# Miura maps for Stäckel systems

Krzysztof Marciniak
Department of Science and Technology
Campus Norrköping, Linköping University
601-74 Norrköping, Sweden
<EMAIL_ADDRESS>Maciej Błaszak
Faculty of Physics, Department of of Mathematical Physics and Computer
Modelling,
A. Mickiewicz University, 61-614 Poznań, Poland
<EMAIL_ADDRESS>
###### Abstract

We introduce the concept of Miura maps between parameter-dependent algebraic
curves of hyper-elliptic type. These Miura maps induce Miura maps between
Stäckel systems defined (on the extended phase space) by the considered
algebraic curves. This construction yields a new way of generating multi-
Hamiltonian representations for Stäckel systems.

##  1 Introduction

Stäckel systems are Hamiltonian systems that are generated by separation
relations affine in Hamiltonians. They are Liouville integrable and separable,
hence integrable in quadratures.

An important class of Stäckel systems is obtained from algebraic curves of
hyperelliptic-type on a plane (see [10] and the literature therein). Such
Stäckel systems are bi-Hamiltonian on the extended phase space [8]. A subclass
of this class are Stäckel systems that are of Benenti type [3, 4] and it is
these systems (or rather their multi-parameter extensions, see below) that we
will consider in this article. To shorten the terminology, throughout the
article we will simply call these systems Stäckel systems.

In this article we introduce a new concept of Miura maps between parameter-
dependent algebraic curves of hyper-elliptic type on the plane, each of them
defining a parameter-dependent Stäckel system, i.e. a Stäckel system in an
extended phase space. These maps in turn generate Miura maps between the
corresponding Stäckel systems themselves in the following sense: we show that
each pair of these Stäckel systems, when considered on the phase space
extended by their respective parameters (that play the role of Casimir
variables in the extended phase space) can be identified by a non-canonical
map [1], being a finite-dimensional counterpart of Miura maps known from the
theory of integrable nonlinear field systems (see for example [12, 2, 11]).
Further, with the help of these finite-dimensional Miura maps, we construct
multi-Hamiltonian representation of considered Stäckel systems in all
coordinate sets defined by our equivalent multi-parameter separation curves.

Stäckel systems generated by different separation curves can sometimes be
identified using an appropriately chosen Stäckel transform [13, 9]. The Miura
maps that we construct in this article do not constitute any Stäckel transform
as Stäckel transform in general requires transforming times (the independent
variables) as well. Our construction can therefore not be reduced to any
Stäckel transform.

##  2 Stäckel systems

Consider the following algebraic (hyperelliptic-type) curve in the
$(\lambda,\mu)$-plane

$\sigma(\lambda)+\sum_{r=1}^{n}h_{r}\lambda^{n-r}=f(\lambda)\mu^{2}$ (2.1)

(with $\lambda$ and $\mu$ $\in\mathbf{R}$) where $f$ and $\sigma$ are
arbitrary Laurent polynomials in $\lambda$. Taking $n$ copies of (2.1) at
points $(\lambda_{i},\mu_{i})$, $i=1,\dotsc,n$, we obtain a system of $n$
linear equations (separation relations) for $h_{r}$:

$\sigma(\lambda_{i})+\sum_{r=1}^{n}h_{r}\lambda_{i}^{n-r}=f(\lambda_{i})\mu_{i}^{2}\text{,
\ \ }i=1,\ldots,n.$

Solving this system (by inverting the Vandermonde matrix $\lambda_{i}^{n-r}$)
yields $n$ functions (Hamiltonians)

$h_{r}=E_{r}+V_{r}^{(\sigma)},\quad r=1,\dotsc,n$ (2.2)

on a $2n$-dimensional manifold (phase space) $M=\mathbf{R}^{2n}=T^{\ast}Q$
with $Q=\mathbf{R}^{n}$. The manifold $M$ is parametrized by the coordinates
$(\lambda_{1},\ldots,\lambda_{n},\mu_{1},\ldots\mu_{n})$ in such a way that
$\lambda_{i}$ are coordinates on $Q$ while $\mu_{i}$ are fiber coordinates in
$T^{\ast}Q$. The geodesic parts $E_{r}$ of (2.2) are given by

$E_{r}=\mu^{T}K_{r}G\mu\text{, \ \ }r=1,\ldots,n,\text{\ \ }$

with $G\,$ treated as a contravariant metric tensor on the configurational
space $Q$, $K_{r}$ ($K_{1}=\operatorname{Id}$) are $(1,1)$-Killing tensors of
$G$ and $V_{r}^{(\sigma)}$ are separable potentials on $Q$. In the coordinates
$\lambda_{i}$ on $Q$ the geometric objects $G$, $K$ and $V_{r}^{(\sigma)}$ are
explicitly given by

$G^{ij}=\frac{f(\lambda_{i})}{\Delta_{i}}\delta^{ij}\text{, \
}(K_{r})_{j}^{i}=-\frac{\partial\rho_{r}}{\partial\lambda_{i}}\delta_{j}^{i}\text{,
\ \
}V_{r}^{(\sigma)}=\sum_{j=1}^{n}\frac{\partial\rho_{r}}{\partial\lambda_{j}}\frac{\sigma(\lambda_{j})}{\Delta_{j}}$
(2.3)

(no summation in the above formulas unless explicitly stated) where
$\Delta_{j}={\textstyle\prod\nolimits_{k\neq j}}(\lambda_{j}-\lambda_{k})$ and
$\rho_{r}=(-1)^{r}s_{r}$ where $s_{r}$ are elementary symmetric polynomials in
$n$ variables $\lambda_{i}$. The coordinates $\lambda_{i}$ are thus orthogonal
coordinates for the metric $G$.

By construction, all the Hamiltonian functions $h_{r}$ are in involution

$\\{h_{r},h_{s}\\}\equiv\pi(dh_{r},dh_{s})=0,\ \ \ \ \ \ \ r,s=1,...,n$ (2.4)

with respect to the Poisson bracket
$\pi={\textstyle\sum_{i=1}^{n}}\frac{\partial}{\partial\lambda_{i}}\wedge\frac{\partial}{\partial\mu_{i}}$
on $M$, i.e. the Hamiltonians (2.2) constitute a Liouville integrable system.
By construction they also separate in coordinates
$(\lambda_{i},\mu_{i})_{i=1,...,n}$ (in the sense of Hamilton-Jacobi theory).
The Hamiltonians (2.2) are known in literature as Stäckel Hamiltonians of
Benenti type. The Hamiltonians (2.2) yield the system of commuting Hamiltonian
flows on $M$:

$\frac{d\xi}{dt_{r}}=X_{r}=\pi dh_{r}\text{, \ \ \ \ }r=1,\ldots,n.$ (2.5)

(where $\xi\in M$). The geometric properties of such systems were investigated
by Benenti et al. in [3, 4, 5, 6]; cf. also [14].

##  3 Miura maps between Stäckel systems in the extended phase space

In this section we consider $1\leq N\leq n$ specific extensions of the curve
(2.1) by $N$ additional real parameters each. Each such extended curve defines
a Stäckel system in the phase space $\mathcal{M=}\mathbf{R}^{2n+N}$ extended
by these parameters treated as new variables. With each curve we associate, in
a very natural way, a degenerated Poisson tensor and the extra parameters
(extra variables) become Casimirs of this Poisson tensors.

We show that these extended curves can be related by specific transformations
which we will call _Miura maps_ , since they induce finite dimensional
counterparts of Miura maps (known from soliton theory) in the extended phase
space $\mathbf{R}^{2n+N}$ between the corresponding Stäckel systems generated
by these curves. These Miura maps on $\mathcal{M=}\mathbf{R}^{2n+N}$ are non-
canonical transformations. We thus prove that all the extended Stäckel systems
actually represent the same system, written in different coordinates,
connected by Miura maps. This in turn will lead to a multi-Hamiltonian
formulation of the considered Stäckel system.

Let us thus consider the curve (2.1) extended by $N$ additional real
parameters $c_{i}$ where $1\leq N\leq n$

$\sigma(\lambda)+c_{N}\lambda^{n+N-1}+\ldots+c_{1}\lambda^{n}+h_{1}\lambda^{n-1}+\ldots+h_{n}=f(\lambda)\mu^{2}.$
(3.1)

It defines the parameter-dependent Stäckel system

$\frac{d\xi}{dt_{r}}=X_{r}\equiv\pi dh_{r}(\lambda,\mu,c)\text{, \
}r=1,\ldots,n\text{, \ \
}\pi={\textstyle\sum_{i=1}^{n}}\frac{\partial}{\partial\lambda_{i}}\wedge\frac{\partial}{\partial\mu_{i}}.$
(3.2)

(with $\xi\in M$) Consider also, for any $s\in\left\\{1,\ldots,N\right\\}$,
the $N$-parameter curve

$\sigma(\bar{\lambda})\bar{\lambda}^{-s}+\bar{c}_{N}\bar{\lambda}^{n+N-s-1}+\ldots+\bar{c}_{s+1}\bar{\lambda}^{n}+\bar{h}_{1}\bar{\lambda}^{n-1}+\ldots+\bar{h}_{n}+\bar{c}_{s}\bar{\lambda}^{-1}+\ldots+\bar{c}_{1}\bar{\lambda}^{-s}=f(\bar{\lambda})\bar{\lambda}^{s}\bar{\mu}^{2}.$
(3.3)

It defines the parameter-dependent Stäckel system

$\frac{d\bar{\xi}}{dt_{r}}=\bar{X}_{r}\equiv\bar{\pi}d\bar{h}_{r}(\bar{\lambda},\bar{\mu},\bar{c})\text{,
\ \ }r=1,\ldots,n\text{, \ \
}\bar{\pi}={\textstyle\sum_{i=1}^{n}}\frac{\partial}{\partial\bar{\lambda}_{i}}\wedge\frac{\partial}{\partial\bar{\mu}_{i}}$
(3.4)

(with $\bar{\xi}\in M$). Note that with each curve $s$ we associate a
degenerated Poisson tensor $\bar{\pi}$ that is canonical in the variables
$(\bar{\lambda}_{i},\bar{\mu}_{i},\bar{c}_{j})_{i=1,...,n;\ j=1,...,N}$.

We will now attempt to find a map between the variables
$(\lambda_{i},\mu_{i},c_{j})_{i=1,...,n;\ j=1,...,N}$ and the variables
$(\bar{\lambda}_{i},\bar{\mu}_{i},\bar{c}_{j})_{i=1,...,n;\ j=1,...,N}$ that
maps the system (3.2) to the system (3.4). Consider thus the following map in
$\mathbf{R}^{2}$:

$\bar{\lambda}=\lambda,\text{ \ \ }\bar{\mu}=\lambda^{-s}\mu.$ (3.5)

This map transforms (algebraically) the curve (3.1) into the curve (3.3),
provided that

$\begin{array}[c]{ll}\bar{c}_{i}=h_{n-i+1},&i=1,\ldots,s\\\
\bar{c}_{i}=c_{i},&i=s+1,\ldots,N\\\ \bar{h}_{i}=c_{s-i+1},&i=1,\ldots,s\\\
\bar{h}_{i}=h_{i-s},&i=s+1,\ldots,n.\end{array}$ (3.6)

We will thus call the pair (3.5)-(3.6) a _Miura map between curves_ (3.1) and
(3.3). The relations (3.6) can be inverted to

$\begin{array}[c]{ll}c_{i}=\bar{c}_{i},&i=s+1,\ldots,N\\\
c_{i}=\bar{h}_{s-i+1},&i=1,\ldots,s\\\ h_{i}=\bar{h}_{s+i},&i=1,\ldots,n-s\\\
h_{i}=\bar{c}_{n-i+1},&i=n-s+1,\ldots,n.\end{array}$ (3.7)

The Miura map (3.5)-(3.6) induces the following _Miura map on the extended
phase space_ $M_{0\rightarrow
s}:\mathbf{R}^{2n+N}\rightarrow\mathbf{R}^{2n+N}$

$\begin{array}[c]{ll}\bar{\lambda}_{i}=\lambda_{i},&i=1,\ldots,n\\\
\bar{\mu}_{i}=\lambda_{i}^{-s}\mu_{i},&i=1,\ldots,n\\\
\bar{c}_{i}=h_{n-i+1}(\lambda,\mu,c),&i=1,\ldots,s\\\
\bar{c}_{i}=c_{i},&i=s+1,\ldots,N,\end{array}$ (3.8)

with the inverse $M_{0\rightarrow
s}^{-1}:\mathbf{R}^{2n+N}\rightarrow\mathbf{R}^{2n+N}$

$\begin{array}[c]{ll}\lambda_{i}=\bar{\lambda}_{i},&i=1,\ldots,n\\\
\mu_{i}=\bar{\lambda}_{i}^{s}\bar{\mu}_{i},&i=1,\ldots,n\\\
c_{i}=\bar{h}_{s-i+1}(\bar{\lambda},\bar{\mu},\bar{c}),&i=1,\ldots,s\\\
c_{i}=\bar{c}_{i},&i=s+1,\ldots,N.\end{array}$ (3.9)

###### Theorem 1

For any $s\in\\{1,\ldots,N\\}$ we have

$X_{r}=\bar{X}_{r},\text{ \ \ }r=1,\ldots,n.$

The proof of this theorem is in the Appendix.

Thus, all the Hamiltonian vector fields $X_{r}$ and $\bar{X}_{r}$ pairwise
coincide on $\mathcal{M}$. This means that all the Stäckel systems (3.4),
generated by the $N$ curves (3.3) (one for each value of $s$ between $1$ and
$N$), represent on the extended phase space $\mathcal{M=}\mathbf{R}^{2n+N}$
the same Stäckel system as (3.2), written in different coordinates, connected
by the corresponding invertible Miura map (3.8).

