel{\mathop{\mathchar 58\relax}}=f\ ({\rm mod}\
\widetilde{\mathfrak{p}})=1$. Thus
$(\mathcal{H}_{A})_{\overline{f}}\cong\mathcal{H}_{A}$. Note that
${\widetilde{\pi}}_{f}$ is a $(\mathcal{H}_{B})_{f}$-algebra map.

Hence by the control theorem, we have

$\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{\mathrm{Ad}},\mathbf{D}_{\mathrm{Ad}})\cong\mathbf{R\Gamma}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3})\otimes^{\mathbf{L}}_{\mathcal{H}_{B}}\mathcal{H}_{A}\cong\mathbf{R\Gamma}(\mathbf{V}_{3}^{\prime},\mathbf{D}_{3}^{\prime})\otimes^{\mathbf{L}}_{(\mathcal{H}_{B})_{f}}\mathcal{H}_{A},$

and the result follows from the functoriality of taking determinants. ∎

######  Remark 7.10.

Geometrically this means that there is a neighborhood $U_{f}$ of
$\Delta(\mathrm{Spm}(A))\times_{E}W$ in $\mathrm{Spm}(B)\times_{E}W$ such that
the coherent sheaf $H^{1}_{\rm Iw}(\mathbf{V}_{3},\mathbf{D}_{3})$ (resp.
$H^{1}_{\rm Iw}(\mathbf{V}_{3}^{*}(1),\mathbf{D}_{3}^{\perp})$) restricted to
$U_{f}$ vanishes (where $W$ is the component of the weight space $\mathcal{W}$
that we have fixed). Even though this neighborhood is not of the form
$\widetilde{U}\times_{E}W$ for some neighborhood $\widetilde{U}$ of
$\Delta(\mathrm{Spm}(A))$ in $\mathrm{Spm}(B)$, this is sufficient for our
purposes. Unfortunately we cannot conclude that $H^{1}_{\rm
Iw}(\mathbf{V}_{3},\mathbf{D}_{3})$ is finitely generated, since submodules of
finitely generated modules are not finitely generated in general (which is
however true for modules over Noetherian rings), we hope to be able to remove
this assumption in the future.

