# Congruences for Siegel modular forms of nonquadratic nebentypus mod $p$

Siegfried Böcherer and Toshiyuki Kikuta

###### Abstract

We prove that weights of two Siegel modular forms of nonquadratic nebentypus
should satisfy some congruence relations if these modular forms are congruent
to each other. Applying this result, we prove that there are no mod $p$
singular forms of nonquadratic nebentypus. Here we consider the case where the
Fourier coefficients of the modular forms are algebraic integers, and we
emphasize that $p$ is a rational prime. Moreover, we construct some examples
of mod $\mathfrak{p}$ singular forms of nonquadratic nebentypus using the
Eisenstein series studied by Takemori.

2020 Mathematics subject classification: Primary 11F33 $\cdot$ Secondary 11F46
Key words: mod $p^{m}$ singular, congruences for modular forms.

##  1 Introduction

Serre [15] proved that the congruence $f_{1}\equiv f_{2}$ mod $p^{r}$
($f_{1}\not\equiv 0$ mod $p$) implies $k_{1}\equiv k_{2}$ mod $(p-1)p^{r-1}$,
where $p$ is an odd prime, $f_{i}$ ($i=1$, $2$) are modular forms for ${\rm
SL}_{2}(\mathbb{Z})$ of weights $k_{i}$. Later, this property was extended by
Katz [8] to the case of congruence subgroups of the type $\Gamma_{1}(N)$ with
$p\nmid N$. In Rasmussen [11], the case with algebraic integer Fourier
coefficients is studied in detail, and the case of the group
$\Gamma_{0}(p)\cap\Gamma_{1}(N)$ is also considered.

All of the above results are for the case of elliptic modular forms. On the
other hand, a generalization to the case of Siegel modular forms was given by
Ichikawa [6] in an algebraic geometrical way and later given in an elementary
way by the first author and Nagaoka [4]. Here, the groups which they
considered are of types $\Gamma(N)$ and $\Gamma_{1}(N)$ respectively ($p\nmid
N$).

The first goal of this paper is to generalize the results mentioned above to
the case of Siegel modular forms $F_{i}$ for a group
$\Gamma_{0}(p^{m})\cap\Gamma_{1}(N)$ ($m\geq 1$, $p\nmid N$) having a
nebentypus characters $\chi_{i}$ mod $p^{m}$. The impetus for considering this
congruence for weights in the case of nonquadratic nebentypus was to see if it
could be applied to discuss the existence of a mod $p$ singular modular form
of nonquadratic nebentypus. We explain this more precisely.

Over $\mathbb{C}$, singular Siegel modular forms are defined by the
nonvanishing of all their Fourier coefficients of maximal rank; the properties
of such modular forms were extensively studied by Freitag (see e.g. [5]); in
particular, he showed that there are no such modular forms for $\Gamma_{0}(N)$
with nonquadratic character $\chi$ mod $N$.

In the works of the authors [1, 2] on mod $p$ singular forms we only
considered the case of quadratic characters. By applying our results on
congruence with respect to weights as above, we find that there is no mod $p$
singular form of nonquadratic nebentypus. Here we consider the case where the
Fourier coefficients of the modular forms are algebraic integers, and we
emphasize that $p$ is rational prime. In fact, there exists a mod
$\mathfrak{p}$ singular modular form of nonqudratic nebentypus. We will show
that it can be constructed from the Eisenstein series of nebentypus in some
cases.

##  2 Preliminaries

###  2.1 Siegel modular forms

Let $n$ be a positive integer and $\mathbb{H}_{n}$ the Siegel upper half space
of degree $n$ defined as

$\mathbb{H}_{n}:=\left\\{X+iY\;|\;X,\ Y\in\mathrm{Sym}_{n}(\mathbb{R}),\
Y>0\right\\},$

where $Y>0$ means that $Y$ is positive definite, and $\mathrm{Sym}_{n}(R)$ is
the set of symmetric matrices of size $n$ with components in $R$.

The Siegel modular group $\Gamma_{n}$ of degree $n$ is defined by

$\Gamma_{n}:=\left\\{\gamma\in\mathrm{GL}_{2n}(\mathbb{Z})\;|\;{}^{t}\gamma
J_{n}\gamma=J_{n}\right\\},$

where $J_{n}=\left(\begin{smallmatrix}0_{n}&-1_{n}\\\
1_{n}&0_{n}\end{smallmatrix}\right)$ and $0_{n}$ (resp. $1_{n}$) is the zero
matrix (resp. the identify matrix) of size $n\times n$.

We define an action of $\Gamma_{n}$ on $\mathbb{H}_{n}$ by $\gamma
Z:=(AZ+B)(CZ+D)^{-1}$ for $Z\in\mathbb{H}_{n}$,
$\gamma=\left(\begin{smallmatrix}A&B\\\
C&D\end{smallmatrix}\right)\in\Gamma_{n}$. For a holomorphic function
$F:\mathbb{H}_{n}\longrightarrow\mathbb{C}$ and a matrix
$\gamma=\left(\begin{smallmatrix}A&B\\\
C&D\end{smallmatrix}\right)\in\Gamma_{n}$, we define a slash operator by

$F|_{k}\;\gamma:=\det(CZ+D)^{-k}F(\gamma Z).$

Let $N$ be a positive integer. In this paper, we deal with congruence
subgroups of $\Gamma_{n}$ defined as

$\displaystyle\Gamma_{1}(N):=\left\\{\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\Gamma_{n}\>\Big{|}\>C\equiv 0_{n}\bmod{N},\ A\equiv
D\equiv 1_{n}\bmod{N}\right\\},$
$\displaystyle\Gamma_{0}(N):=\left\\{\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\Gamma_{n}\>\Big{|}\>C\equiv 0_{n}\bmod{N}\right\\}.$

Here $A$, $B$, $C$, $D$ are $n\times n$ matrices.

Let $N$ be coprime to $p$. For a natural number $k$ and a Dirichlet character
$\chi:(\mathbb{Z}/p^{m}\mathbb{Z})^{\times}\rightarrow\mathbb{C}^{\times}$,
the space $M_{k}(\Gamma_{0}^{(n)}(p^{m})\cap\Gamma_{1}^{(n)}(N),\chi)$ of
Siegel modular forms of weight $k$ with character $\chi$ (or nebentypus)
consists of all of holomorphic functions
$F:\mathbb{H}_{n}\rightarrow\mathbb{C}$ satisfying

$(F|_{k}\>\gamma)(Z)=\chi(\det
D)F(Z)\quad\text{for}\quad\gamma=\begin{pmatrix}A&B\\\
C&D\end{pmatrix}\in\Gamma_{0}(p^{m})\cap\Gamma_{1}(N).$

If $n=1$, the usual condition in the cusps should be added. When $\chi$ is a
trivial character, we write simply
$M^{n}_{k}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N))$ for
$M^{n}_{k}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N),\chi)$.

Any $F\in M_{k}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N),\chi)$ has a Fourier
expansion of the form

$F(Z)=\sum_{0\leq T\in\Lambda_{n}}a_{F}(T)q^{T},\quad q^{T}:=e^{2\pi i{\rm
tr}(TZ)},\quad Z\in\mathbb{H}_{n},$

where

$\Lambda_{n}:=\\{T=(t_{ij})\in{\rm
Sym}_{n}(\mathbb{Q})\;|\;t_{ii},\;2t_{ij}\in\mathbb{Z}\\}.$

We put $\Lambda_{n}^{+}:=\\{T\in\Lambda_{n}\;|\;T>0\\}$.

For a subring $R$ of $\mathbb{C}$, let
$M^{n}_{k}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N),\chi)(R)$ (resp.
$M^{n}_{k}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N))(R)$) denote the $R$-module of
all modular forms in $M^{n}_{k}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N),\chi)$
(resp. $M^{n}_{k}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N))$) whose Fourier
coefficients are in $R$.

###  2.2 Congruences for modular forms

Let $K$ be an algebraic number field, ${\mathcal{O}}_{K}$ the ring of integers
in $K$. We define $\nu_{\mathfrak{p}}(\gamma):=e_{\mathfrak{p}}$ for
$\gamma\in K$, where $e_{\mathfrak{p}}\in\mathbb{Z}$ is defined by
$\gamma{\mathcal{O}}_{K}=\prod\mathfrak{p}^{e_{\mathfrak{p}}}$ (prime ideal
factorization). Let $F$ be a formal power series of the form

$\displaystyle F=\sum_{T\in\Lambda_{n}}a_{F}(T)q^{T}$

with $a_{F}(T)\in K$ for all $T\in\Lambda_{n}$. We define

$\nu_{\mathfrak{p}}(F):=\inf\\{\nu_{\mathfrak{p}}(a_{F}(T))\;|\;T\in\Lambda_{n}\\}.$

Let $F_{i}=\sum_{T\in\Lambda_{n}}a_{F_{i}}(T)q^{T}$ ($i=1$, $2$) be two formal
power series as above. For an ideal
$\mathfrak{a}=\prod\mathfrak{p}^{e_{\mathfrak{p}}}$ in ${\mathcal{O}}_{K}$,
suppose that $\nu_{\mathfrak{p}}(F_{i})\geq 0$ for all $\mathfrak{p}$ with
$\mathfrak{p}\mid\mathfrak{a}$. We write $F_{1}\equiv F_{2}$ mod
$\mathfrak{a}$ if $\nu_{\mathfrak{p}}(F_{1}-F_{2})\geq e_{\mathfrak{p}}$ for
all prime ideals $\mathfrak{p}$ with $\mathfrak{p}\mid\mathfrak{a}$.

Let $p$ be an odd rational prime and $\mathfrak{p}$ a prime ideal in
${\mathcal{O}}_{K}$ such that $\mathfrak{p}\mid p{\mathcal{O}}_{K}$. We denote
by $e$ the ramification index $e=e(\mathfrak{p}/p)$ of $\mathfrak{p}$.
Following Rasmussen [11], we take the Galois closure $L$ of $K$ and a prime
ideal $\mathfrak{P}$ in ${\mathcal{O}}_{L}$ with
$\mathfrak{P}\mid\mathfrak{p}{\mathcal{O}}_{L}$. We put

$\beta(r):=\max\left\\{\Big{\lceil}\frac{r}{e}\Big{\rceil}-s-1,0\right\\},$

where $s:=\nu_{p}(\tilde{e})$ and $\tilde{e}:=e(\mathfrak{P}/p)$ is the
ramification index of $\mathfrak{P}$. We remark that $\beta(r)=r-1$ for
$K=\mathbb{Q}$.