By composing two appropriate Miura maps of type (3.8) we can now easily relate
Stäckel systems generated by two curves of type (3.3), i.e

$\sigma(\lambda)\lambda^{-r}+c_{N}\lambda^{n+N-r-1}+\ldots+c_{r+1}\lambda^{n}+h_{1}\lambda^{n-1}+\ldots+h_{n}+c_{r}\lambda^{-1}+\ldots+c_{1}\lambda^{-r}=f(\lambda)\lambda^{r}\mu^{2}$
(3.10)

and

$\sigma(\bar{\lambda})\bar{\lambda}^{-s}+\bar{c}_{N}\bar{\lambda}^{n+N-s-1}+\ldots+\bar{c}_{s+1}\bar{\lambda}^{n}+\bar{h}_{1}\bar{\lambda}^{n-1}+\ldots+\bar{h}_{n}+\bar{c}_{s}\bar{\lambda}^{-1}+\ldots+\bar{c}_{1}\bar{\lambda}^{-s}=f(\bar{\lambda})\bar{\lambda}^{s}\bar{\mu}^{2}$
(3.11)

(with $0\leq r<s\leq N\leq n$), by a Miura map (note how the parameters
$c_{i}$ respectively $\bar{c}_{i}$ are distributed in both curves). Consider
thus the following map in the plane $\mathbf{R}^{2}$

$\bar{\lambda}=\lambda,\text{ \ }\bar{\mu}=\lambda^{r-s}\mu.$ (3.12)

This map transforms the curve (3.10) into the curve (3.11), provided that

$\begin{array}[c]{ll}\bar{c}_{i}=h_{n+r-i+1},&i=r+1,\ldots,s\\\
\bar{c}_{i}=c_{i},&i=1,\ldots,r,s+1,\ldots,N\\\
\bar{h}_{i}=c_{s-i+1},&i=1,\ldots,s-r\\\
\bar{h}_{i}=h_{i-(s-r)},&i=s-r+1,\ldots,n.\end{array}$ (3.13)

The maps (3.12)-(3.13) constitute a Miura map between the curves (3.10) and
(3.11). The relations (3.13) can be inverted to

$\begin{array}[c]{ll}c_{i}=\bar{c}_{i},&i=1,\ldots,r,s+1,\ldots,N\\\
c_{i}=\bar{h}_{s-i+1},&i=r+1,\ldots,s\\\
h_{i}=\bar{h}_{s-r+i},&i=1,\ldots,n-(s-r)\\\
h_{i}=\bar{c}_{n+r-i+1},&i=n-(s-r)+1,\ldots,n.\end{array}$ (3.14)

Note that these maps properly reduce to the maps (3.6) and (3.7),
respectively, in the case that $r=0$.

The Miura map (3.12)-(3.13) induces the following Miura map $M_{r\rightarrow
s}$ on $\mathbf{R}^{2n+N}$

$\begin{array}[c]{ll}\bar{\lambda}_{i}=\lambda_{i},&i=1,\ldots,n\\\
\bar{\mu}_{i}=\lambda_{i}^{r-s}\mu_{i},&i=1,\ldots,n\\\
\bar{c}_{i}=h_{n+r-i+1}(\lambda,\mu,c)&i=r+1,\ldots,s\\\
\bar{c}_{i}=c_{i},&i=1,\ldots,r,s+1,\ldots,N\end{array}$ (3.15)

that is nothing else than the composition $M_{r\rightarrow s}=M_{0\rightarrow
s}\circ M_{0\rightarrow r}^{-1}$. This map has the inverse $M_{r\rightarrow
s}^{-1}$

$\begin{array}[c]{ll}\lambda_{i}=\bar{\lambda}_{i},&i=1,\ldots,n\\\
\mu_{i}=\bar{\lambda}_{i}^{s-r}\bar{\mu}_{i},&i=1,\ldots,n\\\
c_{i}=\bar{h}_{s-i+1}(\bar{\lambda},\bar{\mu},\bar{c}),&i=r+1,\ldots,s\\\
c_{i}=\bar{c}_{i},&i=1,\ldots,r,s+1,\ldots,N\end{array}$ (3.16)

that is the composition $M_{r\rightarrow s}^{-1}=M_{0\rightarrow r}\circ
M_{0\rightarrow s}^{-1}$. Thus, again, the systems (3.10) and (3.11) represent
in fact the same system expressed in different separable coordinate systems
connected by the Miura map (3.15). Note also that the Miura maps (3.15) and
(3.16) properly reduce to the Miura maps (3.8) and (3.9), respectively, for
$r=0$.

##  4 Miura maps and multi-Hamiltonian structure of Stäckel systems

In the previous chapter we introduced $N+1\,\ $co-rank $N$ Poisson operators
$\pi_{0},\ldots,\pi_{N}$ on $\mathbf{R}^{2n+N}$ defined as follows. The
operator $\pi_{0}$ is canonical with respect to the separation variables
$(\lambda_{i},\mu_{i},c_{j})_{i=1,...,n;\ j=1,...,N}$ of the curve (3.1)

$\pi_{0}=\sum_{i=1}^{n}\frac{\partial}{\partial\lambda_{i}}\wedge\frac{\partial}{\partial\mu_{i}}$

and for a given $s\in\left\\{1,\ldots,N\right\\}$ by $\pi_{s}$ we denote the
operator

$\pi_{s}=\sum_{i=1}^{n}\frac{\partial}{\partial\bar{\lambda}_{i}}\wedge\frac{\partial}{\partial\bar{\mu}_{i}},$

that is canonical with respect to the separation variables
$(\bar{\lambda}_{i},\bar{\mu}_{i},\bar{c}_{j})_{i=1,...,n;\ j=1,...,N}$ of the
curve (3.3) with fixed $s$. Both sets of variables are related by the Miura
map (3.8).

In this section we use Miura maps, introduced in the previous section, to
find, in a novel way, multi-Hamiltonian structure of the considered Stäckel
system on $\mathcal{M}$. We start by formulating a theorem that clarifies how
all the operators $\pi_{r}$, with $r\in\\{0,\ldots,N\\}$, can be expressed by
the base vector fields of the coordinates $(\bar{\lambda},\bar{\mu},\bar{c})$
(those associated with the curve $s$ i.e. the coordinates in which $\pi_{s}$
is canonical).

###### Theorem 2

For $r\leq s$

$\pi_{r}=\sum_{i=1}^{n}\bar{\lambda}_{i}^{r-s}\frac{\partial}{\partial\bar{\lambda}_{i}}\wedge\frac{\partial}{\partial\bar{\mu}_{i}}\text{
}+\sum_{j=1}^{s-r}X_{n-j+1}\wedge\frac{\partial}{\partial\bar{c}_{r+j}}$ (4.1)

while for $r\geq s$

$\pi_{r}=\sum_{i=1}^{n}\bar{\lambda}_{i}^{r-s}\frac{\partial}{\partial\bar{\lambda}_{i}}\wedge\frac{\partial}{\partial\bar{\mu}_{i}}\text{
}+\sum_{j=1}^{r-s}X_{j}\wedge\frac{\partial}{\partial\bar{c}_{r-j+1}}.$ (4.2)

Note that for $r=s$ the formulas (4.1) and (4.2) coincide and yield then the
same (canonical in $(\bar{\lambda},\bar{\mu},\bar{c})$) Poisson operator
$\pi_{s}={\textstyle\sum_{i=1}^{n}}\frac{\partial}{\partial\bar{\lambda}_{i}}\wedge\frac{\partial}{\partial\bar{\mu}_{i}}$.

Theorem 2 is equivalent to the statement that the matrix representation
$\pi_{r}(\bar{\lambda},\bar{\mu},\bar{c})$ of the Poisson operator $\pi_{r}$
in the variables $(\bar{\lambda},\bar{\mu},\bar{c})$ (associated with the
curve $s$) is: for $r<s$

$\pi_{r}(\bar{\lambda},\bar{\mu},\bar{c})=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\
\ 0&\Lambda^{r-s}\\\
-\Lambda^{r-s}&0\end{array}&\overset{r}{\overbrace{0...0}}\text{ \ }X_{n}\
\cdots\ X_{n+r-s+1}\text{ }\overset{N-s}{\overbrace{0...0}}\\\
\ast&\end{array}\right)$ (4.3)

where $\Lambda=\mathrm{diag}(\bar{\lambda}_{1},\ldots,\bar{\lambda}_{n}),$
while for $r>s$

$\pi_{r}(\bar{\lambda},\bar{\mu},\bar{c})=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\
\ 0&\Lambda^{r-s}\\\
-\Lambda^{r-s}&0\end{array}&\overset{s}{\overbrace{0...0}}\text{ \ }X_{r-s}\
\cdots\ X_{1}\text{ }\overset{N-r}{\overbrace{0...0}}\\\
\ast&\end{array}\right)$ (4.4)

(where $\ast$ denote the elements that make the matrix $\bar{\pi}_{r}$
antisymmetric). In both matrices (4.3) and (4.4) above by $X_{i}$ we denote
the columns consisting of components of the vector field in coordinates
$(\bar{\lambda},\bar{\mu},\bar{c})$. Note that for $r=s$ both matrices (4.3)
and (4.4) reduce to the same matrix of the canonical - in the variables
$(\bar{\lambda},\bar{\mu},\bar{c})$ \- operator
$\pi_{s}={\textstyle\sum_{i=1}^{n}}\frac{\partial}{\partial\bar{\lambda}_{i}}\wedge\frac{\partial}{\partial\bar{\mu}_{i}}$,
with $N$ Casimir functions $\bar{c}_{1},\ldots,\bar{c}_{N}$.

Proof. Let us first prove (4.3). To prove this we will use the direct Miura
map (3.15) between the variables associated with curve $r$ (denoted there and
in this proof by $(\lambda,\mu,c)$) and the variables
$(\bar{\lambda},\bar{\mu},\bar{c})$ associated with our fixed curve $s$. The
upper left $2n\times 2n\,\ $block in (4.3) is easily proven by a direct
calculation from the first part of (3.15). The column $2n+i,$ $i=1,\ldots,N,$
of $\pi_{r}$ in (4.3) consists of components of the vector fields
$\pi_{r}d\bar{c}_{i}$ in coordinates $(\bar{\lambda},\bar{\mu},\bar{c})$. Due
to (3.15), $\pi_{r}d\bar{c}_{i}=\pi_{r}dc_{i}=0$ for
$i=1,\ldots,r,s+1,\ldots,N$, since $c_{i}$ are in this notation Casimirs of
$\pi_{r}$. Thus all the columns $2n+1,\ldots,2n+r,2n+s+1,\ldots 2n+N$ of
$\pi_{r}$ contain only zeros. Further, due to (3.15) and to Theorem 1,

$\pi_{r}d\bar{c}_{i}=\pi_{r}dh_{n+r-i+1}(\lambda,\mu,c)=X_{n+r-i+1}\text{ for
}i=r+1,\ldots,s$

which yields exactly the columns $2n+r+1,\ldots,2n+s$ as in (4.3). The formula
(4.4) can be proved in an analogous way but there we have to use the inverse
Miura map (3.16) with the roles of $r$ and $s$ interchanged, as the formula
(3.15) is formulated for the case $r<s$ only.

From the above considerations it also follows that $N$ Casimirs of $\pi_{r}$,
when written in the variables $(\bar{\lambda},\bar{\mu},\bar{c})$ associated
with our fixed curve $s$, are: for $r<s$

$\begin{array}[c]{l}\bar{c}_{i}\text{,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
}i=1,\ldots,r,s+1,\ldots,N,\\\
\bar{h}_{s+1-i}(\bar{\lambda},\bar{\mu},\bar{c})\text{,\ \ \ \
}i=r+1,\ldots,s\end{array}$

and for $r>s$

$\begin{array}[c]{l}\bar{c}_{i}\text{,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
}i=1,\ldots,s,r+1,\ldots,N,\\\
\bar{h}_{n-r+i}(\bar{\lambda},\bar{\mu},\bar{c})\text{,\ \ \ \
}i=s+1,\ldots,r.\end{array}$

###### Corollary 3

All the Poisson operators $\pi_{s}$, $s=0,...,N$ are pairwise compatible. It
follows from Theorem (2) and from the fact, that $\pi_{0}$ and $\pi_{s}$ are
compatible, as it has been proved in [7] (in the special case $N=n$ but the
proof in [7] can easily be generalized to the case $N<n$).

Denoting

$h_{j}\equiv c_{1-j}\text{ \ for }j=0,-1,\ldots,-N+1,$

due to Theorem 1 and (3.6), we obtain

$\displaystyle X_{k}$
$\displaystyle=\pi_{0}dh_{k}=\pi_{r}d\bar{h}_{k}=\pi_{r}dc_{r-k+1}=\pi_{r}dh_{k-r},\text{
\ \ }k=1,\ldots r,$ $\displaystyle X_{k}$
$\displaystyle=\pi_{0}dh_{k}=\pi_{r}d\bar{h}_{k}=\pi_{r}dh_{k-r},\text{ \
}k=r+1,\ldots n.$

The vector fields $X_{k},$ $k=1,...,n$ are thus $(N+1)$-Hamiltonian:

$X_{k}=\pi_{0}dh_{k}=\pi_{1}dh_{k-1}=\cdots=\pi_{N}dh_{k-N},\text{ \ \
}k=1,\ldots,N$ (4.5)

and therefore generate $\binom{N+1}{2}$ bi-Hamiltonian chains

$\begin{array}[c]{l}\pi_{k}dh_{-k}\ \ \ =0\\\
\pi_{k}dh_{-k+1}=X_{1}=\pi_{r}dh_{-r+1}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\vdots\\\ \pi_{k}dh_{-k+i}=X_{i}=\pi_{r}dh_{-r+i}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \vdots\\\ \pi_{k}dh_{-k+n}=X_{n}=\pi_{r}dh_{-r+n}\\\ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ 0=\pi_{r}dh_{-r+n+1}\end{array}$ (4.6)

for $0\leq k<r\leq N$, where $h_{i}$ are defined by the $s=0$ curve (3.1).

The multi-Hamiltonian representations (4.6) of Stäckel systems, generated by
separation curves (3.1), was proved for the first time in [7] by explicit
calculations.