## References

  * [AL78] A. O. L. Atkin and Wen Ch’ing Winnie Li, _Twists of newforms and pseudo-eigenvalues of $W$-operators_, Invent. Math. 48 (1978), no. 3, 221–243. MR 508986
  * [AL21] Alessandro Arlandini and David Loeffler, _On the factorisation of the $p$-adic Rankin-Selberg $L$-function in the supersingular case_, 2021.
  * [BB16] Denis Benois and Kâzım Büyükboduk, _On the exceptional zeros of $p$-non-ordinary $p$-adic $L$-functions and a conjecture of Perrin-Riou_.
  * [BB20] by same author, _Critical $p$-adic $L$-functions and interpolation of Beilinson-Kato elements_, 2020.
  * [BC09] Olivier Brinon and Brian Conrad, _CMI Summer School notes on $p$-adic Hodge theory_, 2009.
  * [BCD+14] Massimo Bertolini, Francesc Castella, Henri Darmon, Samit Dasgupta, Kartik Prasanna, and Victor Rotger, _$p$ -adic $L$-functions and Euler systems: a tale in two trilogies_, Automorphic forms and Galois representations. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 414, Cambridge Univ. Press, Cambridge, 2014, pp. 52–101. MR 3444223
  * [BCS23] Kâzım Büyükboduk, Daniele Casazza, and Ryotaro Sakamoto, _On the artin formalism for $p$-adic Garrett–Rankin $L$-functions_, 2023.
  * [BDR15a] Massimo Bertolini, Henri Darmon, and Victor Rotger, _Beilinson-Flach elements and Euler systems I: Syntomic regulators and $p$-adic Rankin $L$-series_, J. Algebraic Geom. 24 (2015), no. 2, 355–378. MR 3311587
  * [BDR15b] by same author, _Beilinson-Flach elements and Euler systems II: the Birch-Swinnerton-Dyer conjecture for Hasse-Weil-Artin $L$-series_, J. Algebraic Geom. 24 (2015), no. 3, 569–604. MR 3344765
  * [Bel12] Joël Bellaïche, _Critical $p$-adic $L$-functions_, Inventiones mathematicae 189 (2012), no. 1, 1–60.
  * [Bel21] Joël Bellaïche, _The eigenbook–eigenvarieties, families of Galois representations, $p$-adic $L$-functions_, Pathways in Mathematics, Birkhäuser/Springer, Cham, 2021. MR 4306639
  * [Ben14] Denis Benois, _Selmer complexes and the $p$-adic Hodge theory_.
  * [Ben21] Denis Benois, _$p$ -adic heights and $p$-adic Hodge theory_, Mém. Soc. Math. Fr. (N.S.) (2021), no. 167, vi + 135. MR 4219074
  * [Ber04] Laurent Berger, _An introduction to the theory of $p$-adic representations_, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 255–292. MR 2023292
  * [Ber11] by same author, _Trianguline representations_ , Bull. Lond. Math. Soc. 43 (2011), no. 4, 619–635. MR 2820149
  * [BLLV19] Kâzım Büyükboduk, Antonio Lei, David Loeffler, and Guhan Venkat, _Iwasawa theory for Rankin–Selberg products of $p$-nonordinary eigenforms_, Algebra Number Theory 13 (2019), no. 4, 901–941. MR 3951583
  * [BLV21] Kâzım Büyükboduk, Antonio Lei, and Guhan Venkat, _Iwasawa theory for symmetric squares of $p$-non-ordinary eigenforms_, Doc. Math. 26 (2021), 1–64. MR 4234458
  * [Bos14] Siegfried Bosch, _Lectures on formal and rigid geometry_ , Lecture Notes in Mathematics, vol. 2105, Springer, Cham, 2014. MR 3309387
  * [Buz07] Kevin Buzzard, _Eigenvarieties_ , London Mathematical Society Lecture Note Series, p. 59–120, Cambridge University Press, 2007.
  * [CM98] R. Coleman and B. Mazur, _The eigencurve_ , London Mathematical Society Lecture Note Series, p. 1–114, Cambridge University Press, 1998.
  * [CPR89] John Coates and Bernadette Perrin-Riou, _On $p$-adic $L$-functions attached to motives over ${\bf Q}$_, Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 23–54. MR 1097608
  * [Das16] Samit Dasgupta, _Factorization of $p$-adic Rankin $L$-series_, Invent. Math. 205 (2016), no. 1, 221–268. MR 3514962
  * [GKZ08] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, _Discriminants, resultants and multidimensional determinants_ , Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008, Reprint of the 1994 edition. MR 2394437
  * [Gro80] Benedict H. Gross, _On the factorization of $p$-adic $L$-series_, Invent. Math. 57 (1980), no. 1, 83–95. MR 564185
  * [HP14] Robert Harron and Jonathan Pottharst, _Iwasawa theory for symmetric powers of cm modular forms at nonordinary primes, ii_ , 2014.
  * [HTU97] Haruzo Hida, Jacques Tilouine, and Eric Urban, _Adjoint modular Galois representations and their Selmer groups_ , vol. 94, 1997, Elliptic curves and modular forms (Washington, DC, 1996), pp. 11121–11124. MR 1491970
  * [KLZ17] Guido Kings, David Loeffler, and Sarah Livia Zerbes, _Rankin–-Eisenstein classes and explicit reciprocity laws_ , Cambridge Journal of Mathematics 5 (2017), no. 1, 1–122.
  * [KPX14] Kiran S. Kedlaya, Jonathan Pottharst, and Liang Xiao, _Cohomology of arithmetic families of $(\varphi,\Gamma)$-modules_, J. Amer. Math. Soc. 27 (2014), no. 4, 1043–1115. MR 3230818
  * [Kü23] Fırtına Küçük, _Iwasawa theory of symmetric Rankin-Selberg products of modular forms and applications_ , PhD Thesis, University College Dublin, May 2023.
  * [LLZ14] Antonio Lei, David Loeffler, and Sarah Livia Zerbes, _Euler systems for Rankin–Selberg convolutions of modular forms_ , Ann. of Math. (2) 180 (2014), no. 2, 653–771. MR 3224721
  * [Loe18] David Loeffler, _A note on $p$-adic Rankin-–Selberg $L$-functions_, Canadian Mathematical Bulletin 61 (2018), no. 3, 608–621.
  * [LZ16] David Loeffler and Sarah Livia Zerbes, _Rankin–-Eisenstein classes in Coleman families_ , Research in the Mathematical Sciences 3 (2016), no. 1.
  * [LZ19] by same author, _Iwasawa theory for the symmetric square of a modular form_ , J. Reine Angew. Math. 752 (2019), 179–210. MR 3975641
  * [Nek06] Jan Nekovář, _Selmer complexes_ , Astérisque, no. 310, Société mathématique de France, 2006 (en). MR 2333680
  * [Och18] Tadashi Ochiai, _Iwasawa main conjecture for $p$-adic families of elliptic modular cuspforms_, arXiv: Number Theory (2018).
  * [Pal17] Bharathwaj Palvannan, _On Selmer groups and factoring $p$-adic $L$-functions_, 2017.
  * [Pot12] Jonathan Pottharst, _Cyclotomic Iwasawa theory of motives_ , preprint (2012), 115.
  * [Pot13] by same author, _Analytic families of finite-slope Selmer groups_ , Algebra Number Theory 7 (2013), no. 7, 1571–1612. MR 3117501
  * [PR94] Bernadette Perrin-Riou, _Théorie d’Iwasawa des représentations $p$-adiques sur un corps local_, Invent. Math. 115 (1994), no. 1, 81–161, With an appendix by Jean-Marc Fontaine. MR 1248080
  * [PR95] by same author, _Fonctions $L$ $p$-adiques des représentations $p$-adiques_, Astérisque (1995), no. 229, 198. MR 1327803
  * [Rub00] Karl Rubin, _Euler Systems_ , Annals of Math. Studies, Vol.147, 2000.
  * [Sch90] A. J. Scholl, _Motives for modular forms_ , Inventiones mathematicae 100 (1990), no. 1, 419–430.
  * [ST03] Peter Schneider and Jeremy Teitelbaum, _Algebras of $p$-adic distributions and admissible representations_, Invent. Math. 153 (2003), no. 1, 145–196. MR 1990669
  * [Sta23] The Stacks project authors, _The stacks project_ , https://stacks.math.columbia.edu, 2023.