######  Theorem 2.1 (Rasmussen [11] Theorem 2.16).

Let $N$ be a natural number with $p\nmid N$ and $F_{i}\in
M_{k_{i}}^{1}(\Gamma_{0}(p)\cap\Gamma_{1}(N))({\mathcal{O}}_{K})$. If
$F_{1}\equiv F_{2}$ mod $\mathfrak{p}^{r}$ and $F_{1}\not\equiv 0$ mod
$\mathfrak{p}$, then we have

$k_{1}\equiv k_{2}\bmod{(p-1)p^{\beta(r)}}.$

######  Remark 2.2.

Rasmussen [11] assumed $N\geq 3$. However the statement for $N\leq 2$ follows
from that for $N\geq 3$, since there is $M$ such that
$\Gamma_{1}(N)\supset\Gamma_{1}(M)$ and $M\geq 3$.

###  2.3 An Eisenstein series congruent to a constant

Let $p$ be an odd rational prime and $\mu_{p-1}$ the group of $(p-1)$th root
of unity in $\mathbb{C}^{\times}$. We consider the prime ideal factorization
of $p$ in the ring $\mathbb{Z}[\mu_{p-1}]$ of integers in
$\mathbb{Q}(\mu_{p-1})$. We take a generator $\zeta_{p-1}$ of $\mu_{p-1}$. Let
$\Phi(X)\in\mathbb{Z}[X]$ be the minimal polynomial of $\zeta_{p-1}$, namely
$\Phi(X)$ is the cyclotomic polynomial having the root $\zeta_{p-1}$. Then
$\Phi(X)$ can be decomposed in the form $\Phi(X)\equiv q_{1}(X)\cdots
q_{r}(X)$ mod $p$, were $r=\varphi(p-1)$ and each $q_{i}(X)$ is a polynomial
of degree $1$ satisfying $q_{i}(X)\not\equiv q_{j}(X)$ mod $p$. If we write
$q_{i}(X)=X-d_{i}$ with $d_{i}\in\mathbb{Z}$, then we have $d_{i}\not\equiv
d_{j}$ mod $p$ because of $q_{i}(X)\not\equiv q_{j}(X)$ mod $p$. Then $p$ is
decomposed as a product of $r$ prime ideals
$\mathfrak{p}_{i}:=(\zeta_{p-1}-d_{i},p)$, namely we have the perfect
decomposition

$p\mathbb{Z}[\mu_{p-1}]=\mathfrak{p}_{1}\cdots\mathfrak{p}_{r}=(\zeta_{p-1}-d_{1},p)\cdots(\zeta_{p-1}-d_{r},p).$

For this fact, we refer to Washington [14], p.15.

We take a prime ideal $\mathfrak{p}_{i}$ from above. We can define a character
$\psi_{i}$ :$(\mathbb{Z}/p\mathbb{Z})^{\times}\to\mathbb{C}^{\times}$ so that

$\psi_{i}(m)m\equiv 1\bmod{\mathfrak{p}_{i}}$

for all $m$ with $1\leq m\leq p-1$. Note that $\psi_{i}(-1)=-1$ and
$\psi_{i}\neq\psi_{j}$ for all $i$, $j$ with $i\neq j$. We shall say that
$\psi_{i}$ is the character corresponding $\mathfrak{p}_{i}$.

Let $E_{1,\psi_{i}}$ be the Eisenstein series of weight $1$ for
$\Gamma_{0}(p)$ with character $\psi_{i}$ of the form

$E_{1,\psi_{i}}=1-\frac{2}{B_{1,\psi_{i}}}\sum_{n=1}^{\infty}\left(\sum_{d\mid
n}\psi_{i}(d)\right)q^{n},$

where $B_{1,\psi_{i}}$ is the first generalized Bernoulli number with
character $\psi_{i}$ described as

$B_{1,\psi_{i}}=\frac{1}{p}\sum_{m=1}^{p-1}\psi_{i}(m)m.$

######  Theorem 2.3 (Lang [10] Theorem 1.2, p.250, Rasmussen [11], p.12).

We have $\nu_{\mathfrak{p}_{i}}(E_{1,\psi_{i}})\geq 0$ and

$E_{1,\psi_{i}}\equiv 1\mod{\mathfrak{p}_{i}}.$

In other words, if we take $\gamma\in\mathbb{Z}[\mu_{p-1}]$ with
$\gamma\not\in\mathfrak{p}_{i}$ such that ${\mathcal{E}}_{1,\psi_{i}}:=\gamma
E_{1,\psi_{i}}\in M_{1}(\Gamma_{0}(p),\psi_{i})(\mathbb{Z}[\mu_{p-1}])$, then
${\mathcal{E}}_{1,\psi_{i}}\equiv\gamma\not\equiv 0$ mod $\mathfrak{p}_{i}$
holds.

######  Remark 2.4.

  1. (1)

The existence of $\gamma$ in the statement follows from the strong
approximation theorem for the number fields.

  2. (2)

As we will see later, we have

$E_{1,\psi_{i}}\not\equiv 1\mod{p\mathbb{Z}[\mu_{p-1}]}$

for any rational prime $p$ with $p\geq 5$. In other words, there exists
$\mathfrak{p}_{j}$ with $i\neq j$ such that

$E_{1,\psi_{i}}\not\equiv 1\mod{\mathfrak{p}_{j}},$

and equivalently $pB_{1,\psi_{i}}\equiv 0$ mod $\mathfrak{p}_{j}$. We do not
know if $\nu_{\mathfrak{p}_{j}}(E_{1,\psi_{i}})\geq 0$ to begin with.

  3. (3)

Lang [10] states that $E_{1,\psi_{i}}\equiv 1$ mod $p$ and Rasmussen [11]
states that $E_{1,\psi_{i}}$ has a Fourier coefficients in
$\mathbb{Z}[\mu_{p-1}]$. Both statements seem to be inaccurate.

##  3 Main results and their proofs

###  3.1 Statements of main results

The first main result concerns congruences for the weights of modular forms of
nonquadratic nebentypus.

######  Theorem 3.1.

Let $p$ be an odd rational prime and $K$ a number field including
$\mathbb{Q}(\mu_{p-1})$. Let $\mathfrak{P}$ be a prime ideal in
${\mathcal{O}}_{K}$ with $\mathfrak{P}\mid p{\mathcal{O}}_{K}$. For a prime
ideal $\mathfrak{p}$ in $\mathbb{Z}[\mu_{p-1}]$ such that $\mathfrak{p}\mid
p\mathbb{Z}[\mu_{p-1}]$ and $\mathfrak{P}\mid\mathfrak{p}{\mathcal{O}}_{K}$,
we take the character $\psi$ corresponding $\mathfrak{p}$ (see Subsection
2.3). Let $\chi_{i}$ ($i=1$, $2$) be two Dirichlet characters mod $p^{m_{i}}$
with conductor $m_{i}^{\prime}$. We put
$m:=\max\\{m_{1}^{\prime},m_{2}^{\prime}\\}$ and take $\alpha_{i}$ with
$0\leq\alpha_{i}\leq p-2$ such that $\chi_{i}^{p^{m-1}}=\psi^{\alpha_{i}}$. If
$F_{i}\in
M^{n}_{k_{i}}(\Gamma_{0}(p^{m_{i}})\cap\Gamma_{1}(N),\chi_{i})(\mathcal{O}_{K})$
satisfies that $F_{1}\equiv F_{2}$ mod $\mathfrak{P}^{r}$ and $F_{1}\not\equiv
0$ mod $\mathfrak{P}$, then we have

$k_{1}\cdot p^{m-1}-\alpha_{1}\cdot p^{r-1}\equiv k_{2}\cdot
p^{m-1}-\alpha_{2}\cdot p^{r-1}\bmod{(p-1)p^{\beta(r)}}.$

######  Remark 3.2.

The theorem above is essentially a statement about characters mod $p$. In
fact, both characters $\chi_{i}^{p^{m-1}}$ ($i=1,2$) are defined mod $p$,
because
$\sharp(\mathbb{Z}/p^{m^{\prime}_{i}}\mathbb{Z})^{\times}=p^{m^{\prime}_{i}-1}(p-1)$
and
$\chi_{i}^{p^{m-1}}=(\chi_{i}^{p^{m^{\prime}_{i}-1}})^{p^{m-m^{\prime}_{i}}}$
($i=1$, $2$). Therefore we can take $\alpha_{i}$ with $0\leq\alpha_{i}\leq
p-2$ such that $\chi_{i}^{p^{m-1}}=\psi^{\alpha_{i}}$.

As a consequence of the above results, it can be shown that the characters are
somewhat restricted when we consider congruences mod rational prime $p$.

######  Corollary 3.3.

  1. (1)

If $F_{i}\in
M^{n}_{k_{i}}(\Gamma_{0}(p)\cap\Gamma_{1}(N),\chi_{i})(\mathcal{O}_{K})$
satisfies that $F_{1}\equiv F_{2}$ mod $p\mathcal{O}_{K}$ and $F_{1}\not\equiv
0$ mod $\mathfrak{p}$ for some $\mathfrak{p}$ with $\mathfrak{p}\mid
p{\mathcal{O}}_{K}$, then we have $\chi_{1}=\chi_{2}\cdot(*/p)^{t}$ for some
$t\in\mathbb{Z}$. Here $(*/p)$ is the unique nontrivial quadratic character
mod $p$.

In particular, if $F_{1}$ is of quadratic nebentypus, then $F_{2}$ is of
quadratic nebentaypus.

  2. (2)

If there exist ${\mathcal{E}}_{k,\chi}\in
M^{n}_{k}(\Gamma_{0}(p),\chi)(\mathcal{O}_{K})$ and $\gamma\in\mathcal{O}_{K}$
such that ${\mathcal{E}}_{k,\chi}\equiv\gamma\not\equiv 0$ mod
$p\mathcal{O}_{K}$, then $\chi$ is quadratic.

######  Remark 3.4.

  1. (1)

Actually, in order to prove (1), we can replace the condition $F_{1}\equiv
F_{2}$ mod $p{\mathcal{O}}_{K}$ by a weaker condition $F_{1}\equiv F_{2}$ mod
$\mathfrak{p}\mathfrak{p}^{\rho}$. Here $\rho$ is the complex conjugate in
${\rm Aut}(\mathbb{C})$. See the proof in Subsection 3.3

  2. (2)

From the assertion (2), Remark 2.4 (2) follows.