###### Example 1

Let us consider the Stäckel system generated by the curve (3.1) with $n=2$,
$N=2$, $\sigma(\lambda)=\lambda^{4}$, $f(\lambda)=1$:

$\lambda^{4}+c_{2}\lambda^{3}+c_{1}\lambda^{2}+h_{1}\lambda+h_{2}=\mu^{2}.$
(4.7)

It can be shown that this system is the third stationary system in the
Dispersive Water Wave (DWW) hierarchy, see [15]. For $s=0$ (that is in the
variables $(\lambda,\mu,c)$ associated with the curve (4.7)) the matrix
representations of the operators $\pi_{r}$ on $\mathcal{M}=R^{6}$ are given by
the formulas (4.3) and (4.4) that specify to

$\pi_{0}(\lambda,\mu,c)=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\ \
0&I\\\ -I&0\end{array}&0\ \ \ 0\\\ \ast&\end{array}\right),\ \ \
\pi_{1}(\lambda,\mu,c)=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\ \
0&\Lambda\\\ -\Lambda&0\end{array}&X_{1}\ \ \ 0\\\ \ast&\end{array}\right),$
$\pi_{2}(\lambda,\mu,c)=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\ \
0&\Lambda^{2}\\\ -\Lambda^{2}&0\end{array}&X_{2}\ \ \ X_{1}\\\
\ast&\end{array}\right),$

where $I=$diag$(1,1)$ and $\Lambda=$diag$(\lambda_{1},\lambda_{2})$, while the
bi-Hamiltonian chains (4.6) attain in these variables the form:

$\begin{array}[c]{l}\pi_{0}dc_{1}=0\\\ \pi_{0}dh_{1}=X_{1}=\pi_{1}dc_{1}\\\
\pi_{0}dh_{2}=X_{2}=\pi_{1}dh_{1}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
0=\pi_{1}dh_{2}\end{array}\ \ \ \ \ \begin{array}[c]{l}\pi_{0}dc_{1}=0\\\
\pi_{0}dh_{1}=X_{1}=\pi_{2}dc_{2}\\\ \pi_{0}dh_{2}=X_{2}=\pi_{2}dc_{1}\\\ \ \
\ \ \ \ \ \ \ \ \ \ \ 0=\pi_{2}dh_{1}\end{array}\ \ \ \ \
\begin{array}[c]{l}\pi_{1}dc_{2}=0\\\ \pi_{1}dc_{1}=X_{1}=\pi_{2}dc_{2}\\\
\pi_{1}dh_{1}=X_{2}=\pi_{2}dc_{1}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
0=\pi_{2}dh_{1}\end{array}\ .$

For $s=1$ this system is represented by the separation curve

$\bar{\lambda}^{3}+\bar{c}_{2}\lambda^{3}+\bar{c}_{1}\bar{\lambda}^{-1}+\bar{h}_{1}\bar{\lambda}+\bar{h}_{2}=\bar{\lambda}\bar{\mu}^{2}$
(4.8)

The variables $(\bar{\lambda},\bar{\mu},\bar{c})$ associated with the curve
(4.8) are connected with the variables $(\lambda,\mu,c)$ through the Miura map
(3.8) that in this case attains the form

$\bar{\lambda}_{1}=\lambda_{1},\text{ }\bar{\lambda}_{2}=\lambda_{2},\text{
}\bar{\mu}_{1}=\lambda_{1}^{-1}\mu_{1},\text{
}\bar{\mu}_{2}=\lambda_{2}^{-1}\mu_{2},\text{
}\bar{c}_{1}=h_{2}(\lambda,\mu,c),\text{ }\bar{c}_{2}=c_{2}.$ (4.9)

In variables (4.9) the matrix form of Poisson operators $\pi_{r}$ are

$\pi_{0}(\bar{\lambda},\bar{\mu},\bar{c})=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\
\ 0&\Lambda^{-1}\\\ -\Lambda^{-1}&0\end{array}&X_{2}\ \ \ 0\\\
\ast&\end{array}\right),\ \ \
\pi_{1}(\bar{\lambda},\bar{\mu},\bar{c})=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\
\ 0&I\\\ -I&0\end{array}&0\ \ \ 0\\\ \ast&\end{array}\right),$
$\pi_{2}(\bar{\lambda},\bar{\mu},\bar{c})=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\
\ 0&\Lambda\\\ -\Lambda&0\end{array}&0\ \ \ X_{1}\\\ \ast&\end{array}\right),$

(with $\Lambda=$diag$(\bar{\lambda}_{1},\bar{\lambda}_{2})$), while the bi-
Hamiltonian chains (4.6) are

$\begin{array}[c]{l}\pi_{0}d\bar{h}_{1}=0\\\
\pi_{0}d\bar{h}_{2}=X_{1}=\pi_{1}d\bar{h}_{1}\\\
\pi_{0}d\bar{c}_{1}=X_{2}=\pi_{1}d\bar{h}_{2}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
0=\pi_{1}d\bar{c}_{1}\end{array}\ \ \ \ \
\begin{array}[c]{l}\pi_{0}d\bar{h}_{1}=0\\\
\pi_{0}d\bar{h}_{2}=X_{1}=\pi_{2}d\bar{c}_{2}\\\
\pi_{0}d\bar{c}_{1}=X_{2}=\pi_{2}d\bar{h}_{1}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
0=\pi_{2}d\bar{h}_{2}\end{array}\ \ \ \ \
\begin{array}[c]{l}\pi_{1}d\bar{c}_{2}=0\\\
\pi_{1}d\bar{h}_{1}=X_{1}=\pi_{2}d\bar{c}_{2}\\\
\pi_{1}d\bar{h}_{2}=X_{2}=\pi_{2}d\bar{h}_{1}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
0=\pi_{2}d\bar{h}_{2}\end{array}\ .$

Finally, for $s=2$ the system is represented by the separation curve

$\tilde{\lambda}^{2}+\tilde{c}_{2}^{-1}\tilde{\lambda}+\tilde{c}_{1}^{-2}\tilde{\lambda}+\tilde{h}_{1}\tilde{\lambda}+\tilde{h}_{2}=\tilde{\lambda}^{2}\tilde{\mu}^{2}.$
(4.10)

The variables $(\tilde{\lambda},\tilde{\mu},\tilde{c})$ associated with the
curve (4.10) are connected with the variables $(\lambda,\mu,c)$ of the curve
(4.7) through the Miura map (3.8), that in this case attains the form

$\tilde{\lambda}_{1}=\lambda_{1},\text{
}\tilde{\lambda}_{2}=\lambda_{2},\text{
}\tilde{\mu}_{1}=\lambda_{1}^{-2}\mu_{1},\text{
}\tilde{\mu}_{2}=\lambda_{2}^{-2}\mu_{2},\text{
}\tilde{c}_{1}=h_{2}(\lambda,\mu,c),\text{
}\tilde{c}_{2}=h_{1}(\lambda,\mu,c)$ (4.11)

and they are also connected with the variables
$(\bar{\lambda},\bar{\mu},\bar{c})$ associated with the curve (4.8) through
the Miura map (3.15) that attains the form

$\tilde{\lambda}_{1}=\bar{\lambda}_{1},\text{ \
}\tilde{\lambda}_{2}=\bar{\lambda}_{2},\text{ \
}\tilde{\mu}_{1}=\bar{\lambda}_{1}^{-1}\bar{\mu}_{1},\text{ \
}\tilde{\mu}_{2}=\bar{\lambda}_{2}^{-1}\bar{\mu}_{2},\text{ \
}\tilde{c}_{1}=\bar{c}_{1},\text{ \
}\tilde{c}_{2}=\bar{h}_{2}(\bar{\lambda},\bar{\mu},\bar{c}).$

The matrix representations of the Poisson operators $\pi_{r}$ in the variables
(4.11) are as follows

$\pi_{0}(\tilde{\lambda},\tilde{\mu},\tilde{c})=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\
\ 0&\Lambda^{-2}\\\ -\Lambda^{-2}&0\end{array}&X_{2}\ \ \ X_{1}\\\
\ast&\end{array}\right),\ \ \
\pi_{1}(\tilde{\lambda},\tilde{\mu},\tilde{c})=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\
\ 0&\Lambda^{-1}\\\ -\Lambda^{-1}&0\end{array}&0\ \ \ X_{2}\\\
\ast&\end{array}\right),$
$\pi_{2}(\tilde{\lambda},\tilde{\mu},\tilde{c})=\left(\begin{array}[c]{cc}\begin{array}[c]{ll}\
\ 0&I\\\ -I&0\end{array}&0\ \ \ 0\\\ \ast&\end{array}\right),$

(with $\Lambda=$diag$(\tilde{\lambda}_{1},\tilde{\lambda}_{2})$) while the bi-
Hamiltonian chains (4.6) are

$\begin{array}[c]{l}\pi_{0}d\tilde{h}_{2}=0\\\
\pi_{0}d\tilde{c}_{2}=X_{1}=\pi_{1}d\tilde{h}_{2}\\\
\pi_{0}d\tilde{c}_{1}=X_{2}=\pi_{1}d\tilde{c}_{2}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
0=\pi_{1}d\tilde{c}_{1}\end{array}\ \ \ \ \
\begin{array}[c]{l}\pi_{0}d\tilde{h}_{2}=0\\\
\pi_{0}d\tilde{c}_{2}=X_{1}=\pi_{2}d\tilde{h}_{1}\\\
\pi_{0}d\tilde{c}_{1}=X_{2}=\pi_{2}d\tilde{h}_{2}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
0=\pi_{2}d\tilde{c}_{2}\end{array}\ \ \ \ \
\begin{array}[c]{l}\pi_{1}d\tilde{h}_{1}=0\\\
\pi_{1}d\tilde{h}_{2}=X_{1}=\pi_{2}d\tilde{h}_{1}\\\
\pi_{1}d\tilde{c}_{2}=X_{2}=\pi_{2}d\tilde{h}_{2}\\\ \ \ \ \ \ \ \ \ \ \ \ \ \
0=\pi_{2}d\tilde{c}_{2}\end{array}\ .$

Thus, all three curves (4.7), (4.8) and (4.10) represent the same Stäckel
system on extended phase space $\mathcal{M}=\mathbf{R}^{6}$ in different
separable coordinates, connected by respective Miura maps (3.15).

### Acknowledgments

The authors would like to thank Mathematical Institute of Silesian University
in Opava for kind hospitality extended to them in the course of their visit to
Opava where a substantial part of the present article was written. The authors
are especially grateful for useful discussions with Prof. Artur Sergyeyev.

## Appendix

We prove here Theorem 1. The Jacobian $J$ of the map (3.9) is

$J=\left(\begin{array}[c]{cc}\begin{array}[c]{ccccc}I_{n}&&&&0_{n\times n}\\\
&&&&\\\ s\Lambda^{s-1}M&&&&\Lambda^{s}\end{array}&0_{2n\times N}\\\ &\\\
\begin{array}[c]{cc}\left[\frac{\partial\bar{h}_{s-i+1}}{\partial\bar{\lambda}_{j}}\right]_{i=1\ldots
s}&\left[\frac{\partial\bar{h}_{s-i+1}}{\partial\bar{\mu}_{j}}\right]_{i=1\ldots
s}\\\ &\\\ 0_{(N-s)\times n}&0_{(N-s)\times
n}\end{array}&\begin{array}[c]{cc}\left[V_{s-i+1}^{(-s+j+1)}\right]_{i=1\ldots
s}&\left[V_{s-i+1}^{(n-s+j+1)}\right]_{i=1\ldots s}\\\ &\\\ 0_{(N-s)\times
n}&I_{N-s}\end{array}\end{array}\right)$

where $\Lambda=$diag$(\bar{\lambda}_{1},\ldots,\bar{\lambda}_{n})$,
$M=$diag$(\bar{\mu}_{1},\ldots,\bar{\mu}_{n})$. Thus, the vector field
$\bar{X}_{r}$ has in the coordinates $(\lambda,\mu,c)$ the components given by

$J\left(\begin{array}[c]{c}\frac{\partial\bar{h}_{r}}{\partial\bar{\mu}}\\\
-\frac{\partial\bar{h}_{r}}{\partial\bar{\lambda}}\\\ 0\\\
0\end{array}\right)=\left(\begin{array}[c]{c}\frac{\partial\bar{h}_{r}}{\partial\bar{\mu}}\\\
s\Lambda^{s-1}M\frac{\partial\bar{h}_{r}}{\partial\bar{\mu}}-\Lambda^{s}\frac{\partial\bar{h}_{r}}{\partial\bar{\lambda}}\\\
\left\\{\bar{h}_{s-i+1},\bar{h}_{r}\right\\}_{\pi_{s}}\\\ 0\end{array}\right)$
(A.1)

where
$\left\\{\bar{h}_{s-i+1},\bar{h}_{r}\right\\}_{\pi_{s}}=\pi_{s}(d\bar{h}_{s-i+1},d\bar{h}_{r})=0$
by construction (cf. (2.4)). Consider now all $n$ separation relations
following from the curve (3.1):

$\sigma(\lambda_{i})+c_{N}\lambda_{i}^{n+N-1}+\ldots+c_{1}\lambda_{i}^{n}+h_{1}\lambda_{i}^{n-1}+\ldots+h_{n}=f(\lambda_{i})\mu_{i}^{2}\text{,
\ \ }i=1,\ldots,n$ (A.2)

These relations become identities with respect to all variables
$(\lambda,\mu,c)$ if we insert the Hamiltonians $h_{i}$ generated by (3.1)
into (A.2). Differentiating each of these identities with respect to $\mu_{p}$
yields

$\frac{\partial h_{1}}{\partial\mu_{p}}\lambda_{i}^{n-1}+\ldots+\frac{\partial
h_{n}}{\partial\mu_{p}}=2f(\lambda_{i})\mu_{i}\delta_{ip},$

so that, by inverting the Vandermonde matrix $\lambda_{i}^{n-j}$ we obtain
that for any $r$ and any $p$

$\frac{\partial
h_{r}}{\partial\mu_{p}}=-\sum_{i=1}^{n}\frac{\partial\rho_{r}}{\partial\lambda_{i}}\frac{2f(\lambda_{i})\mu_{i}\delta_{ip}}{\Delta_{i}}=-\frac{\partial\rho_{r}}{\partial\lambda_{p}}\frac{2f(\lambda_{p})\mu_{p}}{\Delta_{p}}$

