# Instructions for ACL-IJCNLP 2021 Proceedings

First Author
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Second Author
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###### Abstract

This document contains the instructions for preparing a manuscript for the
proceedings of ACL-IJCNLP 2021. The document itself conforms to its own
specifications, and is therefore an example of what your manuscript should
look like. These instructions should be used for both papers submitted for
review and for final versions of accepted papers. Authors are asked to conform
to all the directions reported in this document.

##  1 Credits

This document has been adapted by Roberto Navigli from the instructions for
earlier ACL, NAACL and EMNLP proceedings, including those for EMNLP 2020 by
Yulan He, ACL 2020 by Steven Bethard, Ryan Cotterrell and Rui Yan, ACL 2019 by
Douwe Kiela and Ivan Vulić, NAACL 2019 by Stephanie Lukin and Alla
Roskovskaya, ACL 2018 by Shay Cohen, Kevin Gimpel, and Wei Lu, NAACL 2018 by
Margaret Michell and Stephanie Lukin, 2017/2018 (NA)ACL bibtex suggestions
from Jason Eisner, ACL 2017 by Dan Gildea and Min-Yen Kan, NAACL 2017 by
Margaret Mitchell, ACL 2012 by Maggie Li and Michael White, ACL 2010 by Jing-
Shing Chang and Philipp Koehn, ACL 2008 by Johanna D. Moore, Simone Teufel,
James Allan, and Sadaoki Furui, ACL 2005 by Hwee Tou Ng and Kemal Oflazer, ACL
2002 by Eugene Charniak and Dekang Lin, and earlier ACL and EACL formats
written by several people, including John Chen, Henry S. Thompson and Donald
Walker. Additional elements were taken from the formatting instructions of the
_International Joint Conference on Artificial Intelligence_ and the
_Conference on Computer Vision and Pattern Recognition_.

##  2 Introduction

The following instructions are directed to authors of papers submitted to ACL-
IJCNLP 2021 or accepted for publication in its proceedings. All authors are
required to adhere to these specifications. Authors are required to provide a
Portable Document Format (PDF) version of their papers. The proceedings are
designed for printing on A4 paper.

##  3 Electronically-available resources

ACL provides this description and accompanying style files at

> https://2021.aclweb.org/files/acl-ijcnlp2021-templates.zip

We strongly recommend the use of these style files, which have been
appropriately tailored for the ACL-IJCNLP 2021 proceedings. You can also start
from the Overleaf template linked from the conference website.

####  LaTeX-specific details:

The templates include the LaTeX2e source (acl2021.tex), the LaTeX2e style file
used to format it (acl2021.sty), an ACL bibliography style (acl_natbib.bst),
an example bibliography (acl2021.bib), and the bibliography for the ACL
Anthology (anthology.bib).

##  4 Length of Submission

The conference accepts submissions of long papers and short papers. Long
papers may consist of up to eight (8) pages of content plus unlimited pages
for references and ethics/broader impact statement. Short papers may consist
of up to four (4) pages of content, plus unlimited pages for references and
ethics/broader impact statement. Upon acceptance, final versions of both long
and short papers will be given one additional content page to address the
reviewers’ comments.

For both long and short papers, all illustrations and tables that are part of
the main text must be accommodated within these page limits, observing the
formatting instructions given in the present document. Papers that do not
conform to the specified length and formatting requirements are subject to be
rejected without review.

The conference encourages the submission of additional material that is
relevant to the reviewers but not an integral part of the paper. There are two
such types of material: appendices, which can be read, and non-readable
supplementary materials, often data or code. Additional material must be
submitted as separate files, and must adhere to the same anonymity guidelines
as the main paper. Appendices should be no longer than 4 pages. The paper must
be self-contained: it is optional for reviewers to look at the supplementary
material. The paper should not refer, for further detail, to documents, code
or data resources that are not available to the reviewers. Refer to Section 8
for further information.

Workshop chairs may have different rules for allowed length and whether
supplemental material is welcome. As always, the respective call for papers is
the authoritative source.

##  5 Anonymity

As reviewing will be double-blind, papers submitted for review should not
include any author information (such as names or affiliations). Furthermore,
self-references that reveal the author’s identity, _e.g._ ,

> We previously showed (Gusfield:97) …

should be avoided. Instead, use citations such as

> Gusfield:97 previously showed…

Please do not use anonymous citations and do not include acknowledgements.
Papers that do not conform to these requirements may be rejected without
review.

Any preliminary non-archival versions of submitted papers should be listed in
the submission form but not in the review version of the paper. Reviewers are
generally aware that authors may present preliminary versions of their work in
other venues, but will not be provided the list of previous presentations from
the submission form.

Please do not include github links that reveal the authors’ identities in your
submission. If you feel it is important to include your source code, you can
zip the code and submit it to softconf.

Once a paper has been accepted to the conference, the camera-ready version of
the paper should include the author’s names and affiliations, and is allowed
to use self-references and provide the related github link.

####  LaTeX-specific details:

For an anonymized submission, ensure that `\aclfinalcopy` at the top of this
document is commented out, and that you have filled in the paper ID number
(assigned during the submission process on softconf) where `***` appears in
the `\def\aclpaperid{***}` definition at the top of this document. For a
camera-ready submission, ensure that `\aclfinalcopy` at the top of this
document is not commented out.

##  6 Multiple Submission Policy

ACL-IJCNLP 2021 will not consider any paper that is under review in a journal
or another conference at the time of submission, and submitted papers must not
be submitted elsewhere during the ACL-IJCNLP 2021 review period. This policy
covers all refereed and archival conferences and workshops (e.g., NAACL). The
only exception is that a paper can be dual-submitted to both ACL-IJCNLP 2021
and an ACL-IJCNLP workshop.

In addition, we will not consider any paper that overlaps significantly in
content or results with papers that will be (or have been) published
elsewhere. Authors submitting more than one paper to ACL-IJCNLP 2021 must
ensure that their submissions do not overlap significantly ($>25$%) with each
other in content or results.

##  7 Formatting Instructions

Manuscripts must be in two-column format. Exceptions to the two-column format
include the title, authors’ names and complete addresses, which must be
centered at the top of the first page, and any full-width figures or tables
(see the guidelines in Section 7.5). Type single-spaced. Start all pages
directly under the top margin. The manuscript should be printed single-sided
and its length should not exceed the maximum page limit described in Section
4. Pages should be numbered in the version submitted for review, but pages
should not be numbered in the camera-ready version.

####  LaTeX-specific details:

The style files will generate page numbers when `\aclfinalcopy` is commented
out, and remove them otherwise.

###  7.1 File Format

For the production of the electronic manuscript you must use Adobe’s Portable
Document Format (PDF). Please make sure that your PDF file includes all the
necessary fonts (especially tree diagrams, symbols, and fonts with Asian
characters). When you print or create the PDF file, there is usually an option
in your printer setup to include none, all or just non-standard fonts. Please
make sure that you select the option of including ALL the fonts. Before
sending it, test your PDF by printing it from a computer different from the
one where it was created. Moreover, some word processors may generate very
large PDF files, where each page is rendered as an image. Such images may
reproduce poorly. In this case, try alternative ways to obtain the PDF. One
way on some systems is to install a driver for a postscript printer, send your
document to the printer specifying “Output to a file”, then convert the file
to PDF.

It is of utmost importance to specify the A4 format (21 cm x 29.7 cm) when
formatting the paper. Print-outs of the PDF file on A4 paper should be
identical to the hardcopy version. If you cannot meet the above requirements
about the production of your electronic submission, please contact the
publication chairs as soon as possible.