As mentioned in Introduction, the above results provide a mod $p$ analogue of
Freitag’s result that there are no singular forms (over $\mathbb{C}$) of
nonquadratic nebentypus. We just write down the simplest case.

For a character $\chi$ mod $p$, we put
$\mathbb{Q}(\chi):=\mathbb{Q}(\chi(a)\>|\>a\in\mathbb{Z})$. Let
$\mathbb{Z}[\chi]$ be the ring of integers in $\mathbb{Q}(\chi)$.

######  Proposition 3.5.

Let $\chi$ be a character mod $p$ and $F\in
M_{k}^{n}(\Gamma_{0}(p),\chi)(\mathbb{Z}[\chi])$. Assume that $F$ is strongly
mod $p$ singular of $p$-rank $r$, i.e., $r\leq n/2$, $a_{F}(T)\equiv 0$ mod
$p\mathbb{Z}[\chi]$ for all $T\in\Lambda_{n}$ with ${\rm rank}(T)<r$, and
$a_{F}(T)\not\equiv 0$ mod $p\mathbb{Z}[\chi]$ for some $T\in\Lambda_{n}$ with
${\rm rank}(T)=r$. Then $\chi$ is a quadratic character.

######  Remark 3.6.

We may as well consider mod $\mathfrak{p}$ singular modular forms for prime
ideal $\mathfrak{p}$ in $\mathbb{Z}[\chi]$. Then our method (see Subsection
3.3) does not work when $\chi$ is nonquadratic character. Indeed, if we assume
the existence of a modular form ${\mathcal{E}}\in
M_{l}^{n}(\Gamma_{0}(p),\chi)(\mathbb{Z}[\chi])$ satisfying
${\mathcal{E}}\equiv\gamma\not\equiv 0$ mod $\mathfrak{p}$ with
$\gamma\in\mathbb{Z}[\chi]$, we may switch between trivial nebentypus and
nebentypus $\chi$ just by multiplication by ${\mathcal{E}}$. To the authors’
knowledge, the existence of such ${\mathcal{E}}$ is not clear at all except
degree $1$ and $2$ (see Theorem 2.3 and [9] Theorem 1.2). For the higher
degree case, in an unpublished note, Takemori constructed such ${\mathcal{E}}$
from Eisenstein series with character when $p$ is a regular prime (see also
Remark 4.4).

###  3.2 Proof of Theorem 3.1

We first give two reductions:

Reduction 1: Decreasing the level to achieve
$\boldsymbol{m_{i}=m_{i}^{\prime}}$.
As in the proof of Section 8.2 in [2] we change $F_{i}$ mod $\mathfrak{P}^{r}$
to a form of level $\Gamma_{0}(p^{m_{i}^{\prime}})\cap\Gamma_{1}(N)$ with same
nebentypus character and with a weight $k_{i}^{\prime}$ congruent to $k_{i}$
modulo a multiple of $(p-1)p^{r}$. We should keep in mind that $\beta(r)\leq
r$ always holds.

Reduction 2: Degree ${\boldsymbol{1}}$ case is sufficient.
We may apply the method of “integral extract” from [4]. We shortly describe
the method (and the modifications necessary). As in [4] we may associate to
$F_{i}$ ($i=1$, $2$) as above two elliptic modular forms $f_{i}\in
M_{k_{i}}(\Gamma_{1}(p^{m_{i}}NR^{2}))({\mathcal{O}}_{K})$ satisfying
$f_{1}\equiv f_{2}$ mod $\mathfrak{P}^{r}$. In [4], we ignored questions
concerning nebentypus characters. The reduction is achieved by the observation
that $f_{i}$ has indeed nebentypus character $\chi_{i}$ on
$\Gamma_{0}(p)\cap\Gamma_{1}(NR^{2})$.

We sketch the modifications necessary for the constriction of an “integral
extract” which should keep track of nebentypus characters:

######  Proposition 3.7.

For $F\in M_{k}^{n}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N),\chi)(K)$ there exists,
for any sufficiently large $R\in\mathbb{N}$ an elliptic modular form $f\in
M^{1}_{k}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(NR^{2}),\chi)(K)$ such that

  * •

$f$ is an “integral extract” of $F$, i.e., the Fourier coefficients are finite
sums of Fourier coefficients of $F$,

  * •

$\nu_{\mathfrak{p}}(F)=\nu_{\mathfrak{p}}(f)$.

###### Proof.

We use the construction from [4] and show that it satisfies the necessary
transformation properties for $\Gamma_{0}(p^{m})\cap\Gamma_{1}(NR^{2})$. The
properties for the smaller group $\Gamma_{1}(p^{m}NR^{2})$ hold anyway. We
recall from [4] that $f$ is obtained (for a suitable $T_{0}$) from

$F^{(R,T_{0})}=\frac{1_{n}}{R^{\frac{n(n+1)}{2}}}\sum_{J}F|_{k}\;\begin{pmatrix}1_{n}&\frac{J}{R}\\\
0&1_{n}\end{pmatrix}\cdot e^{-2\pi i{\rm tr}(\frac{T_{0}\cdot J}{R})},$

where $J$ runs over all symmetric integral matrices mod $R$ of size $n$. Let
$\left(\begin{smallmatrix}A&B\\\
C&D\end{smallmatrix}\right)\in\Gamma_{0}(p^{m})\cap\Gamma_{1}^{o}(NR^{2})$.
Then, for any integral symmetric matrices $J$ of size $n$ we have

$\begin{pmatrix}1_{n}&\frac{J}{R}\\\ 0&1_{n}\end{pmatrix}\begin{pmatrix}A&B\\\
C&D\end{pmatrix}=X\cdot\begin{pmatrix}1_{n}&\frac{J}{R}\\\
0&1_{n}\end{pmatrix}$

with

$X=\begin{pmatrix}A+\frac{J}{R}&-\frac{AJ}{R}-\frac{JCJ}{R^{2}}+B+\frac{JD}{R}\\\
C&-\frac{CJ}{R}+D\end{pmatrix}.$

Then $X\in\Gamma_{0}(p^{m})\cap\Gamma_{1}(NR^{2})$ provided that $C\equiv
0_{n}$ mod $R^{2}$; note that $A\equiv 1_{n}\equiv D$ mod $R$. The assertion
follows in a standard way. ∎

######  Proposition 3.8.

Let $\chi_{i}$ be two Dirichlet characters mod $p^{m_{i}}$ and $F_{i}\in
M_{k_{i}}^{n}(\Gamma_{0}(p^{m_{i}})\cap\Gamma_{1}(N),\chi_{i})({\mathcal{O}}_{K})$.
Suppose that $F_{1}\equiv F_{2}$ mod $\mathfrak{p}^{r}$ and $F_{1}\not\equiv
0$ mod $\mathfrak{p}$ for a prime ideal $\mathfrak{p}$ in ${\mathcal{O}}_{K}$.
Then for sufficiently large $R$ with $p\nmid R$, there exist integral extracts
$f_{i}\in
M_{k_{i}}^{1}(\Gamma_{0}(p^{m_{i}})\cap\Gamma_{1}(N),\chi_{i})({\mathcal{O}}_{K})$
such that $f_{1}\equiv f_{2}$ mod $\mathfrak{p}^{r}$ with $f_{1}\not\equiv 0$
mod $\mathfrak{p}$.

###### Proof of Theorem 3.1.

It suffices to prove the case of degree $1$ by Reduction 2. We may assume that
$m_{i}$ is conductors of $\chi_{i}$ because of Reduction 1.

Suppose that $f_{i}\in
M^{1}_{k_{i}}(\Gamma_{0}(p^{m})\cap\Gamma_{1}(N),\chi_{i})(\mathcal{O}_{K})$,
$f_{1}\equiv f_{2}$ mod $\mathfrak{P}^{r}$ and $f_{1}\not\equiv 0$ mod
$\mathfrak{P}$. Then we can find some $\gamma\in{\mathcal{O}}_{K}$ with
$\gamma\not\in\mathfrak{P}$ such that

$\displaystyle\gamma
f_{i}^{p^{m-1}}\\{(E_{1,\psi})^{p-1-\alpha_{i}}\\}^{p^{r-1}}\in M_{k_{i}\cdot
p^{m-1}+(p-1-\alpha_{i})p^{r-1}}(\Gamma_{0}^{(n)}(p^{m})\cap\Gamma_{1}(N))({\mathcal{O}}_{K})$
$\displaystyle\text{and}\quad\gamma
f_{1}^{p^{m-1}}\\{(E_{1,\psi})^{p-1-\alpha_{1}}\\}^{p^{r-1}}\equiv\gamma
f_{2}^{p^{m-1}}\\{(E_{1,\psi})^{p-1-\alpha_{2}}\\}^{p^{r-1}}\bmod{\mathfrak{P}^{r}}.$

Note here that $\gamma
f_{1}^{p^{m-1}}\\{(E_{1,\psi})^{p-1-\alpha_{1}}\\}^{p^{r-1}}\not\equiv 0$ mod
$\mathfrak{P}$. It follows from Theorem 2.1 that

$k_{1}\cdot p^{m-1}-\alpha_{1}\cdot p^{r-1}\equiv k_{2}\cdot
p^{m-1}-\alpha_{2}\cdot p^{r-1}\bmod{(p-1)p^{\beta(r)}}.$

This completes the proof of Theorem 3.1. ∎

###  3.3 Proofs of Corollary 3.3 and Proposition 3.5

###### Proof of Corollary 3.3.

(1) Let $\rho$ be the complex conjugate in ${\rm Aut}(\mathbb{C})$. Taking
sufficiently large field $K$, we may assume that $K^{\rho}=K$.