Performing the analogous operation on separation relations

$\sigma(\bar{\lambda}_{i})+\bar{c}_{N}\bar{\lambda}_{i}^{n+N-1}+\ldots+\bar{c}_{s+1}\bar{\lambda}_{i}^{n+s}+\bar{h}_{1}\bar{\lambda}_{i}^{n+s-1}+\ldots+\bar{h}_{n}\lambda_{i}^{s}+\bar{c}_{s}\bar{\lambda}_{i}^{s-1}+\ldots+\bar{c}_{1}=f(\bar{\lambda}_{i})\bar{\lambda}_{i}^{2s}\bar{\mu}_{i}^{2}$
(A.3)

we obtain that for any $r$ and any $p$

$\frac{\partial\bar{h}_{r}}{\partial\bar{\mu}_{p}}=-\frac{\partial\bar{\rho}_{r}}{\partial\bar{\lambda}_{p}}\frac{2f(\bar{\lambda}_{p})\bar{\lambda}_{p}^{s}\bar{\mu}_{p}}{\bar{\Delta}_{p}}\overset{(\ref{minv})}{=}-\frac{\partial\rho_{r}}{\partial\lambda_{p}}\frac{2f(\lambda_{p})\mu_{p}}{\Delta_{p}}=\frac{\partial
h_{r}}{\partial\mu_{p}}.$ (A.4)

Thus, due to (A.1), $\bar{X}_{r}=$ $X_{r}$ provided that
$s\lambda_{p}^{s-1}\bar{\mu}_{p}\frac{\partial\bar{h}_{r}}{\partial\bar{\mu}_{p}}-\lambda_{p}^{s}\frac{\partial\bar{h}_{r}}{\partial\bar{\lambda}_{p}}=-\frac{\partial
h_{r}}{\partial\lambda_{p}}$ for all $p$ or, due to (A.4), provided that

$\frac{\partial\bar{h}_{r}}{\partial\bar{\lambda}_{p}}=\lambda_{p}^{-s}\frac{\partial
h_{r}}{\partial\lambda_{p}}+s\lambda_{p}^{-s-1}\mu_{p}\frac{\partial
h_{r}}{\partial\mu_{p}}.$ (A.5)

Differentiating (A.2) with respect to $\lambda_{p}$ yields

$\frac{\partial\sigma(\lambda_{i})}{\partial\lambda_{p}}\delta_{ip}+\sum_{k=1}^{N}(n+k-1)c_{k}\delta_{ip}\lambda_{i}^{n+k-2}+\sum_{k=1}^{n}\frac{\partial
h_{k}}{\partial\lambda_{p}}\lambda_{i}^{n-k}+\sum_{k=1}^{n}(n-k)\delta_{ip}h_{k}\lambda_{i}^{n-k-1}=\frac{\partial
f(\lambda_{i})}{\partial\lambda_{p}}\delta_{ip}\mu_{i}^{2},\ \ \ \
i=1,\ldots,n.$

So, for any $r$ and any $p$

$\displaystyle\frac{\partial h_{r}}{\partial\lambda_{p}}$
$\displaystyle=-\sum_{i=1}^{n}\frac{\partial\rho_{r}}{\partial\lambda_{i}}\frac{\delta_{ip}}{\Delta_{i}}\left(\frac{\partial
f(\lambda_{i})}{\partial\lambda_{p}}\mu_{i}^{2}-\frac{\partial\sigma(\lambda_{i})}{\partial\lambda_{p}}-\sum_{k=1}^{N}(n+k-1)c_{k}\lambda_{i}^{n+k-2}-\sum_{k=1}^{n}(n-k)h_{k}\lambda_{i}^{n-k-1}\right)$
$\displaystyle=-\frac{\partial\rho_{r}}{\partial\lambda_{p}}\frac{1}{\Delta_{p}}\left(\frac{\partial
f(\lambda_{p})}{\partial\lambda_{p}}\mu_{p}^{2}-\frac{\partial\sigma(\lambda_{p})}{\partial\lambda_{p}}-\sum_{k=1}^{N}(n+k-1)c_{k}\lambda_{p}^{n+k-2}-\sum_{k=1}^{n}(n-k)h_{k}\lambda_{p}^{n-k-1}\right).$

Performing the analogous operation on separation relations (A.3) and applying
Miura map (3.9) we find, after some calculation, that for any $r$, $p$ and $i$

$\frac{\partial\sigma(\lambda_{i})}{\partial\lambda_{p}}\delta_{ip}+\sum_{k=1}^{N}(n+k-1)c_{k}\delta_{ip}\lambda_{i}^{n+k-2}+\sum_{k=1}^{n}\frac{\partial\bar{h}_{r}}{\partial\bar{\lambda}_{p}}\lambda_{i}^{n+s-k}+\sum_{k=1}^{n}(n-k)\delta_{ip}h_{k}\lambda_{i}^{n-k-1}=\left(\frac{\partial
f(\lambda_{i})}{\partial\lambda_{p}}+2sf(\lambda_{i})\lambda_{i}^{-1}\right)\delta_{ip}\mu_{i}^{2}.$

Hence,

$\displaystyle\frac{\partial\bar{h}_{r}}{\partial\bar{\lambda}_{p}}$
$\displaystyle=-\sum_{i=1}^{n}\frac{\partial\rho_{r}}{\partial\lambda_{i}}\frac{\lambda_{i}^{-s}\delta_{ip}}{\Delta_{i}}\left(\left(\frac{\partial
f(\lambda_{i})}{\partial\lambda_{p}}+2sf(\lambda_{i})\lambda_{i}^{-1}\right)\mu_{i}^{2}-\frac{\partial\sigma(\lambda_{i})}{\partial\lambda_{p}}-\sum_{k=1}^{N}(n+k-1)c_{k}\lambda_{i}^{n+k-2}-\sum_{k=1}^{n}(n-k)h_{k}\lambda_{i}^{n-k-1}\right)$
$\displaystyle=-\frac{\partial\rho_{r}}{\partial\lambda_{p}}\frac{\lambda_{p}^{-s}}{\Delta_{p}}\left(\left(\frac{\partial
f(\lambda_{p})}{\partial\lambda_{p}}+2sf(\lambda_{p})\lambda_{p}^{-1}\right)\mu_{p}^{2}-\frac{\partial\sigma(\lambda_{p})}{\partial\lambda_{p}}-\sum_{k=1}^{N}(n+k-1)c_{k}\lambda_{p}^{n+k-2}-\sum_{k=1}^{n}(n-k)h_{k}\lambda_{p}^{n-k-1}\right)$
$\displaystyle=\lambda_{p}^{-s}\frac{\partial
h_{r}}{\partial\lambda_{p}}-\frac{\partial\rho_{r}}{\partial\lambda_{p}}\frac{\lambda_{p}^{-s}}{\Delta_{p}}2sf(\lambda_{p})\lambda_{p}^{-1}\mu_{p}^{2}=\lambda_{p}^{-s}\frac{\partial
h_{r}}{\partial\lambda_{p}}+s\lambda_{p}^{-s-1}\mu_{p}\frac{\partial
h_{r}}{\partial\mu_{p}},$

that is exactly the condition (A.5).

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k_{n},\sigma_{-1}\exp^{*}z_{n}^{\prime}\rangle_{\mathrm{dif}}.$

Thus it suffices to see that the images of
$\left(\frac{\alpha}{p}-\frac{\beta}{p}\right)^{-1}k_{n}$ and
$\imath^{-}_{n}(w_{n})$ agree in
$D_{\mathrm{dR}}(V(1))/D^{+}_{\mathrm{dR}}(V(1))$.

To this end, we first explicitly describe the realization of $d_{n}:=d_{n,0}$
and $w_{n}$ in (generalized) Kirillov models. If $\phi$ denotes the
realization of $v_{\mathrm{new}}$ in the Kirillov model of
$\pi_{\mathrm{sm}}(V)$, then

$\phi(p^{n})=\begin{cases}\dfrac{(\alpha/p)^{n+1}-(\beta/p)^{n+1}}{(\alpha/p)-(\beta/p)}&n\geq
0,\\\ 0&\text{otherwise.}\end{cases}$

as in section 6.2. Thus,

$\mathcal{K}_{d_{n}}(p^{r}){\bf e}=\begin{cases}\phi(p^{n+r}){\bf e}&r\geq
0,\\\ \varepsilon^{(-r)}\phi(p^{n+r}){\bf e}&-n\leq r\leq 0,\\\
0&r<-n,\end{cases}$

where $\varepsilon^{(m)}=\varepsilon(p^{-m})$ is a $p^{m}$-th root of unity.

Tracing through the proof of Proposition 6.6.1, we then see that

$\xi_{n}(p^{-n})=\\\
\left(\frac{\alpha}{p}-\frac{\beta}{p}\right)^{-1}\left(\sum_{j=0}^{n-1}\varepsilon^{(n-j)}\left(\left(\frac{\alpha}{p}\right)^{j+1}-\left(\frac{\beta}{p}\right)^{j+1}\right)+\frac{(\alpha/p)^{n+1}}{1-\alpha/p}-\frac{(\beta/p)^{n+1}}{1-\beta/p}\right){\mathbf{e}}.$

Since $\omega=e_{\alpha/p}-e_{\beta/p}$, we have

$\displaystyle k_{n}$
$\displaystyle=\sum_{j=0}^{n-1}\varepsilon^{(n-j)}\otimes\varphi^{j+1}\omega+(1-\varphi)^{-1}\varphi^{n+1}\omega$
$\displaystyle=\sum_{j=0}^{n-1}\varepsilon^{(n-j)}\otimes\left(\left(\frac{\alpha}{p}\right)^{j+1}e_{\alpha/p}-\left(\frac{\beta}{p}\right)^{j+1}e_{\beta/p}\right)+\frac{(\alpha/p)^{n+1}}{1-\alpha/p}e_{\alpha/p}-\frac{(\beta/p)^{n+1}}{1-\beta/p}e_{\beta/p}$

and thus

$k_{n}\equiv\left(\frac{\alpha}{p}-\frac{\beta}{p}\right)\imath^{-}_{n}(w_{n})\pmod{D^{+}_{\mathrm{dR}}(V(1))}$

as desired. ∎

######  Remark 7.6.2.

One can even go further and directly check that the classes $\exp(k_{n})$
satisfy Proposition 7.4.1. Nonetheless, we note that the explicit construction
in this section for the crystalline case do not supplant the main arguments
given so far in this paper. Indeed, these explicit constructions with the
exponential map do not give us any integral control of these cohomology
classes. In the next section, we describe how the $c_{n}$ can be normalized so
that they are in $H^{1}(K_{n,p},T(1))$ for all $n\geq 0$ for $T$ some Galois-
stable lattice in $V$. We cannot directly check this property for the classes
$\exp(k_{n})$.

####  7.7. Normalizations

If $T$ is a $G_{\mathbb{Q}_{p}}$-invariant $\mathcal{O}_{L}$-lattice in the
two-dimensional irreducible continuous $G_{\mathbb{Q}_{p}}$-representation $V$
over $L$, then we can form
$\widetilde{\mathbf{D}}^{+}\bigl{(}T(1)\bigr{)}\subset\widetilde{\mathbf{D}}^{+}\bigl{(}V(1)\bigr{)}$
and
$\widetilde{\mathbf{D}}\bigl{(}T(1)\bigr{)}\subset\widetilde{\mathbf{D}}\bigl{(}V(1)\bigr{)}$.
Note that
$\widetilde{\mathbf{D}}^{+}\bigl{(}T(1)\bigr{)}=\widetilde{\mathbf{D}}\bigl{(}T(1)\bigr{)}\cap\widetilde{\mathbf{D}}^{+}\bigl{(}V(1)\bigr{)}$
(the intersection taking place in
$\widetilde{\mathbf{D}}\bigl{(}V(1)\bigr{)}$), so that we also obtain an
embedding

$\pi(T):=\widetilde{\mathbf{D}}\bigl{(}T(1)\bigr{)}/\widetilde{\mathbf{D}}^{+}\bigl{(}T(1)\bigr{)}\hookrightarrow\widetilde{\mathbf{D}}\bigl{(}V(1)\bigr{)}/\widetilde{\mathbf{D}}^{+}\bigl{(}V(1)\bigr{)}=\pi(V),$

which gives an integral structure on the target.

Suppose now, as before, that $V$ is de Rham, with Hodge–Tate weights $0$ and
$1-k<0.$ Then we have the inclusion
$\pi_{\mathrm{sm}}(V)\otimes_{L}(\mathrm{Sym}^{k-2}L^{2})^{*}\hookrightarrow\pi(V)$,
and we may normalize the vector $v_{\mathrm{new}}\in\pi_{\mathrm{sm}}(V)$ (up
to an element of $\mathcal{O}_{L}^{\times}$) by asking that
$v_{\mathrm{new}}\otimes_{\mathcal{O}_{L}}(\mathrm{Sym}^{k-2}\mathcal{O}_{L}^{2})^{*}$
be contained in $\pi(T)$, but not in $\varpi\pi(T)$. (Here $\varpi$ is a
uniformizer of $\mathcal{O}_{L}$.) Further, if we apply $\imath^{-}_{0}$ to
$v_{\mathrm{new}}\otimes v_{\mathrm{hw}}$ (where $v_{\mathrm{hw}}$ is a basis
element for the highest weight space of
$(\mathrm{Sym}^{k-2}\mathcal{O}_{L}^{2})^{*}$), we obtain an element of
$\mathbf{D}_{\mathrm{dR}}(V)/\mathbf{D}^{+}_{\mathrm{dR}}(V),$ well-determined
up to multiplication by an element of $\mathcal{O}_{L}^{\times}$; in other
words, we obtain an $\mathcal{O}_{L}$-structure on the one-dimensional
$L$-vector space
$\mathbf{D}_{\mathrm{dR}}(V)/\mathbf{D}^{+}_{\mathrm{dR}}(V)$. Setting ${\bf
e}$ equal $\imath^{-}_{0}(v_{\mathrm{new}}\otimes v_{\mathrm{hw}})$ then pins
down the elements $d_{n,j}$ (and thus the $c_{n,j}$) up to scaling by a single
element in $\mathcal{O}_{L}^{\times}$. Further, we note that under this
normalization for all $n\geq 0$, we have that $c_{n,j}$ is in
$H^{1}(K_{n,p},T(1+j))$ (as opposed to just being in $H^{1}(K_{n,p},V(1+j))$).

##  Part II The global theory

###  8\. Algebraic $\theta$-elements

####  8.1. Notations and assumptions

Let $f$ be a newform in $S_{k}(\Gamma_{1}(N),\psi,\overline{\mathbb{Q}}_{p})$
for $k\geq 2$, and let $L/\mathbb{Q}_{p}$ denote the field generated by the
Fourier coefficients of $f$. Let
$\rho_{f}:G_{\mathbb{Q}}\to\operatorname{Aut}(V_{f})$ denote the associated
(cohomological) Galois representation; that is, the determinant of $\rho_{f}$
is $\psi^{-1}\varepsilon^{1-k}$ where $\varepsilon$ is the cyclotomic
character. Let $\Sigma$ denote the finite set of primes where $\rho_{f}$
ramifies.