####  LaTeX-specific details:

PDF files are usually produced from LaTeX using the pdflatex command. If your
version of LaTeX produces Postscript files, ps2pdf or dvipdf can convert these
to PDF. To ensure A4 format in LaTeX, use the command
`\special{papersize=210mm,297mm}` in the LaTeX preamble (below the
`\usepackage` commands) and use dvipdf and/or pdflatex; or specify -t a4 when
working with dvips.

###  7.2 Layout

Format manuscripts two columns to a page, in the manner these instructions are
formatted. The exact dimensions for a page on A4 paper are:

  * •

Left and right margins: 2.5 cm

  * •

Top margin: 2.5 cm

  * •

Bottom margin: 2.5 cm

  * •

Column width: 7.7 cm

  * •

Column height: 24.7 cm

  * •

Gap between columns: 0.6 cm

Papers should not be submitted on any other paper size. If you cannot meet the
above requirements about the production of your electronic submission, please
contact the publication chairs above as soon as possible.

###  7.3 Fonts

For reasons of uniformity, Adobe’s Times Roman font should be used. If Times
Roman is unavailable, you may use Times New Roman or Computer Modern Roman.

Table 1 specifies what font sizes and styles must be used for each type of
text in the manuscript.

Type of Text | Font Size | Style
---|---|---
paper title | 15 pt | bold
author names | 12 pt | bold
author affiliation | 12 pt |
the word “Abstract” | 12 pt | bold
section titles | 12 pt | bold
subsection titles | 11 pt | bold
document text | 11 pt |
captions | 10 pt |
abstract text | 10 pt |
bibliography | 10 pt |
footnotes | 9 pt |
Table 1:  Font guide.

####  LaTeX-specific details:

To use Times Roman in LaTeX2e, put the following in the preamble:

>
>     \usepackage{times}
>     \usepackage{latexsym}
>

###  7.4 Ruler

A printed ruler (line numbers in the left and right margins of the article)
should be presented in the version submitted for review, so that reviewers may
comment on particular lines in the paper without circumlocution. The presence
or absence of the ruler should not change the appearance of any other content
on the page. The camera ready copy should not contain a ruler.

#### Reviewers:

note that the ruler measurements may not align well with lines in the paper –
this turns out to be very difficult to do well when the paper contains many
figures and equations, and, when done, looks ugly. In most cases one would
expect that the approximate location will be adequate, although you can also
use fractional references (_e.g._ , this line ends at mark $295.5$).

####  LaTeX-specific details:

The style files will generate the ruler when `\aclfinalcopy` is commented out,
and remove it otherwise.

###  7.5 Title and Authors

Center the title, author’s name(s) and affiliation(s) across both columns. Do
not use footnotes for affiliations. Place the title centered at the top of the
first page, in a 15-point bold font. Long titles should be typed on two lines
without a blank line intervening. Put the title 2.5 cm from the top of the
page, followed by a blank line, then the author’s names(s), and the
affiliation on the following line. Do not use only initials for given names
(middle initials are allowed). Do not format surnames in all capitals (_e.g._
, use “Mitchell” not “MITCHELL”). Do not format title and section headings in
all capitals except for proper names (such as “BLEU”) that are conventionally
in all capitals. The affiliation should contain the author’s complete address,
and if possible, an electronic mail address.

The title, author names and addresses should be completely identical to those
entered to the electronical paper submission website in order to maintain the
consistency of author information among all publications of the conference. If
they are different, the publication chairs may resolve the difference without
consulting with you; so it is in your own interest to double-check that the
information is consistent.

Start the body of the first page 7.5 cm from the top of the page. Even in the
anonymous version of the paper, you should maintain space for names and
addresses so that they will fit in the final (accepted) version.

###  7.6 Abstract

Use two-column format when you begin the abstract. Type the abstract at the
beginning of the first column. The width of the abstract text should be
smaller than the width of the columns for the text in the body of the paper by
0.6 cm on each side. Center the word Abstract in a 12 point bold font above
the body of the abstract. The abstract should be a concise summary of the
general thesis and conclusions of the paper. It should be no longer than 200
words. The abstract text should be in 10 point font.

###  7.7 Text

Begin typing the main body of the text immediately after the abstract,
observing the two-column format as shown in the present document.

Indent 0.4 cm when starting a new paragraph.

###  7.8 Sections

Format section and subsection headings in the style shown on the present
document. Use numbered sections (Arabic numerals) to facilitate cross
references. Number subsections with the section number and the subsection
number separated by a dot, in Arabic numerals.

###  7.9 Footnotes

Put footnotes at the bottom of the page and use 9 point font. They may be
numbered or referred to by asterisks or other symbols.111This is how a
footnote should appear. Footnotes should be separated from the text by a
line.222Note the line separating the footnotes from the text.

###  7.10 Graphics

Place figures, tables, and photographs in the paper near where they are first
discussed, rather than at the end, if possible. Wide illustrations may run
across both columns. Color is allowed, but adhere to Section 9’s guidelines on
accessibility.

#### Captions:

Provide a caption for every illustration; number each one sequentially in the
form: “Figure 1. Caption of the Figure.” “Table 1. Caption of the Table.” Type
the captions of the figures and tables below the body, using 10 point text.
Captions should be placed below illustrations. Captions that are one line are
centered (see Table 1). Captions longer than one line are left-aligned (see
Table 2).

Command | Output
---|---
`{\"a}` | ä
`{\^e}` | ê
`{\‘i}` | ì
`{\.I}` | İ
`{\o}` | ø
`{\’u}` | ú
`{\aa}` | å

Command | Output
---|---
`{\c c}` | ç
`{\u g}` | ğ
`{\l}` | ł
`{\~n}` | ñ
`{\H o}` | ő
`{\v r}` | ř
`{\ss}` | ß

Table 2: Example commands for accented characters, to be used in, _e.g._ ,
BibTeX names.

####  LaTeX-specific details:

The style files are compatible with the caption and subcaption packages; do
not add optional arguments. Do not override the default caption sizes.

###  7.11 Hyperlinks

Within-document and external hyperlinks are indicated with Dark Blue text,
Color Hex #000099.

###  7.12 Citations

Citations within the text appear in parentheses as (Gusfield:97) or, if the
author’s name appears in the text itself, as Gusfield:97. Append lowercase
letters to the year in cases of ambiguities. Treat double authors as in
(Aho:72), but write as in (Chandra:81) when more than two authors are
involved. Collapse multiple citations as in (Gusfield:97; Aho:72).

Refrain from using full citations as sentence constituents. Instead of

> “(Gusfield:97) showed that …”

write

> “Gusfield:97 showed that …”

Output | natbib command | Old ACL-style command
---|---|---
(Gusfield:97) | `\citep` | `\cite`
Gusfield:97 | `\citealp` | no equivalent
Gusfield:97 | `\citet` | `\newcite`
(Gusfield:97) | `\citeyearpar` | `\shortcite`
Table 3:  Citation commands supported by the style file. The style is based on
the natbib package and supports all natbib citation commands. It also supports
commands defined in previous ACL style files for compatibility.

####  LaTeX-specific details:

Table 3 shows the syntax supported by the style files. We encourage you to use
the natbib styles. You can use the command `\citet` (cite in text) to get
“author (year)” citations as in Gusfield:97. You can use the command `\citep`
(cite in parentheses) to get “(author, year)” citations as in (Gusfield:97).
You can use the command `\citealp` (alternative cite without parentheses) to
get “author year” citations (which is useful for using citations within
parentheses, as in Gusfield:97).