For the prime ideal $\mathfrak{p}$ with $F_{1}\not\equiv 0$ mod
$\mathfrak{p}$, we take the character $\psi$ corresponding $\mathfrak{p}$ and
$\alpha_{i}$ with $0\leq\alpha_{i}\leq p-2$ such that
$\chi_{i}=\psi^{\alpha_{i}}$. The condition $F_{1}\equiv F_{2}\not\equiv 0$
mod $\mathfrak{p}$ implies $k_{1}-k_{2}\equiv\alpha_{1}-\alpha_{2}$ mod $p-1$,
because of Theorem 3.1. On the other hand, we have

$\displaystyle F_{i}\cdot F_{i}^{\rho}\in$ $\displaystyle
M^{n}_{2k_{i}}(\Gamma_{0}(p)\cap\Gamma_{1}(N))(\mathcal{O}_{K}),\quad
F_{1}\cdot F_{1}^{\rho}\equiv F_{2}\cdot F_{2}^{\rho}\bmod{p\mathcal{O}_{K}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{and}\quad
F_{1}\cdot F_{1}^{\rho}\not\equiv 0\bmod{p{\mathcal{O}}_{K}}.$

Taking a prime ideal $\mathfrak{p}^{\prime}$ with $\mathfrak{p}^{\prime}\mid
p{\mathcal{O}}_{K}$ such that $F_{1}\cdot F_{1}^{\rho}\not\equiv 0$ mod
$\mathfrak{p}^{\prime}$, and applying Theorem 3.1 (or Theorem 2.1) to
$F_{i}\cdot F_{i}^{\rho}$, we have $2k_{1}-2k_{2}\equiv 0$ mod $p-1$. These
two congruences imply that $\alpha_{1}-\alpha_{2}\equiv 0$ mod
$\frac{p-1}{2}$. We put $\alpha_{1}=\alpha_{2}+\frac{p-1}{2}\cdot t$ with
$t\in\mathbb{Z}$. This shows that

$\displaystyle\chi_{1}=\psi^{\alpha_{1}}=\psi^{\alpha_{2}+\frac{p-1}{2}\cdot
t}=\chi_{2}\cdot\psi^{\frac{p-1}{2}\cdot
t}=\chi_{2}\cdot\left(*/p\right)^{t}.$

We obtain the claim of (1).
(2) The claim follows immediately from (1). ∎

###### Proof of Proposition 3.5.

In the same way as in [2] Corollary 4.3 and Remark 4.4, we can find
$S\in\Lambda_{r}^{+}$ and $h\in
M^{r}_{k}(\Gamma_{0}(p),\chi)(\mathbb{Z}[\chi])$ such that
$\theta_{S}^{r}\equiv h\not\equiv 0$ mod $p\mathbb{Z}[\chi]$. Here, for
$S\in\Lambda_{m}^{+}$ ($m$ even), $\theta_{S}^{n}\in
M_{m/2}^{n}(\Gamma_{0}(L),\chi_{S})$ denotes the theta series

$\displaystyle\theta_{S}^{n}(Z):=\sum_{X\in\mathbb{Z}^{m,n}}e^{2\pi i({\rm
tr}(S[X]Z))}\quad(Z\in\mathbb{H}_{n}),$ (3.1)

$L$ is the level of $S$, $\chi_{S}$ is the associated quadratic character of
$S$, and $\mathbb{Z}^{m,n}$ is the set of $m\times n$ matrices with integral
components. By Corollary 3.3 (1), this can not happen, unless $\chi$ is
quadratic. ∎

##  4 Eisenstein series of nonquadratic nebentypus

We take a prime ideal $\mathfrak{p}_{i}$ in $\mathbb{Z}[\mu_{p-1}]$ and
$q_{i}(X)=X-d_{i}\in\mathbb{Z}[X]$ from Subsection 2.3. Let $\omega$ be the
Teichmüller character on $\mathbb{Z}_{p}$. Then an embedding from
$\mathbb{Q}(\mu_{p-1})$ to $\mathbb{Q}_{p}$ corresponding to
$\mathfrak{p}_{i}$ is determined by $\sigma(\zeta_{p-1})=\omega(d_{i})$. Let
$\chi:(\mathbb{Z}/p\mathbb{Z})^{\times}\rightarrow\mathbb{Q}(\mu_{p-1})$ be a
character mod $p$. Then we can find $\alpha$ with $0\leq\alpha\leq p-2$ such
that $\chi^{\sigma}=\omega^{\alpha}$. We remark that $\chi=\psi_{i}^{-\alpha}$
holds for the character $\psi_{i}$ corresponding $\mathfrak{p}_{i}$, since
$\omega(d_{i})\equiv d_{i}\equiv(\psi_{i}^{-1}(d_{i}))^{\sigma}$ mod $p$.

Suppose that $\chi$ is nonquadratic primitive character mod $p$. For a
positive integer $k$ with $k>n+1$, let $E^{n}_{k}=\sum_{T}a_{k}(T)q^{T}$ be
the Eisenstein series of weight $k$ and degree $n$ for $\Gamma_{n}$, and
$E^{n}_{k,\chi}=\sum_{T}a_{k,\chi}(T)q^{T}$ the Eisenstein series of weight
$k$ and degree $n$ for $\Gamma_{0}(p)$ with character $\chi$ which studied by
Takemori [13]. Here we assumed that $\chi(-1)=(-1)^{k}$ for $E^{n}_{k,\chi}$.
In stead of giving the direct definition of $E^{n}_{k,\chi}$, we just write
down its Fourier expansion below.

Suppose that ${}^{t}UTU={\rm diag}(T^{\prime},0_{n-r})$ with
$T^{\prime}\in\Lambda_{r}^{+}$ for some $U\in{\rm GL}_{n}(\mathbb{Z})$. Recall
from Katsurada [7] and Takemori [13] that

$\displaystyle
a_{k}(T)=2^{[(r+1)/2]}L(1-k,{\boldsymbol{1}})^{-1}\left(\prod_{i=1}^{[r/2]}L(1+2i-2k,{\boldsymbol{1}})^{-1}\right)$
$\displaystyle~{}~{}~{}~{}~{}\times\prod_{q}F_{q}(T^{\prime},q^{k-r-1})\times\begin{cases}1\quad\text{if}\
r\ \text{odd},\\\ L(1+r/2-k,\chi_{T^{\prime}})\quad\text{if}\ r\
\text{even},\end{cases}$ $\displaystyle
a_{k,\chi}(T)=2^{[(r+1)/2]}L^{(p)}(1-k,\chi)^{-1}\left(\prod_{i=1}^{[r/2]}L^{(p)}(1+2i-2k,\chi^{2})^{-1}\right)$
(4.1) $\displaystyle~{}~{}~{}~{}~{}\times\prod_{q\neq
p}F_{q}(T^{\prime},\chi(q)q^{k-r-1})\times\begin{cases}1\quad\text{if}\ r\
\text{odd},\\\ L^{(p)}(1+r/2-k,\chi_{T^{\prime}}\chi)\quad\text{if}\ r\
\text{even}.\end{cases}$ (4.2)

Here we put $L^{(p)}(s,\psi):=(1-\psi(p)p^{-s})L(s,\psi)$, ${\boldsymbol{1}}$
is the trivial character mod $1$, and $F_{q}(T,X)\in\mathbb{Z}[X]$ is a
polynomial with constant term $1$ calculated by Katsurada [7].

Let $L_{p}(s,\psi)$ be the Kubota–Leopoldt $p$-adic $L$-function. Then
$L_{p}(s,\psi)$ is a $p$-adic meromorphic (analytic if
$\psi\neq{\boldsymbol{1}}$) function on $\mathbb{Z}_{p}$ satisfying

$\displaystyle
L_{p}(1-s,\psi)=-(1-\psi\omega^{-s}(p)p^{s-1})\frac{B_{s,\psi\omega^{-s}}}{s}\quad(s\in\mathbb{Z}_{\geq
1}),$ (4.3)

and $L_{p}(s,{\boldsymbol{1}})$ is analytic except for a pole at $s=1$. Here
we regard as $\psi\omega^{-s}={\boldsymbol{1}}$ when $\psi=\omega^{s}$. On
these facts, see Washington [14], p.57.

Then we have

$\displaystyle L^{(p)}(1-s,\psi\omega^{-s})=L_{p}(1-s,\psi)$ (4.4)

for any $s\in\mathbb{Z}_{\geq 1}$.

###  4.1 Mod $p$-power singular forms of nonquadratic nebentypus

######  Theorem 4.1.

Let $l$ be a natural number and $p$ a regular prime with $p>2l+1$. We take
$\alpha\in\mathbb{Z}$ with $1\leq\alpha\leq p-2$ ($\alpha\not\equiv 0$ mod
$\frac{p-1}{2}$) such that $\chi^{\sigma}=\omega^{\alpha}$. Let
$\\{l_{\delta,m}\\}$ ($\delta=0$, $1$) be two sequences of positive integers
such that $l+\frac{p-1}{2^{\delta}}$ is even and

$l_{\delta,m}\to\left(l,l-\alpha+\frac{p-1}{2^{\delta}}\right)\in{\boldsymbol{X}},\quad
l_{\delta,m}\to\infty\quad(m\to\infty).$

Then we have

$\lim_{m\to\infty}(E^{n}_{l_{\delta,m},\chi})^{\sigma}=\widetilde{E}^{n}_{(l,l+(p-1)/2^{\delta})}.$

Here $\widetilde{E}^{n}_{(l,l+(p-1)/2^{\delta})}$ is the $p$-adic Eisenstein
series studied in the authors [3].

###### Proof.

We may assume that $l_{\delta,m}=l+a(m)p^{b(m)}$ with
$a(m)\equiv-\alpha+(p-1)/2^{\delta}$ mod $p-1$, $b(m)\to\infty$
($m\to\infty$). Recall that

$\displaystyle
a_{l_{\delta,m},\chi}(T)^{\sigma}=2^{[(r+1)/2]}L^{(p)}(1-l_{\delta,m},\omega^{\alpha})^{-1}\left(\prod_{i=1}^{[r/2]}L^{(p)}(1+2i-2l_{\delta,m},\omega^{2\alpha})^{-1}\right)$
$\displaystyle~{}~{}~{}~{}~{}\times\prod_{q\neq
p}F_{q}(T^{\prime},\omega^{\alpha}(q)q^{l_{\delta,m}-r-1})\times\begin{cases}1\quad\text{if}\
r\ \text{odd},\\\
L^{(p)}(1+r/2-l_{\delta,m},\chi_{T^{\prime}}\omega^{\alpha})\quad\text{if}\ r\
\text{even}.\end{cases}$

We consider the case where $s=l_{\delta,m}$ and
$\psi\omega^{-s}=\omega^{\alpha}$ in the formula (4.4). Then we have
$\psi=\omega^{\alpha+l_{\delta,m}}=\omega^{l+(p-1)/2^{\delta}}$ and therefore

$\displaystyle
L^{(p)}(1-l_{\delta,m},\omega^{\alpha})=L_{p}(1-l_{\delta,m},\omega^{l+(p-1)/2^{\delta}}).$

It follows that

$\displaystyle\lim_{m\to\infty}L_{p}(1-l_{\delta,m},\omega^{l+(p-1)/2^{\delta}})=L_{p}(1-l,\omega^{l+(p-1)/2^{\delta}})=L^{(p)}\left(1-l,(q/p)^{\delta}\right).$

This shows

$\displaystyle\lim_{m\to\infty}L^{(p)}(1-l_{\delta,m},\omega^{\alpha})=L^{(p)}\left(1-l,(q/p)^{\delta}\right).$

Similarly, we have the following $p$-adic limits.