As $f$ will be fixed for the remainder of the paper, we simply write
$V:=V_{f}$ and $\overline{V}:=V_{\overline{f}}$ where $\overline{f}$ is the
complex conjugate of $f$. Fix $T$ a Galois stable lattice in $V$. Set $A=V/T$
and $A^{*}=V^{*}/T^{*}$. We similarly write $\overline{T}$, $\overline{A}$,
and $\overline{A}^{*}$ for the analogous constructions using $\overline{V}$.
Note that $A^{*}\cong\overline{A}(k-1)$. In what follows, to ease notation, we
will simply write $A^{*}_{r}$ for $A^{*}(1-r)$ for any $r\in\mathbb{Z}$ whose
Tate dual is simply $T(r)$.

Further, set $K_{n}=\mathbb{Q}(\mu_{p^{n}})$,
$K_{\infty}=\mathbb{Q}(\mu_{p^{\infty}})$,
$\Gamma_{n}=\mathop{\mathrm{Gal}}\nolimits(K_{\infty}/K_{n})$,
$\mathcal{G}_{n}=\mathop{\mathrm{Gal}}\nolimits(K_{n}/\mathbb{Q})$,
$\Lambda=\mathcal{O}_{L}[[\mathop{\mathrm{Gal}}\nolimits(K_{\infty}/\mathbb{Q})]]$,
and
$\Lambda_{n}=\Lambda_{\Gamma_{n}}\cong\mathcal{O}_{L}[\mathop{\mathrm{Gal}}\nolimits(K_{n}/\mathbb{Q})]$.
We fix an isomorphism of $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$ with
$\mathcal{G}_{n}$ by mapping $a$ to $\sigma_{a}$ where
$\sigma_{a}(\zeta)=\zeta^{a}$ for $\zeta\in\mu_{p^{n}}$.

Let $\pi^{n+1}_{n}:\Lambda_{n+1}\to\Lambda_{n}$ be the natural projection, and
let $\mathop{\mathrm{cor}}\nolimits^{n+1}_{n}:\Lambda_{n}\to\Lambda_{n+1}$ be
the natural injection which sends a group like element
$\sigma\in\mathcal{G}_{n}$ to $\sum\tau$ where $\tau$ runs over all elements
in $\mathcal{G}_{n+1}$ whose image in $\mathcal{G}_{n}$ equals $\sigma$.

####  8.2. Selmer groups

In this section, we give a construction of algebraic $\theta$-elements
attached to $f$ along the $\mathbb{Z}_{p}^{\times}$-extension
$K_{\infty}/\mathbb{Q}$. We begin by recalling the definition of the relevant
Selmer groups. We have

$H^{1}_{g}(K_{n},A^{*}_{r}):=\ker\left(H^{1}(K_{n},A^{*}_{r})\to\prod_{v\nmid
p}H^{1}(I_{v},A^{*}_{r})\times\frac{H^{1}(K_{n,p},A^{*}_{r})}{H^{1}_{g}(K_{n,p},A^{*}_{r})}\right)$

where $v$ runs over places of $K_{n}$ and $I_{v}$ denotes an inertia group at
$v$, and

$H^{1}_{g}(K_{\infty},A^{*}_{r}):=\ker\left(H^{1}(K_{\infty},A^{*}_{r})\to\prod_{w\nmid
p}H^{1}(I_{w},A^{*}_{r})\right)$

where $w$ runs over places of $K_{\infty}$. We note that no condition at $p$
is imposed in this second Selmer group since
$H^{1}_{g}(K_{\infty,p},A^{*}_{r})=H^{1}(K_{\infty,p},A^{*}_{r})$ by [4,
Theorem A].

The key global input in what follows is the sequence in the following
proposition which precisely describes the failure of the control theorem in
this non-ordinary situation.

######  Proposition 8.2.1.

Assuming (Irred) there is a natural exact sequence

(14) $0\to H^{1}_{g}(K_{n},A^{*}_{r})\to
H^{1}_{g}(K_{\infty},A^{*}_{r})^{\Gamma_{n}}\to\frac{H^{1}(K_{n,p},A^{*}_{r})}{H^{1}_{g}(K_{n,p},A^{*}_{r})}\oplus
B_{n}$

where $B_{n}$ is a finite group with size bounded independent of $n$.

###### Proof.

This sequence is derived via the Snake Lemma as in [14, Chapter 3] or as in
[16, Theorem 3.1]. We note that the local ingredients needed to prove this
are:

  1. (1)

$H^{1}(K_{\infty,w}/K_{n,v},A^{*}_{r}(K_{\infty,w}))$ is finite with size
bounded independent of $n$ for $v\nmid p$.

  2. (2)

$H^{1}_{g}(K_{\infty,p},A^{*}_{r})=H^{1}(K_{\infty,p},A^{*}_{r})$.

The first of these facts can be proven as in [14, Lemma 3.3]. The second fact
follows from [4, Theorem A] by Tate local duality and (Irred). ∎

Dualizing (14) and applying Tate local duality yields

(15) $H^{1}_{e}(K_{n,p},T(r))\oplus
B_{n}^{\vee}\longrightarrow(H^{1}_{g}(K_{\infty},A^{*}_{r})^{\vee})_{\Gamma_{n}}\longrightarrow
H^{1}_{g}(K_{n},A^{*}_{r})^{\vee}\longrightarrow 0.$

Let $\mathbb{Q}_{\Sigma}$ denote the maximal extension of $\mathbb{Q}$
unramified outside of $\Sigma$, and let

$\mathbb{H}^{i}(T^{*}_{r}):=\varprojlim_{n}H^{i}(\mathbb{Q}_{\Sigma}/K_{n},T^{*}_{r}).$

The following is a deep theorem of Kato.

######  Theorem 8.2.2 (Kato).

Assuming (Irred) we have

  1. (1)

$\mathbb{H}^{1}(T^{*}_{r})$ is a free $\Lambda$-module of rank one.

  2. (2)

$H^{1}_{g}(K_{\infty},A^{*}_{r})^{\vee}$ is a rank one $\Lambda$-module.

###### Proof.

The first part is [20, Theorem 12.4.3]. For the second part, we have that the
$\Lambda$-corank of $H^{1}(\mathbb{Q}_{\Sigma}/K_{\infty},A^{*}_{r})$ equals
the $\Lambda$-rank of $\mathbb{H}^{1}(T^{*}_{r})$ by [31, Proposition 1.3.2].
Further, consider the exact sequence

(16) $0\to H^{1}_{g}(K_{\infty},A^{*}_{r})\to
H^{1}(\mathbb{Q}_{\Sigma}/K_{\infty},A^{*}_{r})\to\bigoplus_{w\in\Sigma-\\{p\\}}H^{1}(K_{\infty,w},A^{*}_{r}).$

Since $H^{1}(K_{\infty,w},A^{*}_{r})$ is $\Lambda$-cotorsion if $w\nmid p$
(see [13, Proposition 2]), we have

$\operatorname{\operatorname{corank}}_{\Lambda}H^{1}_{g}(K_{\infty},A^{*}_{r})=\operatorname{\operatorname{corank}}_{\Lambda}H^{1}(\mathbb{Q}_{\Sigma}/K_{\infty},A^{*}_{r})=1$

as desired. ∎

To ease notation, let $X_{r}=H^{1}_{g}(K_{\infty},A^{*}_{r})^{\vee}$. Set
$(X_{r})_{\operatorname{\Lambda{\text{-}tor}}}$ equal to the $\Lambda$-torsion
submodule of $X_{r}$, and let
$Z_{r}=X_{r}/(X_{r})_{\operatorname{\Lambda{\text{-}tor}}}$, a torsion-free
$\Lambda$-module. Let
$R_{\Lambda}(Z_{r}):=\mathop{\mathrm{Hom}}\nolimits_{\Lambda}\bigl{(}\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(Z_{r},\Lambda),\Lambda\bigr{)}$
denote the reflexive hull of $Z_{r}$. Choosing an isomorphism of
$\alpha:R_{\Lambda}(Z_{r})\cong\Lambda$ (via Theorem 8.2.2) yields maps for
each $n\geq 0$

(17)
$(X_{r})_{\Gamma_{n}}\longrightarrow(Z_{r})_{\Gamma_{n}}\longrightarrow\Lambda_{n}.$

Combining (15) and (17), we get a map

$\psi_{n}=\psi_{n,\alpha}:H^{1}_{e}(K_{n,p},T(r))\to
H^{1}_{e}(K_{n,p},T(r))\oplus
B_{n}^{\vee}\longrightarrow(X_{r})_{\Gamma_{n}}\to(Z_{r})_{\Gamma_{n}}\to\Lambda_{n}.$

######  Lemma 8.2.3.

The diagram

$\begin{CD}H^{1}_{e}(K_{n,p},T(r))@>{\psi_{n}}>{}>\Lambda_{n}\\\
@A{\mathop{\mathrm{res}}\nolimits^{n}_{m}}A{}A@A{}A{\mathop{\mathrm{cor}}\nolimits^{n}_{m}}A\\\
H^{1}_{e}(K_{m,p},T(r))@>{\psi_{m}}>{}>\Lambda_{m}\end{CD}$

commutes for $n\leq n$.

###### Proof.

This commutativity can easily be checked. See [16, Proposition 6.3] for a
similar computation. ∎

Recall our local cohomology classes $c_{n,r-1}\in H^{1}_{e}(K_{n,p},T(r))$
defined in section 7.3. For $1\leq r\leq k-1$, we set

$\psi^{\operatorname{alg}}_{n,r}(f)=\psi_{n}(c_{n,r-1})$

and

$\theta^{\operatorname{alg}}_{n,r}(f)=\psi_{n,r}^{\operatorname{alg}}(f)\cdot\operatorname{char}_{\Lambda}((X_{r})_{\operatorname{\Lambda{\text{-}tor}}})$

which is our $n$-th layer algebraic $\theta$-element.

Further, let $I^{\operatorname{alg}}_{n,r}(f)$ (resp.
$J^{\operatorname{alg}}_{n,r}(f)$) denote the ideal of $\Lambda_{n}$ generated
by the elements
$\mathop{\mathrm{cor}}\nolimits^{n}_{m}(\theta^{\operatorname{alg}}_{m,r}(f))$
(resp.
$\mathop{\mathrm{cor}}\nolimits^{n}_{m}(\psi^{\operatorname{alg}}_{m,r}(f))$)
for $0\leq m\leq n$. Note that
$I^{\operatorname{alg}}_{n,r}(f)=J^{\operatorname{alg}}_{n,r}(f)\cdot\operatorname{char}_{\Lambda}(X_{r})_{\operatorname{\Lambda{\text{-}tor}}}$.
Also, changing our fixed isomorphism $\alpha$ has the effect of scaling all of
the $\psi_{n,r}^{\operatorname{alg}}(f)$ by a unit in $\Lambda$. In
particular, the ideals $I^{\operatorname{alg}}_{n,r}(f)$ and
$J^{\operatorname{alg}}_{n,r}(f)$ are independent of this choice.

######  Remark 8.2.4.

This construction of algebraic $\theta$-elements dates back to Perrin-Riou in
her 1990 paper [30].

###  9\. Algebraic results on Fitting ideals

####  9.1. Main algebraic results

The remainder of this section will be occupied with proving the following
purely algebraic result.

######  Theorem 9.1.1.

If (Irred) holds, we have

$\psi_{n}(c)\cdot\operatorname{char}_{\Lambda}(X_{r})_{\operatorname{\Lambda{\text{-}tor}}}\in\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}H^{1}_{g}(K_{n},A^{*}_{r})^{\vee}$

for any $c$ in $H^{1}_{e}(K_{n,p},T(r))$.

We note that as an immediate corollary we get.

######  Corollary 9.1.2.

If (Irred) holds,

$I_{n,r}^{\operatorname{alg}}(f)\subseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}H^{1}_{g}(K_{n},{A^{*}_{r}})^{\vee}.$

###### Proof.

This corollary follows immediately from Theorem 9.1.1 by taking
$c=\mathop{\mathrm{cor}}\nolimits^{n}_{m}(c_{m,r})$ for $0\leq m\leq n$. ∎

####  9.2. Fitting ideal lemmas

######  Lemma 9.2.1.

If $A_{n},B_{n},C_{n}$ are $\Lambda_{n}$-modules, and $Y$ is a
$\Lambda$-module then

  1. (1)

If $A_{n}\to B_{n}$ is surjective, then
$\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(A_{n})\subseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(B_{n})$.

  2. (2)

If $0\to A_{n}\to B_{n}\to C_{n}\to 0$ is exact, then

$\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(B_{n})\supseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(A_{n})\cdot\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(C_{n}).$

  3. (3)

$\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(Y_{\Gamma_{n}})=\pi_{n}(\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}(Y))$
where $\pi_{n}:\Lambda\to\Lambda_{n}$ is the natural map.

###### Proof.

See [27, Appendix: 1,9,4] ∎

######  Lemma 9.2.2.

If $Y$ is a finitely generated torsion $\Lambda$-module with no non-zero
finite submodules, then
$\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}(Y)=\operatorname{char}_{\Lambda}(Y)$.
In particular, $\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}(Y)$ is a principal
ideal.

###### Proof.

See [35, Proposition 1.3.4]. ∎

We thank Cornelius Greither for the following argument.

######  Lemma 9.2.3.

Let $I\subseteq A$ be ideals of $\Lambda$ with $A$ finite-index in $\Lambda$.
Then

$I\subseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}(A/I).$

###### Proof.

We first note that it suffices to assume that $I$ is a principal ideal.
Indeed, assume that we have proven this lemma for all principal ideals, and
let $f$ be any element of $I$. Then we have

$f\Lambda\subseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}(A/f\Lambda)\subseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}(A/I)$

where the second inclusion follows from Lemma 9.2.1. Since this is true for
all $f\in I$, we have
$I\subseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}(A/I)$.