###  7.13 References

Gather the full set of references together under the heading References; place
the section before any Appendices. Arrange the references alphabetically by
first author, rather than by order of occurrence in the text.

Provide as complete a citation as possible, using a consistent format, such as
the one for _Computational Linguistics_ or the one in the _Publication Manual
of the American Psychological Association_ (APA:83). Use full names for
authors, not just initials.

Submissions should accurately reference prior and related work, including code
and data. If a piece of prior work appeared in multiple venues, the version
that appeared in a refereed, archival venue should be referenced. If multiple
versions of a piece of prior work exist, the one used by the authors should be
referenced. Authors should not rely on automated citation indices to provide
accurate references for prior and related work.

The following text cites various types of articles so that the references
section of the present document will include them.

  * •

Example article in journal: (Ando2005).

  * •

Example article in proceedings, with location: (borschinger-
johnson-2011-particle).

  * •

Example article in proceedings, without location: (andrew2007scalable).

  * •

Example arxiv paper: (rasooli-tetrault-2015).

####  LaTeX-specific details:

The LaTeX and BibTeX style files provided roughly follow the American
Psychological Association format. If your own bib file is named acl2021.bib,
then placing the following before any appendices in your LaTeX file will
generate the references section for you:

> `\bibliographystyle{acl_natbib}`
> `\bibliography{acl2021}`

You can obtain the complete ACL Anthology as a BibTeX file from
https://aclweb.org/anthology/anthology.bib.gz. To include both the anthology
and your own bib file, use the following instead of the above.

> `\bibliographystyle{acl_natbib}`
> `\bibliography{anthology,acl2021}`

###  7.14 Digital Object Identifiers

As part of our work to make ACL materials more widely used and cited outside
of our discipline, ACL has registered as a CrossRef member, as a registrant of
Digital Object Identifiers (DOIs), the standard for registering permanent URNs
for referencing scholarly materials.

All camera-ready references are required to contain the appropriate DOIs (or,
as a second resort, the hyperlinked ACL Anthology Identifier) to all cited
works. Appropriate records should be found for most materials in the current
ACL Anthology at http://aclanthology.info/. As examples, we cite (goodman-
etal-2016-noise) to show you how papers with a DOI will appear in the
bibliography. We cite (harper-2014-learning) to show how papers without a DOI
but with an ACL Anthology Identifier will appear in the bibliography.

####  LaTeX-specific details:

Please ensure that you use BibTeX records that contain DOI or URLs for any of
the ACL materials that you reference. If the BibTeX file contains DOI fields,
the paper title in the references section will appear as a hyperlink to the
DOI, using the hyperref LaTeX package.

##  8 Supplementary Materials

Supplementary material may report preprocessing decisions, model parameters,
and other details necessary for the replication of the experiments reported in
the paper. Seemingly small preprocessing decisions can sometimes make a large
difference in performance, so it is crucial to record such decisions to
precisely characterize state-of-the-art methods.

Nonetheless, supplementary material should be supplementary (rather than
central) to the paper. Submissions that misuse the supplementary material may
be rejected without review. Supplementary material may include explanations or
details of proofs or derivations that do not fit into the paper, lists of
features or feature templates, sample inputs and outputs for a system,
hyperparameter values, pseudo-code or source code, and data.

The paper should not rely on the supplementary material: while the paper may
refer to and cite the supplementary material and the supplementary material
will be available to the reviewers, they will not be asked to review the
supplementary material.

Supplementary material can be uploaded as up to three separate files: a single
appendix file in PDF format, a single .tgz or .zip archive containing
software, and/or a single .tgz or .zip archive containing data.

###  8.1 Appendices

Appendices are material that can be read, and include lemmas, formulas,
proofs, and tables that are not critical to the reading and understanding of
the paper. In the submitted version, appendices should be uploaded as a
separate file. Submissions which include appendices in the main paper will be
rejected without review. In the camera ready version, instead, appendices
should directly follow the text and the references.

####  LaTeX-specific details:

In your camera ready version use `\appendix` before any appendix section to
switch the section numbering over to letters.

###  8.2 Software and Data

Submissions may include software and data used in the work and described in
the paper. Any accompanying software and/or data should include licenses and
documentation of research review as appropriate.

##  9 Accessibility

In an effort to accommodate people who are color-blind (as well as those
printing to paper), grayscale readability is strongly encouraged. Color is not
forbidden, but authors should ensure that tables and figures do not rely
solely on color to convey critical distinctions. A simple criterion: All
curves and points in your figures should be clearly distinguishable without
color.

##  10 Translation of non-English Terms

It is also advised to supplement non-English characters and terms with
appropriate transliterations and/or translations since not all readers
understand all such characters and terms. Inline transliteration or
translation can be represented in the order of:

original-form
---
transliteration
“translation”

##  11 LaTeX Compilation Issues

You may encounter the following error during compilation:

> `\pdfendlink` ended up in different nesting level than `\pdfstartlink`.

This happens when pdflatex is used and a citation splits across a page
boundary. To fix this, the style file contains a patch consisting of two
lines: (1) `\RequirePackage{etoolbox}` (line 455 in acl2021.sty), and (2) A
long line below (line 456 in acl2021.sty).

If you still encounter compilation issues even with the patch enabled, disable
the patch by commenting the two lines, and then disable the hyperref package
by loading the style file with the nohyperref option:

`\usepackage[nohyperref]{acl2021}`

Then recompile, find the problematic citation, and rewrite the sentence
containing the citation. (See, e.g., http://tug.org/errors.html)

## Acknowledgments

The acknowledgments should go immediately before the references. Do not number
the acknowledgments section. Do not include this section when submitting your
paper for review.


# On the anticyclotomic Mazur–Tate conjecture for elliptic curves with
supersingular reduction

Chan-Ho Kim  June E Huh Center for Mathematical Challenges, Korea Institute
for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.

In this short note, we study the anticyclotomic analogue of the “weak” main
conjecture of Mazur–Tate on Fitting ideals of Selmer groups for elliptic
curves with supersingular reduction.

###### Key words and phrases:

refined Iwasawa theory, anticyclotomic Iwasawa theory

###### 2010 Mathematics Subject Classification:

11F67, 11G40, 11R23

###### Contents

  1. 1 Introduction
  2. 2 Bertolini–Darmon’s theta elements and anticyclotomic $p$-adic $L$-functions
  3. 3 Anticyclotomic Iwasawa theory for elliptic curves with supersingular reduction
  4. 4 The proof of Theorem 1.3

##  1\. Introduction

###  1.1. The statement of the main result

Let $E$ be an elliptic curve of conductor $N$ over $\mathbb{Q}$ and $p\geq 5$
be a prime of supersingular reduction for $E$, so $a_{p}(E)=0$, such that the
mod $p$ representation
$\overline{\rho}:G_{\mathbb{Q}}=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GL}_{2}(\mathbb{F}_{p})$
is surjective.

Let $K$ be an imaginary quadratic field with $(D_{K},Np)=1$ where $p$ splits
and and write $p=\mathfrak{p}\cdot\overline{\mathfrak{p}}$ in $K$. Write

$N=N^{+}\cdot N^{-}$

where a prime divisor of $N^{+}$ splits in $K$ and a prime divisor of $N^{-}$
is inert in $K$.