$\displaystyle\lim_{m\to\infty}L^{(p)}(1+2i-2l_{\delta,m},\omega^{2\alpha})=\lim_{m\to\infty}L_{p}(1+2i-2l_{\delta,m},\omega^{2l-2i})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=L_{p}(1+2i-2l,\omega^{2l-2i})\quad\text{for\
all\ }i\ \text{with}\ 1\leq i\leq[r/2],$
$\displaystyle\lim_{m\to\infty}L^{(p)}(1+r/2-l_{\delta,m},\chi_{T^{\prime}}\omega^{\alpha})=\lim_{m\to\infty}L_{p}(1+r/2-l_{\delta,m},\chi_{T^{\prime}}\omega^{l+(p-1)/2^{\delta}-r/2}),$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=L_{p}(1+r/2-l,\chi_{T^{\prime}}\omega^{l+(p-1)/2^{\delta}-r/2})\quad\text{for\
}r\ \text{even}.$

Here, due to the convergence of the factor
$L^{(p)}(1+2i-2l_{\delta,m},\omega^{2\alpha})^{-1}$, the condition of
regularity on $p$ is necessary. In fact, if $p$ is a regular prime, then
$L_{p}(1+2i,\omega^{-2i})\neq 0$ for all $i$ with $i\in\mathbb{Z}_{\geq 1}$
and hence

$\lim_{m\to\infty}L^{(p)}(1+2i-2l_{\delta,m},\omega^{2\alpha})=L_{p}(1+2i-2l,\omega^{2l-2i})\neq
0.$

This means that $L^{(p)}(1+2i-2l_{\delta,m},\omega^{2\alpha})^{-1}$ converges.

We consider the factor
$F_{q}(T^{\prime},\omega^{\alpha}(q)q^{l_{\delta,m}-r-1})$. Since

$\displaystyle\omega^{\alpha}(q)q^{l_{\delta,m}-r-1}$
$\displaystyle\equiv\omega^{\alpha}(q)q^{a(m)p^{b(m)}}q^{l-r-1}$
$\displaystyle\equiv\omega^{\alpha}(q)\omega^{a(m)}(q)q^{l-r-1}$
$\displaystyle\equiv\omega^{(p-1)/2^{\delta}}(q)q^{l-r-1}\bmod{p^{b(m)+1}}$

and the constant term of $F_{q}(T^{\prime},X)$ is $1$, we have

$\displaystyle F_{q}(T^{\prime},\omega^{\alpha}(q)q^{l_{\delta,m}-r-1})\equiv
F_{q}(T^{\prime},\omega^{(p-1)/2^{\delta}}(q)q^{l-r-1})\bmod{p^{b(m)+1}}.$
(4.5)

This shows

$\displaystyle\lim_{m\to\infty}F_{q}(T^{\prime},\omega^{\alpha}(q)q^{l_{\delta,m}-r-1})=F_{q}(T^{\prime},(q/p)^{\delta}q^{l-r-1}).$

Summering these formulas, we have the following.

$\displaystyle\lim_{m\to\infty}a_{l_{\delta,m},\chi}(T)^{\sigma}=2^{[(r+1)/2]}L^{(p)}(1-l,(q/p)^{\delta})^{-1}\left(\prod_{i=1}^{[r/2]}L_{p}(1+2i-2l,\omega^{2l-2i})^{-1}\right)$
$\displaystyle~{}~{}~{}~{}~{}\times\prod_{q\neq
p}F_{q}(T^{\prime},(q/p)^{\delta}q^{l-r-1})\times\begin{cases}1\quad\text{if}\
r\ \text{odd},\\\
L_{p}(1+r/2-l,\chi_{T^{\prime}}\omega^{l+(p-1)/2^{\delta}-r/2})\quad\text{if}\
r\ \text{even}.\end{cases}$

We need to check that this convergence is uniform in $T$. The main issue is
dependence on the quadratic character $\chi_{T^{\prime}}$. If the conductor of
$\chi_{T^{\prime}}\omega^{l+(p-1)/2^{\delta}-r/2}$ is divisible by a prime $q$
other than $p$, then there exists a formal power series
$\Phi_{T^{\prime}}(X)\in\mathbb{Z}_{p}[\\![X]\\!]$ such that
$L_{p}(s,\chi_{T^{\prime}}\omega^{l+(p-1)/2^{\delta}-r/2})=\Phi_{T^{\prime}}((1+p)^{s}-1)$
(cf. Washington [14] Theorem 7.10). By this fact and (4.5), we see that the
convergence is uniform in $T$.

On the other hand, we take a sequence $\\{k_{\delta,m}\\}$ of positive even
integers such that $k_{\delta,m}\to(l,l+(p-1)/2^{\delta})\in{\boldsymbol{X}}$
($m\to\infty$). We can confirm similarly that
$L(1-k_{\delta,m},{\boldsymbol{1}})$,
$L(1+2i-2k_{\delta,m},{\boldsymbol{1}})$,
$F_{q}(T^{\prime},q^{k_{\delta,m}-r-1})$ and
$L(1+r/2-k_{\delta,m},\chi_{T^{\prime}})$ have same $p$-adic limits
respectively as above, and also that
$\lim_{m\to\infty}F_{p}(T^{\prime},p^{k_{\delta,m}-r-1})=1$. Therefore we get

$\lim_{m\to\infty}(E^{n}_{l_{\delta,m},\chi})^{\sigma}=\lim_{m\to\infty}E^{n}_{k_{\delta,m}}=\widetilde{E}^{n}_{(l,l+(p-1)/2^{\delta})}.$

∎

######  Remark 4.2.

  1. (1)

We assumed that $p$ is regular. The condition we need actually is
$L_{p}(1+2i-2l,\omega^{2l-2i})\neq 0$ for all $l<j\leq[n/2]$. Of course, if
$p$ is regular, this condition holds.

  2. (2)

Since $\widetilde{E}^{n}_{(l,l+(p-1)/2^{\delta})}$ is mod $p$ singular of rank
$2l$ by the results of authors [3], we know that
$\lim_{m\to\infty}(E_{l_{\delta,m},\chi})^{\sigma}$ is too, but we can also
check directly by calculating the Fourier coefficients as follows. Let $r>2l$.
Then the factor $L^{(p)}(1+2i-2l_{\delta,m},\omega^{2\alpha})^{-1}$ for $i=l$
appears in the Fourier coefficient of $a_{l_{\delta,m},\chi}(T)^{\sigma}$. If
we put $s=2l_{m}-2l$, $\psi=\omega^{2l_{m}-2l+2\alpha}$ in (4.4), then we have
$\psi\omega^{-s}=\omega^{2\alpha}$ and hence

$L^{(p)}(1+2i-2l_{m},\omega^{2\alpha})^{-1}=L_{p}(1+2l-2l_{m},\omega^{2l_{m}-2l+2\alpha})^{-1}.$

Since $\omega^{2l_{m}-2l+2\alpha}={\boldsymbol{1}}$ and
$L_{p}(s,{\boldsymbol{1}})$ has a pole at $s=1$, we have

$\displaystyle\lim_{m\to\infty}L^{(p)}(1+2l-2l_{m},\omega^{2\alpha})^{-1}$
$\displaystyle=\lim_{m\to\infty}L_{p}(1+2l-2l_{m},{\boldsymbol{1}})^{-1}=0.$

On the other hand, by the assumption, we have
$L_{p}(1+2i-2l,\omega^{2l-2i})\neq 0$ for all $1\leq j\leq[n/2]$. Therefore
the other factors have $p$-adic limits in $\mathbb{Q}_{p}$, this shows that
all Fourier coefficients for the higher degree vanish.

By the result for $\widetilde{E}^{n}_{(l,l+(p-1)/2^{j})}$ in [3], we obtain
the following statement.

######  Corollary 4.3.

The following holds under the situation as Theorem 4.1.

  1. (1)

There exists some constant $\mu_{\delta}$ with $\mu_{\delta}\leq 0$ such that

$\displaystyle
p^{-\mu_{\delta}}(E_{l_{\delta,m},\chi}^{n})^{\sigma}\equiv\sum_{\begin{subarray}{c}{\rm
gen}(S)\\\ {\rm level}(S)\mid p\\\
\chi_{S}=(*/p)^{\delta}\end{subarray}}p^{-\mu_{\delta}}a_{\delta}({\rm
gen}(S))\cdot(\Theta^{n}_{{\rm gen}(S)})^{0}\bmod{p^{c(m)}},$ (4.6)

where $c(m)$ is a certain positive integer satisfying $c(m)\to\infty$ if
$m\to\infty$. Here the summation goes over finitely many genera ${\rm gen}(S)$
of $S\in\Lambda_{2l}^{+}/{\rm GL}_{2l}(\mathbb{Z})$ such that ${\rm
level}(S)\mid p$ and $\chi_{S}=(*/p)^{\delta}$.

  2. (2)

We have

$\lim_{m\to\infty}(E^{n}_{l_{\delta,m},\chi})^{\sigma}=\sum_{\begin{subarray}{c}{\rm
gen}(S)\\\ {\rm level}(S)\mid p\\\
\chi_{S}=(*/p)^{\delta}\end{subarray}}a_{\delta}({\rm
gen}(S))\cdot(\Theta^{n}_{{\rm gen}(S)})^{0}.$

Here, the notation is as follows.

  * •

$(\Theta^{n}_{{\rm gen}(S)})^{0}$ is the unnormalized genus theta series
defined by

$(\Theta^{n}_{{\rm gen}(S)})^{0}:=\sum_{S\in{\rm
gen}(S)}\frac{1}{\epsilon(S)}\theta^{n}_{S},$

  * •

$\theta^{n}_{S}$ is as in (3.1),

  * •

$\epsilon(S)$ is the order of automorphism group of $S$,

  * •

$(*/p)^{0}$ is understood as the trivial character ${\boldsymbol{1}}_{p}$ mod
$p$,

  * •

$a_{\delta}({\rm gen}(S))\in\mathbb{Q}$ are certain constants coming from
Fourier coefficients of the Eisenstein series of level $1$.