Thus, we will now assume that $I=f\Lambda$. We then have an exact sequence

$0\to A/f\Lambda\to\Lambda/f\Lambda\to\Lambda/A\to 0.$

Since $A$ has finite index in $\Lambda$, for any height one prime
$\mathfrak{p}$ in $\Lambda$, we have

$(A/f\Lambda)_{\mathfrak{p}}\cong(\Lambda/f\Lambda)_{\mathfrak{p}}.$

Thus,

$\left(\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{\mathfrak{p}}}(A/f\Lambda)\right)_{\mathfrak{p}}=\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{\mathfrak{p}}}\left((A/f\Lambda)_{\mathfrak{p}}\right)=\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{\mathfrak{p}}}\left((\Lambda/f\Lambda)_{\mathfrak{p}}\right)=(f\Lambda)_{\mathfrak{p}}.$

By Lemma 9.2.2, we have
$\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}(A/f\Lambda)=g\Lambda$ for some $g$
in $\Lambda$. Thus,

$(g\Lambda)_{\mathfrak{p}}=(f\Lambda)_{\mathfrak{p}}$

for all height one primes $\mathfrak{p}$. Hence $f\Lambda=g\Lambda$ and
$f\Lambda=\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}(A/f\Lambda)$ as desired. ∎

######  Lemma 9.2.4.

Let $A$ be some finite-index ideal of $\Lambda$, and let $I_{n}$ be a
$\Lambda_{n}$-submodule of $A_{\Gamma_{n}}$. Then

$i_{n}(I_{n})\subseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}A_{\Gamma_{n}}/I_{n}$

where $i_{n}$ is the natural map $A_{\Gamma_{n}}\rightarrow\Lambda_{n}$.

###### Proof.

Let $\pi_{n}$ denote the natural map from $\Lambda\to\Lambda_{n}$, and let
$p_{n}$ denote the natural map from $A$ to $A_{\Gamma_{n}}$ so that
$i_{n}\circ p_{n}=\pi_{n}|_{A}$. The map $p_{n}$ induces an isomorphism

$\frac{A}{p_{n}^{-1}(I_{n})}\cong\frac{A_{\Gamma_{n}}}{I_{n}},$

and thus

$\displaystyle\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}\left(\frac{A_{\Gamma_{n}}}{I_{n}}\right)$
$\displaystyle=\pi_{n}\left(\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}\left(\frac{A}{p_{n}^{-1}(I_{n})}\right)\right)$
by Lemma 9.2.1.3 $\displaystyle\supseteq\pi_{n}\left(p_{n}^{-1}(I_{n})\right)$
by Lemma 9.2.3 $\displaystyle=i_{n}(I_{n})$

as desired. ∎

We can now prove the main algebraic theorem of this section.

###### Proof of Theorem 9.1.1.

Consider the sequence

$0\to(X_{r})_{\operatorname{\Lambda{\text{-}tor}}}\to X_{r}\to Z_{r}\to 0.$

Taking $\Gamma_{n}$-coinvariants yields

$0\to((X_{r})_{\operatorname{\Lambda{\text{-}tor}}})_{\Gamma_{n}}\to(X_{r})_{\Gamma_{n}}\to(Z_{r})_{\Gamma_{n}}\to
0$

since the kernel of the first map equals $Z_{r}^{\Gamma_{n}}=0$. Let $E_{n}$
denote the image of $H^{1}_{e}(K_{n,p},T(r))\oplus\\{0\\}$ in
$(X_{r})_{\Gamma_{n}}$ under (15), and let $F_{n}$ denote its image in
$(Z_{r})_{\Gamma_{n}}$. If $i_{n}$ denotes the natural map
$(Z_{r})_{\Gamma_{n}}\to\Lambda_{n}$, we recall that by definition
$i_{n}(F_{n})$ equals the image of $\psi_{n}$.

We have

$0\to\frac{((X_{r})_{\operatorname{\Lambda{\text{-}tor}}})_{\Gamma_{n}}}{((X_{r})_{\operatorname{\Lambda{\text{-}tor}}})_{\Gamma_{n}}\cap
E_{n}}\to\frac{(X_{r})_{\Gamma_{n}}}{E_{n}}\to\frac{(Z_{r})_{\Gamma_{n}}}{F_{n}}\to
0$

Note that (15) equates a quotient of the middle term of this sequence with
$H^{1}_{g}(K_{n},A^{*}_{r})^{\vee}$. Thus,

$\displaystyle\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}$
$\displaystyle(H^{1}_{g}(K_{n},A^{*}_{r})^{\vee})$
$\displaystyle\supseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}\frac{(X_{r})_{\Gamma_{n}}}{E_{n}}$
by Lemma 9.2.1.1
$\displaystyle\supseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}\left(\frac{((X_{r})_{\operatorname{\Lambda{\text{-}tor}}})_{\Gamma_{n}}}{((X_{r})_{\operatorname{\Lambda{\text{-}tor}}})_{\Gamma_{n}}\cap
E_{n}}\right)\cdot\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}\left(\frac{(Z_{r})_{\Gamma_{n}}}{F_{n}}\right)$
by Lemma 9.2.1.2
$\displaystyle\supseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}\left(((X_{r})_{\operatorname{\Lambda{\text{-}tor}}})_{\Gamma_{n}}\right)\cdot\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}\left(\frac{(Z_{r})_{\Gamma_{n}}}{F_{n}}\right)$
by Lemma 9.2.1.1
$\displaystyle=\pi_{n}(\mathop{\mathrm{Fitt}}\nolimits_{\Lambda}\left((X_{r})_{\operatorname{\Lambda{\text{-}tor}}}\right))\cdot\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}\left(\frac{(Z_{r})_{\Gamma_{n}}}{F_{n}}\right)$
by Lemma 9.2.1.3
$\displaystyle\supseteq\pi_{n}(\operatorname{char}_{\Lambda}\left((X_{r})_{\operatorname{\Lambda{\text{-}tor}}}\right))\cdot\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}\left(\frac{(Z_{r})_{\Gamma_{n}}}{F_{n}}\right)$
by Lemma 9.2.2
$\displaystyle\supseteq\pi_{n}(\operatorname{char}_{\Lambda}\left((X_{r})_{\operatorname{\Lambda{\text{-}tor}}}\right))\cdot
i_{n}(F_{n})$ by Lemma 9.2.4
$\displaystyle\supseteq\pi_{n}(\operatorname{char}_{\Lambda}\left((X_{r})_{\operatorname{\Lambda{\text{-}tor}}}\right))\cdot\psi_{n}(H^{1}_{e}(K_{n,p},T(r)))$

as desired.

We note that in applying Lemma 9.2.2, the fact that
$(X_{r})_{\operatorname{\Lambda{\text{-}tor}}}$ has no non-zero finite
submodules is a theorem of Greenberg (see [15, Proposition 4.1.1]). ∎

###  10\. Mazur-Tate elements and Kato’s Euler system

####  10.1. Mazur-Tate elements and statement of main results

Following [26], we define

$\lambda(f,z^{j},a,m):=2\pi i\int_{\infty}^{-a/m}f(z)(mz+a)^{j}~{}\\!dz.$

Setting
$\lambda^{\pm}(f,z^{j},a,m):=\lambda(f,z^{j},a,m)\pm\lambda(f,z^{j},a,m)$, we
further define

$\varphi(f,z^{j},a,m):=\frac{\lambda^{+}(f,z^{j},a,m)}{\Omega_{f}^{+}}+\frac{\lambda^{-}(f,z^{j},a,m)}{\Omega_{f}^{-}}.$

With this notation in hand, we define our Mazur-Tate elements of $f$ over
$K_{n}$:

$\theta^{\operatorname{an}}_{n,r}(f):=\sum_{\sigma_{a}\in\mathcal{G}_{n}}\varphi(f,z^{r-1},a,p^{n})\sigma_{a}^{-1}.$

For $\chi$ a primitive character of $\mathcal{G}_{n}$ we have

(18) $\chi(\theta^{\operatorname{an}}_{n,r}(f))=\chi(-1)\cdot(r-1)!\cdot
p^{n(r-1)}\cdot\tau(\chi^{-1})\cdot\frac{L(f,\chi,r)}{(-2\pi
i)^{r-1}\Omega_{f}^{\pm}}.$

where the sign $\pm$ equals the sign of $(-1)^{r-1}\chi(-1)$.

######  Remark 10.1.1.

Our definition here is a bit non-standard as Mazur and Tate’s original
definition would replace $\sigma_{a}^{-1}$ above with $\sigma_{a}$. That is,
if $\iota$ is the involution on $\Lambda_{n}$ sending $\sigma$ to
$\sigma^{-1}$, then in our notation above
$\theta^{\operatorname{an}}_{n,r}(f)^{\iota}$ is the more standard definition
of Mazur-Tate elements. Nonetheless, the methods of this paper naturally show
that our $\theta^{\operatorname{an}}_{n,r}(f)$ belong to a Fitting ideal of a
Selmer group and so we chose to state our results in this more natural form.

The Mazur-Tate elements satisfy the relations:

(19)
$\pi^{n+1}_{n}(\theta^{\operatorname{an}}_{n+1,r}(f))=\begin{cases}a_{p}(f)\cdot\theta^{\operatorname{an}}_{n,r}(f)-\psi(p)p^{k-2}\cdot\mathop{\mathrm{cor}}\nolimits^{n}_{n-1}(\theta^{\operatorname{an}}_{n-1,r}(f))&n\geq
1,\\\ ~{}\\\
(a_{p}-1-\psi(p)p^{k-2})\cdot\theta^{\operatorname{an}}_{0,r}(f)&n=0,\end{cases}$

where $\pi^{n+1}_{n}$ is the natural projection from $\Lambda_{n+1}$ to
$\Lambda_{n}$. These formulas follow from [26, (4.2)]. See also [34,
Proposition 2.5] for the case $r=1$.

We have the following conjecture relating these analytically defined Mazur-
Tate elements to Selmer groups.

######  Conjecture 10.1.2.

(Mazur-Tate [25]) For each $n\geq 0$, we have

$\theta^{\operatorname{an}}_{n,r}(f)\in\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(H^{1}_{g}(K_{n},A^{*}_{r})^{\vee}).$

We note that Mazur and Tate only formulated this conjecture for elliptic
curves, but in that formulation, the conjecture was much more general,
covering Selmer groups over all abelian extensions of $\mathbb{Q}$.

######  Remark 10.1.3.

The functional equation for $L$-values gives an equality

$\theta^{\operatorname{an}}_{n,r}(f)=C\cdot\theta^{\operatorname{an}}_{n,k-r}(\overline{f})^{\iota}$

where $C$ is some constant. Thus, it may be equally reasonable to conjecture
that $\theta^{\operatorname{an}}_{n,k-r}(\overline{f})^{\iota}$ belongs to the
above Fitting ideal. Moreover, since
$A^{*}_{r}=A^{*}(1-r)\cong\overline{A}(k-r)$, that second conjecture
equivalently reads

$\theta^{\operatorname{an}}_{n,r}(f)^{\iota}\in\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(H^{1}_{g}(K_{n},A(r))^{\vee})$

which might be a more familiar form of this conjecture for some (noting the
presence of $\iota$ is caused by our non-standard definition of the Mazur-Tate
element as in Remark 10.1.1).

However, as the above constant $C$ need not be integral, neither conjecture is
stronger nor weaker than the other. We do note that for $n$ large enough, the
constant $C$ is a $p$-adic unit and thus the two conjectures (for $n$ large)
are equivalent. Also, showing that these Fitting ideals were stable under the
$\iota$-involution (which seems very plausible), would give a neat explanation
of the equivalence of the two conjectures.

We set $I^{\operatorname{an}}_{n}(f)$ to be the ideal of $\Lambda_{n}$
generated by the elements
$\mathop{\mathrm{cor}}\nolimits^{n}_{m}(\theta^{\operatorname{an}}_{m,r}(f))$
for $0\leq m\leq n$. The remainder of the paper will be devoted to proving the
following divisibility.

######  Theorem 10.1.4.

If (Irred), (No pole at $s=r$), and (Euler) hold, then there exists a non-zero
constant $C\in\mathcal{O}_{L}$ (independent of $n$ and $r$) such that

$C\cdot I^{\operatorname{an}}_{n,r}(f)\subseteq
I^{\operatorname{alg}}_{n,r}(f)$

for all $n\geq 0$.

From Corollary 9.1.2 and Theorem 10.1.4, we get our main result.

######  Corollary 10.1.5.

If (Irred), (No pole at $s=r$), and (Euler) hold, then there exists a non-zero
constant $C\in\mathcal{O}_{L}$ (independent of $n$ and $r$) such that

$C\cdot
I^{\operatorname{an}}_{n,r}(f)\subseteq\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(H^{1}_{g}(K_{n},A^{*}_{r})^{\vee}).$

for all $n\geq 0$. In particular,
$C\cdot\theta^{\operatorname{an}}_{n,r}(f)\in\mathop{\mathrm{Fitt}}\nolimits_{\Lambda_{n}}(H^{1}_{g}(K_{n},A^{*}_{r})^{\vee})$
for all $n\geq 0$.

####  10.2. Kato’s Euler system

In this section, we describe how to recover the Mazur-Tate elements from
Kato’s Euler system and the local classes $c_{n,j}$ constructed in section
7.3.

Let
$z_{\operatorname{K}}=(z_{\operatorname{K},n})_{n}\in\mathbb{H}^{1}(T^{*}_{r})=\mathbb{H}^{1}(\overline{T}(k-r))$
denote Kato’s Euler system where we recall that $\overline{T}$ is a lattice in
the Galois representation attached to $\overline{f}$, the complex conjugate of
$f$. We now state Kato’s theorem relating $z_{\operatorname{K}}$ to
$L$-values. To this end, let $\mathbf{e}$ continue to denote our fixed basis
from section 7.1 of
$\mathbf{D}_{\mathrm{dR}}(V(1))/\mathbf{D}^{+}_{\mathrm{dR}}(V(1))$, and let
$\omega_{f}$ denote the element of $\mathbf{D}_{\mathrm{dR}}^{+}(V^{*})$
canonically identified with $\overline{f}$, via Eichler–Shimura theory and the
comparison theorems.