######  Assumption 1.1 (odd).

Assume that $N^{-}$ is the square-free product of an odd number of primes.

Let $K_{\infty}$ be the anticyclotomic $\mathbb{Z}_{p}$-extension of $K$ and
$K_{n}$ be the subextension of $K$ in $K_{\infty}$ of degree $p^{n}$ for
$n\geq 0$. Let
$\Lambda_{n}=\mathbb{Z}_{p}[\mathrm{Gal}(K_{n}/K)]\simeq\mathbb{Z}_{p}[X]/((1+X)^{p^{n}}-1)$
be the finite layer Iwasawa algebra. Under Assumption 1.1, denote by

$\theta_{n}(E/K)=\sum_{\sigma\in\mathrm{Gal}(K_{n}/K)}a_{\sigma}\cdot\sigma\in\Lambda_{n}$

Bertolini–Darmon’s theta element of $E$ over $K_{n}$ reviewed in Section 2.
Let $\iota$ be the involution on $\Lambda_{n}$ defined by inverting group-like
elements. Then we have
$\iota(\theta_{n}(E/K))=\sum_{\sigma\in\mathrm{Gal}(K_{n}/K)}a_{\sigma}\cdot\sigma^{-1}$
and define

$L_{p}(E/K_{n})=\theta_{n}(E/K)\cdot\iota(\theta_{n}(E/K)).$

######  Assumption 1.2.

We assume the following hypotheses throughout this article:

  * (ram)

$K_{\infty}/K$ is totally ramified at every prime lying above $p$.

  * (CR)

If a prime $\ell$ divides $N^{-}$ and $\ell^{2}\equiv 1\pmod{p}$, then
$\overline{\rho}$ is ramified at $\ell$.

The goal of this article is to prove the following anticyclotomic analogue of
the “weak” main conjecture of Mazur and Tate [MT87, Conj. 3].

######  Theorem 1.3.

Under (odd), (ram), and (CR), we have

$L_{p}(E/K_{n})\in\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}(K_{n},E[p^{\infty}])^{\vee}\right).$

where $(-)^{\vee}$ means the Pontryagin dual.

Let $\chi:\mathrm{Gal}(K_{n}/K)\to\overline{\mathbb{Q}}^{\times}_{p}$ be a
character and we extend it to $\Lambda_{n}$ linearly. Let
$I_{\chi}:=\mathrm{ker}\left(\chi:\Lambda_{n}\to\overline{\mathbb{Q}}_{p}\right)\subseteq\Lambda_{n}$
be the augmentation ideal at $\chi$ and
$(E(K_{n})\otimes\overline{\mathbb{Q}}_{p})^{\chi}$ be the $\chi$-isotypic
subspace of $E(K_{n})\otimes\overline{\mathbb{Q}}_{p}$. Thanks to [MT87, Prop.
3], we obtain the “weak” vanishing conjecture.

######  Corollary 1.4.

Under (odd), (ram), and (CR), we have

$L_{p}(E/K_{n})\in I^{r_{\chi}}_{\chi}/I^{r_{\chi}+1}_{\chi}$

where
$r_{\chi}=\mathrm{dim}_{\overline{\mathbb{Q}}_{p}}(E(K_{n})\otimes\overline{\mathbb{Q}}_{p})^{\chi}$.

In particular, $L_{p}(E/K_{n})$ encodes an upper bound of
$\mathrm{rk}_{\mathbb{Z}}E(K)$. In the ordinary case, Corollary 1.4 can be
deduced from the one-sided divisibility of the anticyclotomic main conjecture
more directly.

######  Remark 1.5.

In [MT87], Mazur and Tate formulated “Iwasawa theory for elliptic curves over
finite abelian extensions” and several related conjectures. This finite layer
setting is much more delicate since the linear algebra techniques for the
usual Iwasawa theory (over $\mathbb{Z}_{p}$-extensions) are not applicable and
we cannot ignore “finite errors” in the computation. The formulation of the
“main conjecture” also becomes more subtle since the notion of characteristic
ideals is not available. The “weak” main conjecture is formulated in terms of
Fitting ideals. See [Kur02, Conj. 0.3] for the strong version.

###  1.2. Acknowledgement

It is a great pleasure to dedicate this paper to Massimo Bertolini on the
occasion of his 60th birthday for his great work and wonderful inspiration. In
my graduate years, the celebrated article [BD05] by Bertolini and Darmon was
one of the very first serious research articles I read. In particular, my
thesis [Kim17] can be explained as “a Greenberg–Vatsal type result for the
Bertolini–Darmon setting”. It is evident that the content of [BD05] became an
important part of my research.

I deeply thank Matteo Longo, Marco Seveso, Stefano Vigni, and Rodolfo
Venerucci for inviting me for his birthday conference and for giving me a
chance to contribute this volume.

Indeed, the topic of this paper was exactly my original thesis problem
suggested by Rob Pollack, but I could not find how to attack it at that time.
Later, I found a way to resolve it in the middle of writing the paper with
Masato Kurihara [KK21]. I deeply appreciate Rob Pollack and Masato Kurihara.
This is the sequel to [Kim], which studies the ordinary case.

This research was partially supported by a KIAS Individual Grant (SP054103)
via the Center for Mathematical Challenges at Korea Institute for Advanced
Study and by the National Research Foundation of Korea(NRF) grant funded by
the Korea government(MSIT) (No. 2018R1C1B6007009).

##  2\. Bertolini–Darmon’s theta elements and anticyclotomic $p$-adic
$L$-functions

We quickly review the construction of Gross points of conductor $p^{n}$ and
signed anticyclotomic $p$-adic $L$-functions. See [CH15, CH18, Kim19] for
details.

###  2.1. Gross points

Let $K$ be the imaginary quadratic field of discriminant $-D_{K}<0$. Define

$\vartheta:=\left\\{\begin{array}[]{ll}\dfrac{D_{K}-\sqrt{-D_{K}}}{2}&\textrm{
if }2\nmid D_{K}\\\ \dfrac{D_{K}-2\sqrt{-D_{K}}}{4}&\textrm{ if }2\mid
D_{K}\end{array}\right.$

so that $\mathcal{O}_{K}=\mathbb{Z}+\mathbb{Z}\vartheta$. Let $B_{N^{-}}$ be
the definite quaternion algebra over $\mathbb{Q}$ of discriminant $N^{-}$.
Then there exists an embedding of $K$ into $B_{N^{-}}$ [Vig80]. More
explicitly, we choose a $K$-basis $(1,J)$ of $B_{N^{-}}$ so that
$B_{N^{-}}=K\oplus K\cdot J$ such that $\beta:=J^{2}\in\mathbb{Q}^{\times}$
with $\beta<0$, $J\cdot t=\overline{t}\cdot J$ for all $t\in K$,
$\beta\in\left(\mathbb{Z}^{\times}_{q}\right)^{2}$ for all $q\mid pN^{+}$, and
$\beta\in\mathbb{Z}^{\times}_{q}$ for all $q\mid D_{K}$. Fix a square root
$\sqrt{\beta}\in\overline{\mathbb{Q}}$ of $\beta$. For a $\mathbb{Z}$-module
$A$, write $\widehat{A}=A\otimes\widehat{\mathbb{Z}}$. Fix an isomorphism

$i:=\prod
i_{q}:\widehat{B}^{(N^{-})}_{N^{-}}\simeq\mathrm{M}_{2}(\mathbb{A}^{(N^{-}\infty)})$

as follows:

  * •

For each finite place $q\mid N^{+}p$, the isomorphism
$i_{q}:B_{N^{-},q}\simeq\mathrm{M}_{2}(\mathbb{Q}_{q})$ is defined by

$\textstyle{{i_{q}(\vartheta)=\left(\begin{matrix}\mathrm{trd}(\vartheta)&-\mathrm{nrd}(\vartheta)\\\
1&0\end{matrix}\right)},}$$\textstyle{{i_{q}(J)=\sqrt{\beta}\cdot\left(\begin{matrix}-1&\mathrm{trd}(\vartheta)\\\
0&1\end{matrix}\right)}}$

where $\mathrm{trd}$ and $\mathrm{nrd}$ are the reduced trace and the reduced
norm on $B$, respectively.