######  Remark 4.4.

  1. (1)

It is shown by Takemori in his unpublish note that there exists
$E^{n}_{l,\chi}\in M^{n}_{l}(\Gamma_{0}(p),\chi)$ such that
$(E^{n}_{l,\chi})^{\sigma}\equiv 1$ mod $p$ for some $l$, if $p$ is a regular
prime. For usual mod $p$ singular form $F\in M^{n}_{k}(\Gamma_{0}(p))$, by
multiplying $(E^{n}_{l,\chi})^{\sigma}$, we can construct mod $p$ singular
form $F(E^{n}_{l,\chi})^{\sigma}\in M^{n}_{k+l}(\Gamma_{0}(p),\chi^{\sigma})$
of nonquadratic nebentypus.

  2. (2)

The embedding $\sigma$ gives an isomorphism of fields from
$\mathbb{Q}(\mu_{p-1})$ to $\mathbb{Q}(\omega(d))$ ($\subset\mathbb{Q}_{p}$)
over $\mathbb{Q}$, where $d=d_{i}$ is as in the beginning of Section 4. Then
“mod $p^{c(m)}$” in (4.6) means “mod $p^{c(m)}\mathbb{Z}_{(p)}[\omega(d)]$”.

  3. (3)

Let $\mathfrak{p}$ be a prime ideal in $\mathbb{Z}[\mu_{p-1}]$ such that the
corresponding character is $\psi$ for which $(\psi^{-1})^{\sigma}=\omega$.
Then (4.6) corresponds to $\lambda_{m}\cdot
p^{-\mu_{\delta}}E^{n}_{l_{\delta,m},\chi}$ being mod $\mathfrak{p}^{c(m)}$
singular of $\mathfrak{p}$-rank $2l$, where $\lambda_{m}$ is a rational
integer with $v_{p}(\lambda_{m})=0$ such that $\lambda_{m}\cdot
p^{-\mu_{\delta}}E^{n}_{l_{\delta,m},\chi}$ has Fourier coefficients in
$\mathbb{Z}[\mu_{p-1}]$. In fact, we can confirm this as follows. We define
$\sigma^{-1}:\mathbb{Q}(\omega(d))\to\mathbb{Q}(\mu_{p-1})$ by
$\sigma^{-1}(\omega(d)):=\zeta_{p-1}$. Then it is easy to see that
$\sigma^{-1}(p\mathbb{Z}[\omega(d)])=p\mathbb{Z}[\mu_{p-1}]\subset\mathfrak{p}$.
Hence we may apply $\sigma^{-1}$ to the both sides of (4.6).

## Acknowledgment

This work was supported by JSPS KAKENHI Grant Number 22K03259.

## Data availability

During the work on this publication, no data sets were generated, used or
analyzed. Thus, there is no need for a link to a data repository.

## References

  * [1] S. Böcherer, T. Kikuta, On mod $p$ singular modular forms. Forum Math. 28 (2016), no. 6, 1051-1065.
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  * [6] T. Ichikawa, Congruences between Siegel modular forms. Math. Ann. 342 (2008), no.3, 527-532.
  * [7] H. Katsurada, An explicit formula for Siegel series. Amer. J. Math. 121 (1999), no. 2, 415-452.
  * [8] N. M. Katz, $p$-adic properties of modular schemes and modular forms. Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 69-190 Lecture Notes in Math., Vol. 350 Springer-Verlag, Berlin-New York, 1973
  * [9] T. Kikuta, On p-adic Siegel modular forms of non-real Nebentypus. Acta Arith. 152 (2012), no.2, 175-183.
  * [10] S. Lang, Introduction to modular forms. With appendixes by D. Zagier and Walter Feit. Corrected reprint of the 1976 original Grundlehren Math. Wiss., 222[Fundamental Principles of Mathematical Sciences] Springer-Verlag, Berlin, 1995. x+261 pp. ISBN:3-540-07833-9
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Siegfried Böcherer

Kunzenhof 4B

79177 Freiburg, Germany

Email<EMAIL_ADDRESS>
Toshiyuki Kikuta

Faculty of Information Engineering

Department of Information and Systems Engineering

Fukuoka Institute of Technology

3-30-1 Wajiro-higashi, Higashi-ku, Fukuoka 811-0295, Japan

E-mail<EMAIL_ADDRESS>


# Hall effect induced by topologically trivial target skyrmions

Tan Dao  Department of Physics, Harvard University, Cambridge, MA 02138, USA
Department of Physics and Astronomy, University of New Hampshire, Durham, New
Hampshire 03824, USA  Sergey S. Pershoguba  Department of Physics and
Astronomy, University of New Hampshire, Durham, New Hampshire 03824, USA
Jiadong Zang<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of New Hampshire, Durham, New Hampshire 03824, USA  Materials
Science Program, University of New Hampshire, Durham, New Hampshire 03824, USA

###### Abstract

Electrons moving through a noncoplanar magnetic texture acquire a Berry phase,
which can be described as an effective magnetic field. This effect is known as
the topological Hall effect and has been observed in topological spin
textures. Motivated by recent experimental realizations, here we study the
Hall effect in a nontopological magnetic texture known as a target skyrmion.
We start from a simplified semiclassical picture and show that the Hall signal
is a nonmonotonic function of both the electronic energy and target skyrmion
radius. That observation carries over to the fully quantum mechanical
treatment in a Landauer-Büttiker formalism in a mesoscopic setting. Our
conclusion challenges the popular opinion in the community that the Hall
effect in such structures necessarily requires a nonzero skyrmion number.

##  I Introduction

In contrast to the classical Hall effect [1] induced by a nonzero external
magnetic field, the anomalous Hall effect (AHE) represents a component of the
Hall signal proportional to the magnetization of a magnet [2, 3]. Therefore it
can exist even in the absence of the external magnetic field. In frustrated
chiral magnets, there is an additional contribution to AHE arising from the
noncoplanar structure of spin magnetic moments [4, 5, 6, 7, 8]. In these
materials, electrons acquire a Berry phase, therefore generating an anomalous
velocity which can be understood as the Lorentz force of an effective magnetic
field in momentum space [2, 3, 4, 5]. More interestingly, in the adiabatic
regime where itinerant electrons are strongly coupled to the spins, the Berry
curvature in real space can give rise to the AHE effect [9, 10, 11]. This
effect is also known as the topological Hall effect (THE). As the real space
Berry phase is directly related to the topology of the spin texture, the THE
is often used as evidence of nontrivial spin texture [12, 13, 14, 15, 16, 17].
One notable example of nontrivial spin texture is the magnetic skyrmion, whose
topological charge (Q = 1) is defined as the integer winding number of spin
$\bm{S(r)}$ on a unit sphere [18].

Since the discovery of skyrmions in MnSi[14], the role of topology and
magnetization has increasingly expanded for its potential use in dense and
robust data storage, as well as racetrack memory for quick data access [19,
20, 21]. However, chiral spin textures that are classified as topologically
trivial have remained underexplored in theory and experiment. In particular,
the target skyrmion whose texure is a concentric ring of one skyrmion inside
another skyrmion with reversed spins (see Fig.1a) can be moved linearly with a
current, making it promising for racetrack memory applications [22, 23, 24,
25, 26]. Currently, target skyrmion are being identified by techniques such as
magnetic force microscopy, Lorentz transmission electron miscroscopy, X-ray
photoelectron spectroscopy, and electron hollography, which can not be
integrated to electronic devices [22, 25, 27]. The topological Hall effect,
however, can be used to quickly detect presence of spin texture by measuring
the transverse voltage of devices. Here we show that the topological trivial
target skyrmion can give rise to the Hall effect.

We start with a classical model of electrons traveling in the effective
magnetic field of the target skyrmion and show a non-zero transverse current.
We then performed quantum transport calculations and observed a sign change in
the non-monotonic Hall angle dependence of target skyrmion radii. This
indicates the inner and outer skyrmion shells contribute to the Hall angle
independently. This result is further supported by the calculation of the Hall
angle for each shell; showing that the sum of the Hall angle of each shell is
the same as the target skyrmion’s Hall angle. We also shows that our results
respect the adiabatic approximation. This implies that the Hall effect came
from real space topology, which can be interpreted as the THE.

##  II Target skyrmion configuration

To set the stage, we discuss details of a target-skyrmion texture in this
section. We consider a two-dimensional ferromagnet described by a
magnetization vector $\bm{S}(\bm{r})$ normalized to unity
$|\bm{S}(\bm{r})|=1$. We may use the following parametrization of the target
skyrmion,

$\displaystyle\bm{S}(\bm{r})$
$\displaystyle=\left(\begin{array}[]{c}\sin\phi(\bm{r})\,\sin\theta(\bm{r})\\\
\cos\phi(\bm{r})\,\sin\theta(\bm{r})\\\ \cos\theta(\bm{r})\end{array}\right),$
(4)

where the polar and azimuthal angles

$\displaystyle\phi(\bm{r})$
$\displaystyle=\tan^{-1}\left(\frac{y}{x}\right)-\pi/2$ (5)
$\displaystyle\theta(\bm{r})$
$\displaystyle=2\pi-8\tan^{-1}\left[\exp\left(\frac{4r}{r}_{0}\right)\right]$

are position-$\bm{r}$ dependent. The specific choice of the angle (5) is
motivated by the necessity to eliminate spurious electronic scattering at the
target skyrmion boundary [28, 22].

Figure 1: a) Target skyrmion spins configuration. The arrows indicate the
spins rotation, and the colorbar indicates $S^{z}$ magnitude. b) The effective
magnetic field of a target skyrmion.

Assuming that we are working in the adiabatic regime, we may apply the real
space Berry phase picture by evaluating the effective magnetic field

$B_{z}(\bm{r})=\frac{1}{2}\bm{S}(\bm{r})\cdot[\partial_{x}\bm{S}(\bm{r})\times\partial_{y}\bm{S}(\bm{r})]$
(6)

The effective magnetic field distribution of the target skyrmion is shown in
Fig. 1b. The inner shell’s field is positive and greatest at the center of the
target skyrmion due to a larger gradient of the spins at around the center
spin. The outer shell’s field is negative and is much weaker. The total
magnetic field added up to zero which is consistent with $Q=0$ of the target
skyrmion [22, 23]. We use the effective field of the target skyrmion for our
classical numerical calculations.