######  Theorem 10.2.1 (Kato).

Let $\chi$ denote a primitive character of $\mathcal{G}_{n}$. Then

$\sum_{\sigma\in\mathcal{G}_{n}}\chi(\sigma)\exp^{*}(\sigma(z_{\operatorname{K},n}))=\dfrac{L_{\\{p\\}}(f,\chi,r)}{(-2\pi
i)^{r-1}\Omega_{f}^{\pm}}\cdot t^{r-1}\omega_{f}$

where the sign $\pm$ equals the sign of $\chi(-1)$ and $L_{\\{p\\}}$ denotes
the $L$-value with the Euler factor at $p$ removed.

###### Proof.

See [20, Theorem 12.5]. ∎

The following proposition shows how one can recover Mazur-Tate elements from
Kato’s Euler system and the local classes $c_{n,j}$.

######  Proposition 10.2.2.

If (No pole at $s=r$) holds, then there exists a non-zero constant
$C\in\mathcal{O}_{L}$ (independent of $n$ and $r$) such that

$C\cdot\theta^{\operatorname{an}}_{n,r}(f)=\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n})\right\rangle_{n}\sigma^{-1}.$

###### Proof.

Set

$\theta_{n,r}=\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n})\right\rangle_{n}\sigma^{-1}.$

To prove this proposition it suffices to check that the $\theta_{n,r}$ satisfy
the same interpolation property (up to a constant) as the Mazur-Tate elements
(e.g. (18)) for primitive characters, and that they satisfy the same three-
term relation (e.g. (19)).

To this end, for $n>0$, we have

$\displaystyle\chi(\theta_{n,r})$
$\displaystyle=\sum_{\sigma\in\mathcal{G}_{n}}\chi(\sigma)^{-1}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n})\right\rangle_{n}$
$\displaystyle=(r-1)!\cdot
p^{m(r-1)}\cdot\tau(\chi^{-1})\cdot\sum_{\sigma\in\mathcal{G}_{n}}\chi(\sigma)\big{\langle}t^{1-r}\mathbf{e},\exp^{*}(z_{\operatorname{K},n}^{\sigma})\big{\rangle}_{\mathrm{dR},n}$
$\displaystyle\quad[\text{Proposition~{}}\ref{prop:recip}]$
$\displaystyle=\chi(-1)\cdot(r-1)!\cdot
p^{m(r-1)}\cdot\tau(\chi^{-1})\cdot\dfrac{L_{\\{p\\}}(f,\chi,r)}{(2\pi
i)^{r-1}\Omega_{f}^{\pm}}\langle\mathbf{e},\omega_{f}\rangle_{\mathrm{dR}}$
$\displaystyle\quad[\text{Proposition~{}}\ref{thm:katorecip}]$
$\displaystyle=\chi(\theta^{\operatorname{an}}_{n,r}(f))\cdot\langle\mathbf{e},\omega_{f}\rangle_{\mathrm{dR}}.$

since the local Euler factor at $p$ is trivial for $L(f,\chi,s)$. Note that
the application of Proposition 7.4.1 requires the hypothesis (No pole at
$s=r$).

For $n=0$, we have

$\displaystyle{\bf 1}(\theta_{0,r})$ $\displaystyle=\left\langle
c_{0,r},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},0})\right\rangle_{0}$
$\displaystyle=(r-1)!\cdot\widetilde{Z}(x^{1-r}\phi_{v_{\mathrm{new}}},1)\left\langle
t^{1-r}\mathbf{e},\exp^{*}z_{0}^{\prime}\right\rangle_{\mathrm{dR},0}$
$\displaystyle\quad[\text{Proposition~{}}\ref{prop:recip}]$
$\displaystyle=(r-1)!\cdot\widetilde{Z}(x^{1-r}\phi_{v_{\mathrm{new}}},1)\cdot\dfrac{L_{\\{p\\}}(f,r)}{\Omega_{f}^{+}}\langle\mathbf{e},\omega_{f}\rangle_{\mathrm{dR}}$
$\displaystyle\quad[\text{Proposition~{}}\ref{thm:katorecip}]$
$\displaystyle=(r-1)!\cdot\dfrac{L(f,r)}{\Omega_{f}^{+}}\langle\mathbf{e},\omega_{f}\rangle_{\mathrm{dR}}$
$\displaystyle={\bf
1}(\theta^{\operatorname{an}}_{0,r}(f))\cdot\left\langle\mathbf{e},\omega_{f}\right\rangle_{\mathrm{dR}}.$

Here $\phi_{v_{\mathrm{new}}}$ is the newvector associated to $f$ and
$\widetilde{Z}(x^{1-r}\phi_{v_{\mathrm{new}}},1)$ is the simply the local
Euler factor at $p$ for $L(f,r)$. Thus, $\theta_{n,r}$ and
$\theta^{\operatorname{an}}_{n,r}(f)$ agree at primitive characters (up to a
constant).

To finish the proof, we compute

$\displaystyle\pi^{n+1}_{n}(\theta_{n,r})=\pi^{n+1}_{n}\left(\sum_{\tau\in\mathcal{G}_{n+1}}\left\langle
c_{n+1,r-1}^{\tau},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n+1})\right\rangle_{n+1}\tau^{-1}\right)$
$\displaystyle=\sum_{\sigma\in\mathcal{G}_{n}}\left(\sum_{\stackrel{{\scriptstyle\tau\in\mathcal{G}_{n+1}}}{{\tau\to\sigma}}}\left\langle
c_{n+1,r-1}^{\tau},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n+1})\right\rangle_{n+1}\right)\sigma^{-1}$
$\displaystyle=\sum_{\sigma\in\mathcal{G}_{n}}\left\langle\mathop{\mathrm{cor}}\nolimits^{n+1}_{n}(c_{n+1,r-1}),\mathop{\mathrm{cor}}\nolimits^{n+1}_{n}(\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n+1}))\right\rangle_{n}\sigma^{-1}$
$\displaystyle=\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
a_{p}(f)c_{n,r-1}^{\sigma}-\psi(p)p^{k-2}\mathop{\mathrm{res}}\nolimits^{n}_{n-1}(c_{n-1,r-1}^{\sigma}),\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n})\right\rangle_{n}\sigma^{-1}$
$\displaystyle=a_{p}(f)\cdot\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n})\right\rangle_{n}\sigma^{-1}-\psi(p)p^{k-2}\sum_{\sigma\in\mathcal{G}_{n}}\left\langle\mathop{\mathrm{res}}\nolimits^{n}_{n-1}(c_{n-1,r-1}^{\sigma}),\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n})\right\rangle_{n}\sigma^{-1}$
$\displaystyle=a_{p}(f)\cdot\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n})\right\rangle_{n}\sigma^{-1}-\psi(p)p^{k-2}\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n-1,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n-1})\right\rangle_{n-1}\sigma^{-1}$
$\displaystyle=a_{p}(f)\cdot\theta_{n,r}-\psi(p)p^{k-2}\cdot\mathop{\mathrm{cor}}\nolimits_{n-1}^{n}(\theta_{n-1,r})$

as desired. Here we used Proposition 7.5.2. A similar computation handles the
case $n=0$ as well. ∎

###  11\. Algebraic $\theta$-elements, take II

Throughout this section we assume that (Irred) holds.

####  11.1. An alternative construction of $\theta$-elements

In section 8, we gave a construction of algebraic $\theta$-elements working
with the Galois cohomology of the discrete module $A^{*}_{r}$ which allowed us
to relate these elements to the Fitting ideals of its Selmer group. We now
give a simpler and more direct construction of these elements using the
cohomology of $T^{*}_{r}$. The advantage of this second construction is that
it will be easier to relate to Mazur-Tate elements (via Kato’s results) and
thus to prove Theorem 10.1.4. However, showing that these two constructions
match then takes additional work, and will be carried out in this subsection.

By Theorem 8.2.2, let $w=(w_{n})\in\mathbb{H}^{1}(T^{*}_{r})$ denote some
generator of this free $\Lambda$-module of rank 1. For $c\in
H^{1}_{e}(K_{n,p},T(r))$, set

$\widetilde{\psi}_{n}(c):=\widetilde{\psi}_{n,w}(c):=\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(w_{n})\right\rangle_{n}\sigma\in\Lambda_{n}$

where $\mathop{\mathrm{res}}\nolimits_{p}$ denotes restriction to the unique
place over $p$. The pairing $\langle\cdot,\cdot\rangle_{n}$ is given by Tate
local duality

$H^{1}(K_{n,p},T(r))\times H^{1}(K_{n,p},T^{*}_{r})\to\mathcal{O}_{L}.$

####  11.2. Comparing the two constructions

Let $\iota$ be the involution of $\Lambda_{n}$ which sends a group-like
element $\sigma$ to $\sigma^{-1}$. We will see that the above simple and
direct definition of $\widetilde{\psi}_{n}$ yields a function which, up to
$\iota$, equals $\psi_{n}$ (which was the key input in defining algebraic
$\theta$-elements). However, to make this claim precise, recall that the
definition of $\psi_{n}=\psi_{n,\alpha}$ depended on an isomorphism
$\alpha:R_{\Lambda}(X_{r})\cong\Lambda$ while the definition of
$\widetilde{\psi}_{n}=\widetilde{\psi}_{n,w}$ depended on a choice of a
generator $w$ of $\mathbb{H}^{1}(T^{*}_{r})$. Thus, to relate $\psi_{n}$ to
$\widetilde{\psi}_{n}$, we must first relate these choices.

We now show how to use our fixed $w=(w_{n})\in\mathbb{H}^{1}(T^{*}_{r})$ to
give an explicit identification of $R_{\Lambda}(X_{r})$ with $\Lambda$. Set
$H_{r}=H^{1}(\mathbb{Q}_{\Sigma}/K_{\infty},A^{*}_{r})$ and
$X_{r}^{\Sigma}:=H_{r}^{\vee}$. Dualizing (16) yields

$W\to X_{r}^{\Sigma}\to X_{r}\to 0$

where $W$ is some finitely generated torsion $\Lambda$-module. Thus, applying
$\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(\cdot,\Lambda)$ gives a canonical
isomorphism

$\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(X_{r},\Lambda)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(X_{r}^{\Sigma},\Lambda)$

and hence a canonical isomorphism $R_{\Lambda}(X_{r})\cong
R_{\Lambda}(X_{r}^{\Sigma})$. It therefore suffices to make an identification
of $R_{\Lambda}(X_{r}^{\Sigma})$ with $\Lambda$ which we now do.

From (Irred), we deduce that
$H^{0}(\mathbb{Q}_{p}(\mu_{p}),\overline{\rho}_{f})=0$, and thus the control
theorem holds for $H_{r}$; that is, the restriction map

$r_{n}:H^{1}(\mathbb{Q}_{\Sigma}/K_{n},A^{*}_{r})\to H_{r}^{\Gamma_{n}}$

is an isomorphism for any $n$. Let

$\pi_{n}^{m}:H^{1}(\mathbb{Q}_{\Sigma}/K_{n},T^{*}_{r})\to
H^{1}(\mathbb{Q}_{\Sigma}/K_{n},T^{*}_{r}/p^{m}T^{*}_{r})\cong
H^{1}(\mathbb{Q}_{\Sigma}/K_{n},A^{*}_{r}[p^{m}])\overset{r_{n}}{\to}H_{r}^{\Gamma_{n}}[p^{m}]$

denote the canonical map. Define a $\Lambda$-homomorphism

$\xi_{w}:X_{r}^{\Sigma}\to\Lambda=\underset{n}{\varprojlim}\,\Lambda_{n}$

given, for $\eta:H_{r}\to\mathbb{Q}_{p}/\mathbb{Z}_{p}$, by
$\xi_{w}(\eta)=\left(\sum_{\sigma\in\mathcal{G}_{n}}a_{\sigma}\sigma\right)_{n}$
where

$a_{\sigma}=\varprojlim_{m\geq
n}p^{m}\eta\bigr{(}\pi_{n}^{m}(w_{n}^{\sigma})\bigl{)}\in\mathbb{Z}_{p}.$

Note that $\eta(\pi_{n}^{m}(w_{n}^{\sigma}))$ lies in
$\frac{1}{p^{m}}\mathbb{Z}_{p}/\mathbb{Z}_{p}$, so that the $m^{\text{th}}$
term in this limit lies in $\mathbb{Z}/p^{m}\mathbb{Z}$.

######  Lemma 11.2.1.

$\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(X_{r}^{\Sigma},\Lambda)$ is free of
rank one over $\Lambda$ with generator $\xi_{w}$.

###### Proof.

We have the following $\Lambda$-module homomorphisms:

$\displaystyle\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(X_{r}^{\Sigma},\Lambda)$
$\displaystyle\cong\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(\Lambda^{\vee},(X_{r}^{\Sigma})^{\vee})\cong\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(\Lambda^{\vee},H_{r})\cong\mathop{\mathrm{Hom}}\nolimits_{\Lambda}\bigl{(}\underset{n}{\varinjlim}\,(\Lambda^{\vee})^{\Gamma_{n}}[p^{n}],H_{r}\bigr{)}$
$\displaystyle\cong\underset{n}{\varprojlim}\,\mathop{\mathrm{Hom}}\nolimits_{\Lambda}\bigl{(}(\Lambda^{\vee})^{\Gamma_{n}}[p^{n}],H_{r}\bigr{)}\cong\underset{n}{\varprojlim}\,\mathop{\mathrm{Hom}}\nolimits_{\Lambda}\bigl{(}(\Lambda_{n}/p^{n})^{\vee},H_{r}\bigr{)}$
$\displaystyle\cong\underset{n}{\varprojlim}\,\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(\Lambda_{n}/p^{n},H_{r})\cong\underset{n}{\varprojlim}\,\,H_{r}[p^{n}]^{\Gamma_{n}}\cong\mathbb{H}^{1}(T^{*}_{r}).$

Above we used the self-duality

$\displaystyle\Lambda_{n}/p^{n}\times\Lambda_{n}/p^{n}$
$\displaystyle\to\mathbb{Z}/p^{n}$ $\displaystyle\left<\sum
a_{\sigma}\sigma,\sum b_{\sigma}\sigma\right>$ $\displaystyle\mapsto\sum
a_{\sigma}b_{\sigma^{-1}}$

as this is the duality which induces a $\Lambda$-module isomorphism:
$(\Lambda_{n}/p^{n})^{\vee}\cong\Lambda_{n}/p^{n}$. To prove this lemma, one
then chases through the above maps to see that $\xi_{w}$ in
$\mathop{\mathrm{Hom}}\nolimits_{\Lambda}(X_{r}^{\Sigma},\Lambda)$ maps to the
generator $w$ in $\mathbb{H}^{1}(T^{*}_{r})$. ∎

######  Corollary 11.2.2.

The map

$\Xi_{w}:R_{\Lambda}(X_{r})\stackrel{{\scriptstyle\sim}}{{\to}}R_{\Lambda}(X_{r}^{\Sigma})\to\Lambda$

given by evaluation at $\xi_{w}$ is an isomorphism.