  * •

For each finite place $q\nmid pN^{+}$, the isomorphism
$i_{q}:B_{N^{-},q}\simeq\mathrm{M}_{2}(\mathbb{Q}_{q})$ is chosen so that
$i_{q}\left(\mathcal{O}_{K}\otimes\mathbb{Z}_{q}\right)\subseteq\mathrm{M}_{2}(\mathbb{Z}_{q})$.

Under the fixed isomorphism $i$, for any rational prime $q$, the local Gross
point $\varsigma_{q}\in B^{\times}_{N^{-},q}$ is defined as follows:

  * •

$\varsigma_{q}:=1$ in $B^{\times}_{nN^{-},q}$ for $q\nmid pN^{+}$.

  * •

$\varsigma_{q}:=\frac{1}{\sqrt{D_{K}}}\cdot\left(\begin{matrix}\vartheta&\overline{\vartheta}\\\
1&1\end{matrix}\right)\in\mathrm{GL}_{2}(K_{\mathfrak{q}})=\mathrm{GL}_{2}(\mathbb{Q}_{q})$
for $q\mid N^{+}$ with $q=\mathfrak{q}\overline{\mathfrak{q}}$ in
$\mathcal{O}_{K}$.

  * •

$\varsigma^{(n)}_{p}=\left(\begin{matrix}\vartheta&-1\\\
1&0\end{matrix}\right)\cdot\left(\begin{matrix}p^{n}&0\\\
0&1\end{matrix}\right)\in\mathrm{GL}_{2}(K_{\mathfrak{p}})=\mathrm{GL}_{2}(\mathbb{Q}_{p})$
where $p=\mathfrak{p}\overline{\mathfrak{p}}$ splits in $K$.

By using the fixed embedding of $K$ into $B_{N^{-}}$, we define
$x_{n}:\widehat{K}^{\times}\to\widehat{B}^{\times}_{N^{-}}$ by
$x_{n}(a)=a\cdot\varsigma^{(n)}:=a\cdot\left(\varsigma^{(n)}_{p}\times\prod_{q\neq
p}\varsigma_{q}\right)$. The collection
$\left\\{x_{n}(a):a\in\widehat{K}^{\times}\right\\}$ of points is called the
Gross points of conductor $p^{n}$ on $\widehat{B}^{\times}_{N^{-}}$. The fixed
embedding $K\hookrightarrow B_{N^{-}}$ also induces an optimal embedding of
$\mathcal{O}_{n}=\mathbb{Z}+p^{n}\mathcal{O}_{K}$ into the Eichler order
$B_{N^{-}}\cap\varsigma^{(n)}\widehat{R}_{N^{+}}(\varsigma^{(n)})^{-1}$ where
$R_{N^{+}}$ is the Eichler order of level $N^{+}$ under the fixed isomorphism
$i$.

###  2.2. Theta elements and $p$-adic $L$-functions

#### 2.2.1.

Let
$\phi_{f}:B^{\times}_{N^{-}}\backslash\widehat{B}^{\times}_{N^{-}}/\widehat{R}^{\times}_{N^{+}}\to\mathbb{C}$
be the Jacquet–Langlands transfer of $f$. Since
$B^{\times}_{N^{-}}\backslash\widehat{B}^{\times}_{N^{-}}/\widehat{R}^{\times}_{N^{+}}$
is a finite set and $f$ is a Hecke eigenform, we are able to and do normalize

$\phi_{f}:B^{\times}_{N^{-}}\backslash\widehat{B}^{\times}_{N^{-}}/\widehat{R}^{\times}_{N^{+}}\to\mathbb{Z}_{p}$

such that the image of $\phi_{f}$ does not lie in $p\mathbb{Z}_{p}$. This
normalization is closely related to the $N^{-}$-new congruence ideal [PW11,
Kim17, KO23]. Let

$\widetilde{\theta}_{n}(E/K)=\sum_{[a]\in\mathcal{G}_{n}}\phi_{f}(x_{n}(a))\cdot[a]\in\mathbb{Z}_{p}[\mathcal{G}_{n}]$

where
$\mathcal{G}_{n}=K^{\times}\backslash\widehat{K}^{\times}/\widehat{\mathcal{O}}^{\times}_{n}$
and $[a]$ is the image of $a\in\widehat{K}^{\times}$ in $\mathcal{G}_{n}$.
Then Bertolini–Darmon’s theta element $\theta_{n}(E/K)$ of $E$ over $K_{n}$ is
defined by the image of $\widetilde{\theta}_{n}(E/K)$ in $\Lambda_{n}$

$\textstyle{\mathbb{Z}_{p}[\mathcal{G}_{n}]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{n}}$$\textstyle{\widetilde{\theta}_{n}(E/K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\theta_{n}(E/K).}$

It is known that $\theta_{n}(E/K)$ interpolates “an half of” $L(E,\chi,1)$
where $\chi$ runs over characters on $\mathrm{Gal}(K_{n}/K)$ [CH18, Kim19].

#### 2.2.2.

Let $\omega_{n}=\omega_{n}(X)=(1+X)^{p^{n}}-1$ and
$\Phi_{n}(1+X)=\omega_{n}(X)/\omega_{n-1}(X)$ where $\Phi_{n}$ is the
$p^{n}$-th cyclotomic polynomial. Fix a generator $\gamma$ of
$\mathrm{Gal}(K_{\infty}/K)$ and take a generator $\gamma_{n}$ of
$\mathrm{Gal}(K_{n}/K)$ as the image of $\gamma$. Then we have isomorphisms
$\Lambda_{n}\simeq\mathbb{Z}_{p}[X]/\left(\omega_{n}(X)\right)$ and
$\Lambda\simeq\mathbb{Z}_{p}\llbracket X\rrbracket$ by sending the generators
to $1+X$. Via the latter isomorphism, we also regard $\omega_{n}\in\Lambda$.
Let $\omega^{\pm}_{0}(X):=X$, $\widetilde{\omega}^{\pm}_{0}(X):=1$, and

$\textstyle{{\displaystyle\omega^{+}_{n}=\omega^{+}_{n}(X):=X\cdot\prod_{2\leq
m\leq n,m:\textrm{
even}}\Phi_{m}(1+X)},}$$\textstyle{{\displaystyle\omega^{-}_{n}=\omega^{-}_{n}(X):=X\cdot\prod_{1\leq
m\leq n,m:\textrm{
odd}}\Phi_{m}(1+X)},}$$\textstyle{{\displaystyle\widetilde{\omega}^{+}_{n}=\widetilde{\omega}^{+}_{n}(X):=\prod_{2\leq
m\leq n,m:\textrm{
even}}\Phi_{m}(1+X)},}$$\textstyle{{\displaystyle\widetilde{\omega}^{-}_{n}=\widetilde{\omega}^{-}_{n}(X):=\prod_{1\leq
m\leq n,m:\textrm{ odd}}\Phi_{m}(1+X)}.}$

Then we have
$\omega_{n}(X)=\omega^{\pm}_{n}(X)\cdot\widetilde{\omega}^{\mp}_{n}(X)$,
respectively. We also regard $\omega^{\pm}_{n}$,
$\widetilde{\omega}^{\pm}_{n}$ as elements in $\Lambda_{n}$ or $\Lambda$.