##  III Classical model

Figure 2: (a) Approximate representation of a target skyrmion. The magnetic
fields in the inner (i) circular and outer (o) annular regions have opposite
signs [Fig. 1(b)]. The opposite Lorentz force bends the corresponding
cyclotron trajectories in the corresponding regions in the opposite direction.
The scattering angle $\Phi(\rho)$ is given by Eq. (9).

As a warm-up, we analyze classical scattering of electrons of the target
skyrmion. We approximate the target-skyrmion as a two-shell structure shown in
the Fig. 2. The inner (i) circular shell has a radius $r_{i}$, and the outer
(o) annular region has radius $r_{o}$. The effective magnetic fields have
opposite signs $\bm{B}_{i}=+\hat{\bm{z}}\,B_{i}$ and
$\bm{B}_{o}=-\hat{\bm{z}}\,B_{o}$, which we assume constant within the
respective regions. The specific values are constrained by the condition that
the total magnetic flux piercing through the structure vanishes, i.e.

$\displaystyle B_{i}\pi r_{i}^{2}=B_{o}\pi\left(r_{o}^{2}-r_{i}^{2}\right).$
(7)

In the presence of the effective magnetic fields $B_{i}$ and $B_{o}$, the
particles moves along the cyclotron trajectories with radii

$\displaystyle R_{i}=\frac{p}{B_{i}}\,\,{\rm and}\,\,R_{o}=\frac{p}{B_{o}}$
(8)

determined by the magnetic fields in the respective regions $B_{i}$ and
$B_{o}$ as well as the momentum $p=\sqrt{2mE}$. A representative electronic
trajectory is shown in Fig. 2. By matching the circular trajectories in the
two regions, we find the scattering angle dependence

$\displaystyle\Phi(\rho)=2\,{\rm
Re}\left\\{\,{\rm\,acos}\left[\frac{R_{o}+\rho}{\sqrt{R_{o}^{2}+r_{o}^{2}+2\rho
R_{o}}}\right]\right.$
$\displaystyle\qquad-{\rm\,acos}\left[\frac{2R_{o}^{2}+2\rho
R_{o}+r_{o}^{2}-r_{i}^{2}}{2R_{o}\sqrt{R_{o}^{2}+r_{o}^{2}+2\rho
R_{o}}}\right]$ (9)
$\displaystyle-\left.\,{\rm\,acos}\left[\frac{2R_{i}R_{o}-2\rho
R_{o}+r_{i}^{2}-r_{o}^{2}}{2R_{o}\sqrt{R_{i}^{2}+r_{i}^{2}-2\rho
R_{i}+(r_{i}^{2}-r_{o}^{2})R_{i}/R_{o}}}\right]\right\\}.$

on the impact parameter $\rho$.

To get a feeling of Eq. (9), let us give the low-energy (high-field) limiting
cases of $R_{o}/r_{o}\ll 1$

$\displaystyle\Phi(\rho)\approx 2\,{\rm Re}\left\\{\,{\rm
acos}\left[\frac{\rho}{r_{o}}+\frac{R_{o}}{r_{o}}\left(1-\frac{\rho^{2}}{r_{o}^{2}}\right)\right]\right\\},$
(10)

where only the first term in Eq. (9) corresponding to the outer skyrmion
contributes, as well as the high-energy (low-field) limit $r_{o}/R_{o}\ll 1$

$\displaystyle\Phi(\rho)\approx 2{\rm
Re}\left\\{\frac{1}{R}_{o}\sqrt{r_{0}^{2}-\rho^{2}}-\frac{R_{o}+R_{i}}{R_{o}R_{i}}\sqrt{r_{i}^{2}-r_{0}^{2}}\right\\}.$
(11)

Here, the former and the latter terms represent the effect of the outer and
inner skyrmions, respectively. The opposite sign of those terms correspond to
the opposite direction of the Lorentz force acting in the respective regions.
Then, it is instructive to evaluate the skew-scattering Hall (H) component of
the differential cross-section (having units of length in 2D)

$\displaystyle\sigma_{H}=\int_{-\infty}^{\infty}d\rho\,\sin[\Phi(\rho)].$ (12)

In the rest of the paper, we discuss the Hall resistance $R_{H}$ induced by a
target skyrmion, so there is no confusion between the Hall conductivity and
Eq. (12). Evaluating Eq. (12) for the low-energy (10) and the high-energy (11)
limits, we obtain

$\displaystyle\sigma_{H}$ $\displaystyle\approx\frac{\pi R_{o}}{2}\,\,\,{\rm
for}\,\,\,\frac{R_{o}}{r_{o}}\ll 1,$ (13) $\displaystyle\sigma_{H}$
$\displaystyle\approx\pi\left[\frac{r_{o}^{2}}{R_{o}}-\frac{(R_{i}+R_{o})r_{i}^{2}}{R_{o}R_{i}}\right],\,\,\,{\rm
for}\,\,\,\frac{r_{o}}{R_{o}}\ll 1.$ (14)

It is expected that at low energies, the carriers are deflected by the Lorentz
force and are not able to penetrate into the inner shell of the target
skyrmion. So the skew-scattering cross-section (13) is determined only by the
effective magnetic field of the outer skyrmion. In the high-energy limit, it
is interesting to note that $\sigma_{H}\to 0$ vanishes once the constraint of
equal magnetic fluxes (7) is taken into account. It corresponds to the known
result in the literature [29], that skew-scattering is proportional to the
magnetix flux at high energies. However, the latter vanishes for a target
skyrmion. So, we expect a greater suppression of the Hall effect at higher
energies relative to that of, e.g., conventional skyrmion generating nonzero
magnetic flux. This consideration indicates that the Hall effect in a target
skyrmion is a nonmonotonic function of both the carrier energy and target
skyrmion radius. In the rest of the paper, we delve into that analysis for
more realistic target skyrmion geometries.

Next, we solve a continuous model of a target skyrmion profile illustrated in
Fig. 3. We simulate a classical model of electrons traveling through a target
skyrmion. Electrons obey Lorentz force
$F=q(\textbf{E}+\textbf{v}\times\textbf{B})$, where there is no external
electric field $\textbf{E}=0$ in our model. Using
$\textbf{B}=B_{z}(\textbf{r})$ from Fig.1b, we calculate the transverse
current $I_{\text{y}}=\int d\rho$ $v_{y}(\rho)$. We used Runge-Kutta method to
solve the differential equations of electrons in a magnetic field for
$v_{y}(\rho)$. The initial condition for the simulation were an array of
electrons uniformly distributed along the left lead with a fixed initial
momentum in $\hat{x}$-direction. Here we set the electron charge and mass to
one.

We calculated $I_{\text{y}}$ as a function of target skyrmion radius (Fig. 3a)
and as a function of electron momentum (Fig. 3b). A sign switch in
$I_{\text{y}}(r_{0})$ indicates a competing order between the inner and outer
shell. The trajectories of the electrons traveling through the target skyrmion
were plotted in Fig. 3c,d for $p=5$ and $p=15$, respectively. For small
momentum $p<12$, $I_{\text{y}}$ is positive indicating that most of the
electrons are deflected by the negative magnetic field of the outer shell. As
shown in 3c where $p=5$, the trajectories ended up in the top lead more than
than bottom lead. For $p>12$, we observed a sign switch of $I_{\text{y}}$.
Electrons with high momentum can pass through the outer shell with little
change in their trajectories, but are strongly deflected to the bottom lead by
the inner shell. In the high momentum limits, we see that $I_{\text{y}}$
approaches 0. Classically, we have shown that topological trivial target
skyrmion can give raise to a non-monotonic behavior of the Hall effect due to
variations in the magnetic field. The next section will focus on quantum
transport of the target skrymion.

Figure 3: a) Transverse current as a function of target skyrmion radius for a
fixed momentum $p=15$. b) transverse current as a function of momentum for
$r_{0}=N/2$. c,d) Electron trajectories in a target skyrmion’s effective field
for $p=5$ and $p=15$.

##  IV Numerical modeling.

###  IV.1 Tight binding model

Here we used Kwant, a software package for quantum transport, to calculate the
Hall resistance and Hall angle of target skyrmion [30]. The tight-binding
Hamiltonian of the conduction electron is given by,

$H=-t\sum_{\langle
ij\rangle}c_{i}^{\dagger}c_{j}-J\sum_{i}c_{i}^{\dagger}\bm{\sigma}\cdot\bm{S}c_{i},$
(15)

where $c_{i}^{\dagger}$ and $c_{i}$ are the fermionic creation and
annihilation operators of an electron at site $i$, $t=1$ is the nearest
neighbor hopping, and $J$ is the Hund’s coupling strength between the
itinerant electrons and the background spin texture $\bm{S}$. The band is
symmetric with a band width of $8t+2J$ for $J<4$. Above $J>4$, the band is
gapped with a gap of $2J-8t$. Here we focus on $J<4t$, and the transport
energy right below the edge of the band overlap region $E\approx 4t-J$. In
this range, we observed a non-monotonic behavior of the Hall effect as a
function of the target skyrmion radius (Fig. 4a), where it is monotonic for
the skyrmion in Ref. [28].

###  IV.2 Hall resistance and Hall angle

We simulate four ferromagnetic leads attached to the sample and calculate the
transmission probabilities. The leads are fully attached to the left, right,
top and bottom of the system. We impose current ($I$) from the left lead to
the right lead and measure the voltage ($V$) across the top and bottom lead.
We applied Landauer-Büttiker formalism to obtain the Hall resistance and Hall
angle. The equation for Hall resistance is computed as [31],

$R_{\text{H}}=\frac{V}{I}=\frac{h}{2e^{2}}\frac{R^{2}-L^{2}}{(R+L)(R^{2}+L^{2}+2F(F+R+L))},$
(16)

where we set $h/e^{2}$ to unity. See appendix.A for leads configuration and
derivation of the Hall resistance.