We have thus built an explicit identification of $R_{\Lambda}(X_{r})$ with
$\Lambda$ given a generator of $\mathbb{H}^{1}(T^{*}_{r})$.

######  Theorem 11.2.3.

For $c\in H^{1}_{e}(K_{n,p},T(r))$, we have

$\psi_{n,\Xi_{w}}(c)=\widetilde{\psi}_{n,w}(c)^{\iota}$

In particular,

$\theta^{\operatorname{alg}}_{n,r}(f)=\left(\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(w_{n})\right\rangle_{n}\sigma^{-1}\right)\cdot\operatorname{char}_{\Lambda}((X_{r})_{\operatorname{\Lambda{\text{-}tor}}})$

###### Proof.

Let $\Xi_{w,n}:X_{\Gamma_{n}}\to\Lambda_{n}$ denote the
$\Gamma_{n}$-coinvariants of the composition of the canonical map
$\mathrm{ev}:X_{r}\to R_{\Lambda}(X_{r})$ and $\Xi_{w}$. Recall the
isomorphisms:

(20)
$(X_{r}^{\Sigma})_{\Gamma_{n}}=(H_{r}^{\vee})_{\Gamma_{n}}\overset{\sim}{\longrightarrow}(H_{r}^{\Gamma_{n}})^{\vee}\overset{\sim}{\longrightarrow}H^{1}(\mathbb{Q}_{\Sigma}/K_{n},A^{*}_{r})^{\vee}.$

Thus, we may define an element in $(X_{r}^{\Sigma})_{\Gamma_{n}}$ by defining
it as the functional on $H^{1}(\mathbb{Q}_{\Sigma}/K_{n},A^{*}_{r})$.

For $c\in H^{1}_{e}(K_{n,p},T(r))$, consider the element
$\varphi^{\Sigma}_{c}\in(X_{r}^{\Sigma})_{\Gamma_{n}}$ defined by first
restricting to $p$ and then pairing (via Tate local duality) against $c$. That
is, for $h\in H^{1}(\mathbb{Q}_{\Sigma}/K_{n},A^{*}_{r})$, set

$\varphi^{\Sigma}_{c}(h)=\left<c,\mathop{\mathrm{res}}\nolimits_{p}(h)\right>.$

Set $\varphi_{c}\in(X_{r})_{\Gamma_{n}}$ equal to the image of
$\varphi^{\Sigma}_{c}$ under the natural surjection $X_{r}^{\Sigma}\to X_{r}$.
Then, by definition, $\psi_{n,\Xi_{w}}(c)=\Xi_{w,n}(\varphi_{c})$.

However, we can also compute $\Xi_{w,n}(\varphi_{c})$ directly as the image of
$\xi_{w}(\varphi^{\Sigma}_{c})$ in $\Lambda_{n}$. By the definition of
$\xi_{w}$, this image equals $\sum_{\sigma\in G_{n}}b_{\sigma}\sigma$ where

$b_{\sigma}=\varprojlim_{m\geq
n}p^{m}\left<c,\mathop{\mathrm{res}}\nolimits_{p}(\pi_{n}^{m}(w_{n}^{\sigma}))\right>_{n}.$

But, by definition, this limit equals

$\left<c,\mathop{\mathrm{res}}\nolimits_{p}(w_{n}^{\sigma})\right>_{n}=\left<c^{\sigma^{-1}},\mathop{\mathrm{res}}\nolimits_{p}(w_{n})\right>_{n}$

as desired. ∎

###  12\. The main results

In light of Proposition 10.2.2 and Theorem 11.2.3, to compare
$\theta^{\operatorname{an}}_{n,r}(f)$ and
$\theta^{\operatorname{alg}}_{n,r}(f)$ we need to understand the index of
Kato’s Euler system in $\mathbb{H}^{1}(T^{*}_{r})$ compared to
$\operatorname{char}_{\Lambda}X_{\operatorname{\Lambda{\text{-}tor}}}$. This
relation will follow from Kato’s proof of half of the main conjecture (without
$p$-adic $L$-functions).

####  12.1. Euler system result

Let $\mathbb{H}^{i}(T)=\varprojlim_{n}H^{i}(\mathbb{Q}_{\Sigma}/K_{n},T)$ and
$\mathbb{H}^{i}_{\mathrm{Iw},w}(T)=\varprojlim_{n}H^{i}(K_{n,v_{n}},T)$ where
$w$ is some place of $\mathbb{Q}_{\infty}$ and $v_{n}$ is the place of $K_{n}$
below $w$. Further, set

$\mathbb{H}^{2}_{P}(T)=\ker\left(\mathbb{H}^{2}(T)\to\prod_{w\in\Sigma_{\infty}}\mathbb{H}^{2}_{\mathrm{Iw},w}(T)\right)$

where $\Sigma_{\infty}$ is the set of places of $K_{\infty}$ sitting over
$\Sigma$.

######  Theorem 12.1.1 (Kato).

We have

  1. (1)

$\mathbb{H}^{2}_{P}(T^{*}_{r})$ is a torsion $\Lambda$-module;

  2. (2)

if (Euler) holds,

$\operatorname{char}\mathbb{H}^{2}_{P}(T^{*}_{r})\text{~{}divides~{}}\operatorname{char}_{\Lambda}(\mathbb{H}^{1}(T^{*}_{r})/\langle
z_{\operatorname{K}}\rangle).$

###### Proof.

This is [20, Theorems 12.4 and 12.5]. See also [24, pg. 217] where the étale
cohomology groups in [20] are written in terms of Galois cohomology with local
conditions (e.g. $\mathbb{H}^{2}_{P}$). ∎

We must now relate $\mathbb{H}^{2}_{P}(T^{*}_{r})$ to
$(X_{r})_{\operatorname{\Lambda{\text{-}tor}}}$. We begin with a lemma. If $M$
is a $\Lambda$-module, we write $M^{\iota}$ for the $\Lambda$-module whose
underlying set is $M$, but the group-like element $\sigma$ acts by
$\sigma^{-1}$.

######  Lemma 12.1.2.

If $X$ is a finitely generated torsion $\Lambda$-module, then
$\mathrm{Ext}^{1}_{\Lambda}(X,\Lambda)$ is pseudo-isomorphic to
$(X_{\operatorname{\Lambda{\text{-}tor}}})^{\iota}$.

###### Proof.

The exact sequence

$0\to X_{\operatorname{\Lambda{\text{-}tor}}}\to X\to Z\to 0$

induces

$\mathrm{Ext}^{1}_{\Lambda}(Z,\Lambda)\to\mathrm{Ext}^{1}_{\Lambda}(X,\Lambda)\to\mathrm{Ext}^{1}_{\Lambda}(X_{\operatorname{\Lambda{\text{-}tor}}},\Lambda)\to\mathrm{Ext}^{2}_{\Lambda}(Z,\Lambda).$

We first claim that $\mathrm{Ext}^{i}_{\Lambda}(Z,\Lambda)$ is pseudo-null for
$i=1,2$.

To see this, let $R_{\Lambda}(Z)$ denote the reflective-hull of $Z$ and
consider the exact sequence

$0\to Z\to R_{\Lambda}(Z)\to H\to 0$

with $H$ finite. We then have

$0=\mathrm{Ext}^{i}_{\Lambda}(R_{\Lambda}(Z),\Lambda)\to\mathrm{Ext}^{i}_{\Lambda}(Z,\Lambda)\to\mathrm{Ext}^{i}_{\Lambda}(H,\Lambda)$

since $R_{\Lambda}(Z)$ is free. So it suffices to show that
$\mathrm{Ext}^{i}_{\Lambda}(H,\Lambda)$ is finite for $i=1,2$. To see this,
note that $\operatorname{Ann}_{\Lambda}(H)$ is finite-index in $\Lambda$ and
thus the same is true for the annihilator of
$\mathrm{Ext}^{i}_{\Lambda}(H,\Lambda)$. But since this module is finitely-
generated, it must then be finite as desired.

Thus, $\mathrm{Ext}^{1}_{\Lambda}(X,\Lambda)$ is pseudo-isomorphic to
$\mathrm{Ext}^{1}_{\Lambda}(X_{\operatorname{\Lambda{\text{-}tor}}},\Lambda)$.
Lastly, by [37, pg. 474], $\mathrm{Ext}^{1}_{\Lambda}(Y,\Lambda)$ is pseudo-
isomorphic to $Y^{\iota}$ for any finitely generated $\Lambda$-module which
completes the proof. ∎

######  Theorem 12.1.3.

Assume (Irred) holds. If $X_{\operatorname{\Lambda{\text{-}tor}}}$ denotes the
$\Lambda$-torsion submodule of $X$, we have

$\operatorname{char}_{\Lambda}(X_{r})_{\operatorname{\Lambda{\text{-}tor}}}=\operatorname{char}_{\Lambda}(\mathbb{H}^{2}_{P}(T^{*}_{r})^{\iota}).$

###### Proof.

By the two exact sequences in [28] appearing right after (0.13.2), we have a
pseudo-isomorphism

$(\widetilde{H}^{1}_{f,\mathrm{Iw}}(A^{*}_{r})^{\vee})_{\operatorname{\Lambda{\text{-}tor}}}\cong\mathrm{Ext}^{1}_{\Lambda}(\widetilde{H}^{2}_{f,\mathrm{Iw}}(\mathbb{Q}_{\infty}/\mathbb{Q},T^{*}_{r}),\Lambda)$

Here $\widetilde{H}^{i}_{f,\mathrm{Iw}}$ are Selmer complexes where, for
$A^{*}_{r}$, there is no local condition at $p$ (and thus for $T^{*}_{r}$ we
impose the harshest local condition). Relating these back to classical Selmer
groups (as in [28, (0.10)], we have a pseudo-isomorphism

$X_{\operatorname{\Lambda{\text{-}tor}}}\cong\mathrm{Ext}^{1}_{\Lambda}(\mathbb{H}^{2}_{P}(T^{*}_{r}),\Lambda).$

By Theorem 12.1.1 and Lemma 12.1.2, we then have a pseudo-isomorphisms

$X_{\operatorname{\Lambda{\text{-}tor}}}\cong\mathbb{H}^{2}_{P}(T^{*}_{r})^{\iota}$

as desired. ∎

####  12.2. Divisibility

We begin with a simple lemma whose proof we leave to the reader.

######  Lemma 12.2.1.

For $\lambda\in\Lambda$, we have

$\lambda\cdot\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(w_{n})\right\rangle_{n}\sigma=\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(\lambda
w_{n})\right\rangle_{n}\sigma.$

We conclude with our main theorem which immediately implies Theorem 10.1.4.

######  Theorem 12.2.2.

If (Irred), (No pole at $s=r$), and (Euler) hold, then there exists a non-zero
constant $C\in\mathcal{O}_{L}$ (independent of $n$ and $r$) such that

$\theta^{\operatorname{alg}}_{n,r}(f)~{}|~{}C\cdot\theta^{\operatorname{an}}_{n,r}(f)$

in $\Lambda_{n}$ for all $n\geq 0$.

###### Proof.

Recall, $w=(w_{n})$ is our fixed generator of $\mathbb{H}^{1}(T^{*}_{r})$. By
Theorem 11.2.3, we have

$\theta^{\operatorname{alg}}_{n,r}(f)=\operatorname{char}_{\Lambda}\left((X_{r})_{\operatorname{\Lambda{\text{-}tor}}}\right)\cdot\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(w_{n})\right\rangle_{n}\sigma^{-1}$

Applying the involution $\iota:\Lambda_{n}\to\Lambda_{n}$ which sends group-
like elements to their inverse, yields

(21) $\displaystyle\iota(\theta^{\operatorname{alg}}_{n,r}(f))$
$\displaystyle=\operatorname{char}_{\Lambda}\mathbb{H}_{P}^{2}(T^{*}_{r})\cdot\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(w_{n})\right\rangle_{n}\sigma$

by Theorem 12.1.3. Now write $z_{\operatorname{K}}=\alpha\cdot w$ for some
$\alpha\in\Lambda$. Thus,
$\alpha=\operatorname{char}_{\Lambda}(\mathbb{H}^{1}(T^{*}_{r})/\langle
z_{K}\rangle)$, and
$\operatorname{char}_{\Lambda}\mathbb{H}^{2}_{P}(T^{*}_{r})$ divides $\alpha$
by Theorem 12.1.1.

Scaling (21) by $\alpha$, applying Lemma 12.2.1 and then Proposition 10.2.2
gives

$\displaystyle\alpha\cdot\iota(\theta^{\operatorname{alg}}_{n,r}(f))$
$\displaystyle=\operatorname{char}_{\Lambda}\mathbb{H}_{P}^{2}(T^{*}_{r})\cdot\sum_{\sigma\in\mathcal{G}_{n}}\left\langle
c_{n,r-1}^{\sigma},\mathop{\mathrm{res}}\nolimits_{p}(z_{\operatorname{K},n})\right\rangle_{n}\sigma$
$\displaystyle=\operatorname{char}_{\Lambda}\mathbb{H}_{P}^{2}(T^{*}_{r})\cdot
C\cdot\iota(\theta^{\operatorname{an}}_{n,r}(f))$

Thus

$\iota(\theta^{\operatorname{alg}}_{n,r}(f))\cdot\frac{\alpha}{\operatorname{char}_{\Lambda}\mathbb{H}^{2}_{P}(T^{*}_{r})}=C\cdot\iota(\theta^{\operatorname{an}}_{n,r}(f))$

for some non-zero $C$ independent of $n$ and $r$. In particular,
$\theta^{\operatorname{alg}}_{n,r}(f)$ divides
$C\cdot\theta^{\operatorname{an}}_{n,r}(f)$ in $\Lambda_{n}$. ∎

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