######  Proposition 2.1.

Let $\epsilon$ be the sign of $(-1)^{n}$. Then:

  1. (1)

$\omega^{\epsilon}_{n}\theta_{n}(E/K)=0$.

  2. (2)

There exists a unique element $\theta^{\epsilon}_{n}(E/K)$ in
$\Lambda/\omega^{\epsilon}_{n}\Lambda$ such that

(2.1)
$\theta_{n}(E/K)=\widetilde{\omega}^{-\epsilon}_{n}\cdot\theta^{\epsilon}_{n}(E/K).$

###### Proof.

See [DI08, Prop. 2.8]. ∎

Thanks to [DI08, Lem. 2.9],
$\left\\{(-1)^{n/2}\cdot\theta^{+}_{n}(E/K):n\textrm{ is even}\right\\}$ and
$\left\\{(-1)^{(n+1)/2}\cdot\theta^{-}_{n}(E/K):n\textrm{ is odd}\right\\}$
form compatible sequences with respect to $\Lambda/\omega^{+}_{n}\Lambda$ with
even $n$ and $\Lambda/\omega^{-}_{n}\Lambda$ with odd $n$, respectively. The
signed $p$-adic $L$-functions for $E$ and $K$ are defined by

(2.2)
$\displaystyle\begin{split}L^{+}_{p}(E/K_{\infty})&:=\varprojlim_{n,\textrm{
even}}\left((-1)^{n/2}\cdot\theta^{+}_{n}(E/K)\cdot(-1)^{n/2}\cdot\iota(\theta^{+}_{n}(E/K))\right)\\\
&=\varprojlim_{n,\textrm{
even}}\left(\theta^{+}_{n}(E/K)\cdot\iota(\theta^{+}_{n}(E/K))\right)\in\Lambda,\\\
L^{-}_{p}(E/K_{\infty})&:=\varprojlim_{n,\textrm{
odd}}\left((-1)^{(n+1)/2}\cdot\theta^{-}_{n}(E/K)\cdot(-1)^{(n+1)/2}\cdot\iota(\theta^{-}_{n}(E/K))\right)\\\
&=\varprojlim_{n,\textrm{
odd}}\left(\theta^{-}_{n}(E/K)\cdot\iota(\theta^{-}_{n}(E/K))\right)\in\Lambda.\end{split}$

##  3\. Anticyclotomic Iwasawa theory for elliptic curves with supersingular
reduction

###  3.1. Basic objects of $\pm$-Iwasawa theory

We quickly recall the basic objects of $\pm$-Iwasawa theory, which is
initiated by S. Kobayashi [Kob03] and R. Pollack [Pol03]. See also [IP06].

####  3.1.1. Local conditions at $p$

Let $E$ be an elliptic curve over $\mathbb{Q}$ with $a_{p}(E)=0$. Since
$p=\mathfrak{p}\cdot\overline{\mathfrak{p}}$ in $K$ and the (ram) condition,
we are able to write

(3.1) $K_{n,p}=K_{n,\mathfrak{p}}\oplus
K_{n,\overline{\mathfrak{p}}}\simeq\mathbb{Q}_{n,p}\oplus\mathbb{Q}_{n,p}$

where $K_{n,p}=K_{n}\otimes\mathbb{Q}_{p}$ and $K_{n,v}$ is the completion of
$K_{n}$ at $v\in\\{\mathfrak{p},\overline{\mathfrak{p}}\\}$. Then we define

$\displaystyle E^{+}(K_{n,p})$ $\displaystyle:=\\{P\in
E(K_{n,p}):\mathrm{Tr}_{n/m+1}(P)\in E(K_{m,p})\textrm{ for even }m\ (0\leq
m<n)\\}$ $\displaystyle E^{-}(K_{n,p})$ $\displaystyle:=\\{P\in
E(K_{n,p}):\mathrm{Tr}_{n/m+1}(P)\in E(K_{m,p})\textrm{ for odd }m\ (0\leq
m<n)\\}$

where $\mathrm{Tr}_{n/m+1}:E(K_{n,p})\to E(K_{m+1,p})$ is the trace map.

####  3.1.2. The norm subgroups

Let $\widehat{E}$ be the formal group associated to $E$ and $\mathfrak{m}_{n}$
be the maximal ideal of $\mathbb{Q}_{n,p}$. We define

$\displaystyle\widehat{E}^{+}(\mathfrak{m}_{n})$
$\displaystyle:=\\{P\in\widehat{E}(\mathfrak{m}_{n}):\mathrm{Tr}_{n/m+1}(P)\in\widehat{E}(\mathfrak{m}_{m})\textrm{
for even }m\ (0\leq m<n)\\}$ $\displaystyle\widehat{E}^{-}(\mathfrak{m}_{n})$
$\displaystyle:=\\{P\in\widehat{E}(\mathfrak{m}_{n}):\mathrm{Tr}_{n/m+1}(P)\in\widehat{E}(\mathfrak{m}_{m})\textrm{
for odd }m\ (0\leq m<n)\\}$

where
$\mathrm{Tr}_{n/m+1}:\widehat{E}(\mathfrak{m}_{n})\to\widehat{E}(\mathfrak{m}_{m+1})$
is the trace map.

####  3.1.3. $\pm$-Selmer groups

Following [Kim13], we define the $\pm$-Selmer groups of $E[p^{\infty}]$ over
$K_{n}$ by

$\displaystyle\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}]):=$
$\displaystyle\mathrm{ker}\left(\mathrm{Sel}(K_{n},E[p^{\infty}])\to\dfrac{\mathrm{H}^{1}(K_{n,p},E[p^{\infty}])}{E^{\pm}(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}\right)$

where $\mathrm{Sel}(K_{n},E[p^{\infty}])$ is the standard Selmer group of
$E[p^{\infty}]$ over $K_{n}$. The $\pm$-Selmer groups of $E[p^{\infty}]$ over
$K_{\infty}$ is defined by

$\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}]):=\varinjlim_{n}\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}]),$

respectively.

###  3.2. Signed Iwasawa theory

####  3.2.1. Signed main conjectures

We recall the Euler system divisibility of the signed main conjectures.

######  Theorem 3.1.

Under (odd), (ram), and (CR), we have the following statements:

  1. (1)

$\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])$ is $\Lambda$-cotorsion.

  2. (2)

$\left(L^{\pm}_{p}(E/K_{\infty})\right)\subseteq\mathrm{char}_{\Lambda}\left(\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])^{\vee}\right)$.

###### Proof.

See [Vat02, DI08, PW11, KPW17, BBL]. ∎

The signed main conjecture claims that

(3.2)
$\left(L^{\pm}_{p}(E/K_{\infty})\right)=\mathrm{char}_{\Lambda}\left(\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])^{\vee}\right)$

####  3.2.2. Non-existence of proper $\Lambda$-submodules of finite index

We recall B.D.Kim’s result on the non-existence of proper $\Lambda$-submodules
of finite index. See [Shi] for the $p$-inert case.