The Hall angle is defined as the resultant angle between the Hall voltage
($E_{H}$) and the longitudinal voltage ($E_{x}$). Since $E_{H}$ contributed by
the spin texture is very small compared to $E_{x}$, the Hall angle thus can be
expressed as the ratio between $E_{H}$ and $E_{x}$ (Eq 17). Where $E_{H}$
($E_{x}$) is the difference between the top and bottom terminal voltages (left
and right terminal voltages). Detailed derivations of the terminal voltages
can be found in Ref. [28, 32].

$\theta_{\text{H}}=\frac{E_{H}}{E_{x}}=\frac{V_{T}-V_{B}}{V_{R}-V_{L}},$ (17)

##  V Results

###  V.1 Transport results for a target skyrmion

Figure 4: (a) Target skyrmion and skyrmion Hall angle as a function of energy
for $N$x$N$ = $96\times 96$, $r_{0}=48$, and $J=1/3$. Broken plot to help
focus on the non-zero Hall angle region. (b) $S^{z}$ profile taken across the
center of the target skyrmion for various radii. The spins are rotated by
$2\pi$ with respect to the center of spin. (c) Zoomed in plot for Hall
resistance and Hall angle as a function of the transport energy for various
target skyrmion radii.(d) The Hall resistance and Hall angle as a function of
target skyrmion radius for $E=3.64$.

Here we start with a big picture by comparing the Hall angles between the
normal skyrmion and the target skyrmion for $J=1/3$. We calculated the Hall
angle for system size $N$x$N$ = 96$a_{0}$x96$a_{0}$ where $a_{0}=1$ is the
lattice constant, and large initial $r_{0}=48$ to satisfy the adiabatic
approximation for the skyrmion [28]. In the next section we show that
$r_{0}=48$ also satisfies the adiabatic approximation for the target skyrmion.
By sweeping the energy from $E=0$ to $E=4t+J$, we observed a zero Hall angle
for $|E|<J$ in the target skyrmion, contrasting to a constant Hall angle for
the skyrmion. For $J<E<4t-J$ where the spin up and spin down electrons are
mixed due to the band overlap, the skyrmion has a zero Hall angle, while the
target skyrmion has a peak at the edge of the band overlap where $E\approx
4t-J$. The zero Hall angle phenomena for the skyrmion has been thoroughly
investigated in Ref.[28, 32].

We look closely at the peak at $E\approx 4t-J$ for the target skyrmion and
study its radius dependence. Figure 4b shows $S^{z}$ along a line cut through
the middle of the target skyrmion. As the radius increases, the target
skyrmion transitions from a sharp spin texture to a slowly varying spins
texture. Fig 4c shows the Hall angle and resistance for various radii. The
peak disappears as the radius decreases. For small radius, $r_{0}<10$, the
Hall angle at full energy range is similar to the skyrmion. We fixed the
transport energy at the peak position $E=3.64$ for $r_{0}=48$ and record the
Hall angle and resistance.

Fig 4d shows a non-monotonic behavior of $R_{\text{H}}(r_{0})$ and
$\theta_{\text{H}}(r_{0})$. For $r_{0}=0$, the system is in a perfect
ferromagnetic state, therefore does not exhibit any Hall effects. For
$r_{0}<24$, $R_{\text{H}}$ and $\theta_{\text{H}}$ are negative. We also see a
rough behavior for $r_{0}<10$. This can be understood as when the inner shell
is too small (i.e. its radius is less than 4$a_{0}$), it can hardly be
considered as a skyrmion. Therefore, the system as a whole is one large
skyrmion with spin defects at the center, which explains the negative Hall
effect in this region. As the radius increases, the inner shell starts to
expand into a skyrmion. The outer and inner shell scatter electrons in
opposite directions, and eventually they both cancel out for $r_{0}\approx
24$. For $r_{0}>24$, $R_{\text{H}}(r_{0})$ and $\theta_{\text{H}}(r_{0})$ are
positive, indicating that Hall effect is contributed by the inner shell. For
comparison with a skyrmion, the Hall resistance and Hall angle become constant
and independent of skyrmion’s radii for $r_{0}>8$ [28]. Here, we show that the
Hall effect sign switches from negative to positive owing to the topological
Hall effect of the outer and inner shell, respectively.

###  V.2 Adiabatic approximation: J dependence

To demonstrate that the results respect the adiabatic approximation, we show
that the adiabatic parameter $\lambda_{a}>>1$, where $\lambda_{a}$ is defined
as [10, 33]

$\lambda_{a}=\frac{\omega_{s}\xi}{v_{F}},$ (18)

where $v_{F}$ is the Fermi velocity electrons in the lead.
$\omega_{s}=J/\hbar$ is the electron precession frequency. $\xi$ is the
characteristic length such that $S^{z}$ is rotated by an angle of $\pi$. We
calculated $\lambda_{a}$ for various exchange coupling at the edge of the band
overlap region to confirm that $\lambda_{a}$ is greater than one.

Figure 5: (a) Hall angle for the inner and outer shell, with parameters
$r_{0}=48$ and $N=96$. (b) Zoom in plot near the edge of the band overlap
region, and plot $|\theta_{\text{H}}|$. (c) Hall angle plot of a target
skyrmion (green plots) and combined Hall angle of the inner and outer shell
(blue plots).

Without loss of generality, we focus only on the $\theta_{\text{H}}$ for the
results presented in this section. We calculated the Hall angle of the target
skyrmion for multiple $J$ with $J<4t$, and observed the magnitude of Hall
angles decrease as $J$ increases (Fig. 5c). This may be seen as problematic as
it indicates a vanishing Hall angle. Here we resolve this issue by
demonstrating that the magnitude of the Hall angles are set by the finite Hall
angle of a skyrmion in the adiabatic regime. To characterize
$\theta_{\text{H}}$ dependence of the exchange coupling, we isolate the inner
and outer shell of the target skyrmion and placed them in a ferromagnetic
background with both having the same system size ($N=96$).

We placed the inner shell in a spin down ferromagnet to prevent scattering
from an abrupt domain wall. Similarly, we placed the outer shell in a spin up
ferromagnetic background. Fig. 5a shows $\theta_{\text{H}}$ of the inner and
outer shell, and the inset figures show $S^{z}$ of the inner and outer shell.
Since the shells have opposite topological charge, their Hall angles also have
the opposite signs: the inner shell has positive $\theta_{\text{H}}$ for
$E>4t-J$, and the outer shell has negative $\theta_{\text{H}}$. We zoom in on
the energy range near the edge of the bands overlap and compare
$|\theta_{\text{H}}|$ (Fig. 5b). We observed both shells have the same
magnitude for $4t-J<E$. $\theta_{\text{H}}$ of the inner shell has a soft
decrease to zero starting from $E=4t-J$, whereas the outer shell has an abrupt
jump to zero. This difference gives raise to the Hall angle and the magnitude
is set by the single skyrmion (Fig.5c). In the adiabatic limits, all skyrmion
Hall angle becomes constant right above the band overlap region, and thus the
target skyrmion’s Hall angle never vanishes for $J<4t$. We compared the total
Hall angle obtained by calculating the contribution from each shells versus
the calculation of one target skyrmion. Here we demonstrated that the Hall
angle of the target skyrmion is the sum of the inner and outer shell’s Hall
angles, as the Hall angles superimposed one another in Fig.5c. Furthermore,
our results respect the adiabatic approximation which mean that the Hall angle
arises from real space topology and can be classified as the topological Hall
effect despite it the spin texture has trivial topology.

##  VI Conclusion

In this work, we investigated the Hall effect in a recently discovered target
skyrmion spin configuration [22, 24, 25, 26, 27]. The target skyrmion is
characterized by a zero skyrmion (topological) charge. We started from a
highly-simplified semiclassical model of a target skyrmion and found that the
Hall effect is a non-monotonic function of both the carrier energy and target
skyrmion radius. On top of that, we found that the Hall effect changes sign,
which indicates competition between opposite Hall effect produced by the inner
and outer shell of the target skyrmion. This observation carries over to the
fully quantum mechanical treatment, where we observed a nonmotonic Hall angle
and Hall resistance as a function of starget skyrmion radius. We developed a
method to better understand the transport contribution in the target skyrmion
by isolating the shells in ferromagnetic backgrounds. We showed that, in the
broad region of energies, the Hall effect can be decomposed into a
superposition of the Hall effects produced by the distinct shells of the
target skyrmion. This method could be applied to more complex spin textures,
which may facilitate understanding of the result of transport theory
calculations. Our results suggest that it may possible to detect the presence
of target skyrmions in experiments and call for further experimental
investigation of the Hall effect in such textures. Finally, our result
challenges the popular belief in the community that the Hall effect
necessarily implies a nonzero skyrmion number.

##  VII Acknowledgement

This work was supported by the U.S. Department of Energy, Basic Energy Science
under award No. DE-SC0020221.

##  Appendix A Hall resistance

Figure 6: The leads (red) are fully attached to the scattering region (blue).

We ran a current from lead 1 to lead 3, and measure the voltage across lead 2
and 4. Without loss of generality we set $V_{4}=0$ and measure $V_{2}$. Fig. 6
shows the leads configuration. We start with $\bm{I}=\bar{\bm{G}}\bm{V}$,
where $\bar{\bm{G}}$ is the the conductance matrix defined as

$\bar{\bm{G}}=\frac{2e^{2}}{h}\begin{bmatrix}T_{0}&-T_{12}&-T_{14}\\\
-T_{21}&T_{0}&-T_{24}\\\ -T_{41}&-T_{42}&T_{0}\end{bmatrix}.$ (19)

Since four leads are completely symmetric, we can define forward transmission,
right turning, and left turning as follows:

$\begin{matrix}F=T_{13}=T_{31}=T_{24}=T_{42}\\\
R=T_{12}=T_{23}=T_{34}=T_{41}\\\ L=T_{14}=T_{43}=T_{32}=T_{21}\end{matrix}$
(20)

and

$T_{0}=F+R+L$ (21)

Solve for the voltage $\bm{V}=\bar{\bm{R}}\bm{I}$ where $\bar{\bm{R}}$ is the
resistance matrix (i.e. inverse of the conductance matrix). Using the current
$I_{1}=I=-I_{3}$ and $I_{2}=0$, we obtained

$V_{2}=\frac{I}{|\bar{\bm{G}}|}(R^{2}-L^{2}),$ (22)

where

$|\bar{\bm{G}}|=\frac{2e^{2}}{h}(R+L)(R^{2}+L^{2}+2F(F+R+L))$ (23)

We arrive at the Hall resistance $R_{\text{H}}=V_{2}/I$ which is Eq.16.

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