######  Theorem 3.2.

If $\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])$ is $\Lambda$-cotorsion, then
$\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])$ has no proper
$\Lambda$-submodule of finite index, respectively.

###### Proof.

See [Kim13, Thm. 1.1]. ∎

######  Corollary 3.3.

Under (odd), (ram), and (CR), $\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])$
has no proper $\Lambda$-submodule of finite index; thus,

$\mathrm{char}_{\Lambda}\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])=\mathrm{Fitt}_{\Lambda}\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}]).$

###### Proof.

It follows from Theorems 3.1 and 3.2. ∎

####  3.2.3. Signed control theorems

We recall the signed version of the control theorem.

######  Theorem 3.4.

The restriction map

$\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])[\omega^{\pm}_{n}]\to\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])[\omega^{\pm}_{n}]$

is injective with the finite cokernel whose size is bounded independently of
$n$. If we further assume that $p\nmid\mathrm{Tam}(E)$, then the restriction
map is an isomorphism.

###### Proof.

The $a_{p}(E)=0$ condition ensures that $E(K)[p]$ is trivial, and it implies
that the restriction map is injective as in [Kob03, Lem. 9.1]. The failure of
the surjectivity comes only from prime-to-$p$ local conditions; thus, the
situation coincides with the ordinary case. The $p\nmid\mathrm{Tam}(E)$
condition ensures that the failure vanishes. See [Kob03, Thm. 9.3], [Gre99,
Prop. 3.8], and [IP06, Thm. 6.8] for detail. ∎

###  3.3. The consequence

######  Corollary 3.5.

Under (odd), (ram), and (CR), we have

$\left(\widetilde{\omega}^{\mp}_{n}\cdot L^{\pm}_{p}(E/K_{\infty})\
(\text{mod}\
\omega_{n})\right)\subseteq\left(\widetilde{\omega}^{\mp}_{n}\right)\cdot\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])^{\vee}\right)$

in $\Lambda_{n}$, respectively.

###### Proof.

By applying Theorem 3.1 and Corollary 3.3, we have

$\left(L^{\pm}_{p}(E/K_{\infty})\right)\subseteq\mathrm{Fitt}_{\Lambda}\left(\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])^{\vee}\right).$

Taking the quotient by $\omega^{\pm}_{n}$, we obtain

$\left(L^{\pm}_{p}(E/K_{\infty})\ (\text{mod}\
\omega^{\pm}_{n})\right)\subseteq\mathrm{Fitt}_{\Lambda_{n}/\omega^{\pm}_{n}}\left(\left(\mathrm{Sel}^{\pm}(K_{\infty},E[p^{\infty}])[\omega^{\pm}_{n}]\right)^{\vee}\right)$

in $\Lambda_{n}/\omega^{\pm}_{n}$, respectively. By the signed control theorem
(Theorem 3.4), we obtain

$\left(L^{\pm}_{p}(E/K_{\infty})\ (\text{mod}\
\omega^{\pm}_{n})\right)\subseteq\mathrm{Fitt}_{\Lambda_{n}/\omega^{\pm}_{n}}\left(\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])[\omega^{\pm}_{n}]\right)^{\vee}\right)$

in $\Lambda_{n}/\omega^{\pm}_{n}$, respectively. Since Fitting ideals are
compatible with base change, we have the equality

$\displaystyle\mathrm{Fitt}_{\Lambda_{n}/\omega^{\pm}_{n}}\left(\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])[\omega^{\pm}_{n}]\right)^{\vee}\right)$
$\displaystyle=\dfrac{\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])^{\vee}\right)+(\omega^{\pm}_{n})}{(\omega^{\pm}_{n})}$

in $\Lambda_{n}/\omega^{\pm}_{n}$. Thus, we have inclusions

$\left(L^{\pm}_{p}(E/K_{\infty})\ (\text{mod}\
\omega_{n})\right)+(\omega^{\pm}_{n})\subseteq\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])^{\vee}\right)+(\omega^{\pm}_{n})$

in $\Lambda_{n}$, respectively. Multiplying $\widetilde{\omega}^{\mp}_{n}$,
the conclusion immediately follows. ∎

######  Remark 3.6.

If we further assume $p\nmid\mathrm{Tam}(E)$ and the $\pm$-main conjectures
(3.2), then the inclusion in Corollary 3.5 becomes an equality.

##  4\. The proof of Theorem 1.3

By using the global Poitou–Tate duality, consider the exact sequence of
$\Lambda_{n}$-modules (cf. [KO18, (4.2)])

$\textstyle{\left(\dfrac{E(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}{E^{\pm}(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}\right)^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{\pm}}$$\textstyle{\mathrm{Sel}(K_{n},E[p^{\infty}])^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Then we have

$\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\dfrac{E(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}{E^{\pm}(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}\right)^{\vee}/\mathrm{ker}(\iota^{\pm})\right)\cdot\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])^{\vee}\right)\subseteq\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}(K_{n},E[p^{\infty}])^{\vee}\right)$

thanks to the behavior of Fitting ideals in the short exact sequence. Due to
the behavior of Fitting ideals under the quotient map, we also have

$\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\dfrac{E(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}{E^{\pm}(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}\right)^{\vee}\right)\subseteq\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\dfrac{E(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}{E^{\pm}(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}\right)^{\vee}/\mathrm{ker}(\iota^{\pm})\right).$

Applying [KO18, Lem. 3.14] with (3.1), we observe that

$\displaystyle\left(\dfrac{E(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}{E^{\pm}(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}\right)^{\vee}$
$\displaystyle\simeq\left(\dfrac{\widehat{E}(\mathfrak{m}_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}{\widehat{E}^{\pm}(\mathfrak{m}_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}\right)^{\vee}\oplus\left(\dfrac{\widehat{E}(\mathfrak{m}_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}{\widehat{E}^{\pm}(\mathfrak{m}_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}\right)^{\vee}$
$\displaystyle\simeq\left(\dfrac{\widehat{E}(\mathfrak{m}_{n})}{\widehat{E}^{\pm}(\mathfrak{m}_{n})}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\vee}\oplus\left(\dfrac{\widehat{E}(\mathfrak{m}_{n})}{\widehat{E}^{\pm}(\mathfrak{m}_{n})}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\vee}.$

Due to [IP06, Prop. 4.11], we have the following exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widehat{E}(p\mathbb{Z}_{p})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\simeq}$$\textstyle{\widehat{E}^{+}(\mathfrak{m}_{n})\oplus\widehat{E}^{-}(\mathfrak{m}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{\simeq}$$\textstyle{\widehat{E}(\mathfrak{m}_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$[IP06,
Prop.
5.8]$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\omega}^{+}_{n}\widetilde{\omega}^{-}_{n}\Lambda_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\omega}^{-}_{n}\Lambda_{n}\oplus\widetilde{\omega}^{+}_{n}\Lambda_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left(\widetilde{\omega}^{+}_{n},\widetilde{\omega}^{-}_{n}\right)\Lambda_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $f$ is the diagonal embedding and $g:(a,b)\mapsto a-b$. Note that
$\widetilde{\omega}^{+}_{n}\widetilde{\omega}^{-}_{n}\Lambda_{n}\simeq\Lambda_{n}/X\Lambda_{n}\simeq\mathbb{Z}_{p}$.
Since
$\widehat{E}(\mathfrak{m}_{n})/\widehat{E}^{\pm}(\mathfrak{m}_{n})\simeq\left(\widetilde{\omega}^{+}_{n},\widetilde{\omega}^{-}_{n}\right)\Lambda_{n}/\widetilde{\omega}^{\mp}_{n}\Lambda_{n}$,
we have

$\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\frac{\left(\widetilde{\omega}^{+}_{n},\widetilde{\omega}^{-}_{n}\right)\Lambda_{n}}{\widetilde{\omega}^{\mp}_{n}\Lambda_{n}}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\vee}\right)=\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\dfrac{\widehat{E}(\mathfrak{m}_{n})}{\widehat{E}^{\pm}(\mathfrak{m}_{n})}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\vee}\right).$

######  Proposition 4.1.

$\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\frac{\left(\widetilde{\omega}^{+}_{n},\widetilde{\omega}^{-}_{n}\right)\Lambda_{n}}{\widetilde{\omega}^{\mp}_{n}\Lambda_{n}}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\vee}\right)=\widetilde{\omega}^{\mp}_{n}\Lambda_{n},$

respectively.

###### Proof.

See [KK21, Prop. 4.1]. ∎

To sum up, we have

$\displaystyle\left(\widetilde{\omega}^{\mp}_{n}\right)^{2}\cdot\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])^{\vee}\right)$
$\displaystyle=\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\frac{\left(\widetilde{\omega}^{+}_{n},\widetilde{\omega}^{-}_{n}\right)\Lambda_{n}}{\widetilde{\omega}^{\mp}_{n}\Lambda_{n}}\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\vee}\right)^{2}\cdot\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])^{\vee}\right)$
$\displaystyle=\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\widehat{E}(\mathfrak{m}_{n})/\widehat{E}^{\pm}(\mathfrak{m}_{n})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)^{\vee}\right)^{2}\cdot\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])^{\vee}\right)$
$\displaystyle\subseteq\mathrm{Fitt}_{\Lambda_{n}}\left(\left(\dfrac{E(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}{E^{\pm}(K_{n,p})\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}}\right)^{\vee}/\mathrm{ker}(\iota^{\pm})\right)\cdot\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}^{\pm}(K_{n},E[p^{\infty}])^{\vee}\right)$
$\displaystyle\subseteq\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}(K_{n},E[p^{\infty}])^{\vee}\right).$

By Corollary 3.5 with the multiplication by $\widetilde{\omega}^{\mp}_{n}$, we
have

$\left(\left(\widetilde{\omega}^{\mp}_{n}\right)^{2}\cdot
L^{\pm}_{p}(E/K_{\infty})\ (\text{mod}\
\omega_{n})\right)\subseteq\mathrm{Fitt}_{\Lambda_{n}}\left(\mathrm{Sel}(K_{n},E[p^{\infty}])^{\vee}\right).$

By using (2.1), (2.2), and the functional equation for Bertolini–Darmon’s
theta elements (see [BD96, Prop. 2.13] and [BD05, Lem. 1.5]), we have

$\displaystyle\left(\left(\widetilde{\omega}^{\mp}_{n}\right)^{2}\cdot
L^{\pm}_{p}(E/K_{\infty})\ (\text{mod}\ \omega_{n})\right)$
$\displaystyle=\left(\left(\widetilde{\omega}^{\mp}_{n}\right)^{2}\cdot\theta^{\pm}(E/K_{n})\cdot\iota(\theta^{\pm}(E/K_{n}))\right)$
$\displaystyle=\left(\left(\widetilde{\omega}^{\mp}_{n}\right)^{2}\cdot\theta^{\pm}(E/K_{n})\cdot\theta^{\pm}(E/K_{n})\right)$
$\displaystyle=\left(\theta(E/K_{n})\cdot\theta(E/K_{n})\right)$
$\displaystyle=\left(\theta(E/K_{n})\cdot\iota(\theta(E/K_{n}))\right)$
$\displaystyle=\left(L_{p}(E/K_{n})\right).$

Thus, Theorem 1.3 follows.

## References

  * [BBL] A. A. Burungale, K. Büyükboduk, and A. Lei, _Anticyclotomic Iwasawa theory of abelian arieties of $\mathrm{GL}_{2}$-type at non-ordinary primes_, preprint, arXiv:2211.03722.
  * [BD96] M. Bertolini and H. Darmon, _Heegner points on Mumford–Tate curves_ , Invent. Math. 126 (1996), no. 3, 413–456.
  * [BD05] by same author, _Iwasawa’s main conjectures for elliptic curves over anticyclotomic $\mathbb{Z}_{p}$-extensions_, Ann. of Math. (2) 162 (2005), no. 1, 1–64.
  * [CH15] M. Chida and M.-L. Hsieh, _On the anticyclotomic Iwasawa main conjecture for modular forms_ , Compos. Math. 151 (2015), no. 5, 863–897.
  * [CH18] by same author, _Special values of anticyclotomic $L$-functions for modular forms_, J. Reine Angew. Math. 741 (2018), 87–131.
  * [DI08] H. Darmon and A. Iovita, _The anticyclotomic main conjecture for elliptic curves at supersingular primes_ , J. Inst. Math. Jussieu 7 (2008), no. 2, 291–325.
  * [Gre99] R. Greenberg, _Iwasawa theory for elliptic curves_ , Arithmetic theory of elliptic curves (Cetraro, 1997) (Berlin) (C. Viola, ed.), Lecture Notes in Math., vol. 1716, Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, Springer-Verlag, 1999, Lectures from the 3rd C.I.M.E. Session held in Cetraro, July 12-–19, 1997, pp. 51––144.
  * [IP06] A. Iovita and R. Pollack, _Iwasawa theory of elliptic curves at supersingular primes over $\mathbb{Z}_{p}$-extensions of number fields_, J. Reine Angew. Math. 598 (2006), 71–103.
  * [Kim] C.-H. Kim, _An anticyclotomic Mazur–Tate conjecture for modular forms_ , submitted, arXiv:1612.03743.
  * [Kim13] B. D. Kim, _The plus/minus Selmer groups for supersingular primes_ , J. Aust. Math. Soc. 95 (2013), no. 2, 189–200.
  * [Kim17] C.-H. Kim, _Anticyclotomic Iwasawa invariants and congruences of modular forms_ , Asian J. Math. 21 (2017), no. 3, 499–530.
  * [Kim19] by same author, _Overconvergent quaternionic forms and anticyclotomic $p$-adic $L$-functions_, Publ. Mat. 63 (2019), no. 2, 727–767.
  * [KK21] C.-H. Kim and M. Kurihara, _On the refined conjectures on Fitting ideals of Selmer groups of elliptic curves with supersingular reduction_ , Int. Math. Res. Not. IMRN 2021 (2021), no. 14, 10559–10599.
  * [KO18] T. Kitajima and R. Otsuki, _On the plus and the minus Selmer groups for elliptic curves at supersingular primes_ , Tokyo J. Math. 41 (2018), no. 1, 273–303.
  * [KO23] C.-H. Kim and K. Ota, _On the quantitative variation of congruence ideals and integral periods of modular forms_ , Res. Math. Sci. 10 (2023), no. 22, 34 pages.
  * [Kob03] S. Kobayashi, _Iwasawa theory for elliptic curves at supersingular primes_ , Invent. Math. 152 (2003), no. 1, 1–36.
  * [KPW17] C.-H. Kim, R. Pollack, and T. Weston, _On the freeness of anticyclotomic Selmer groups of modular forms_ , Int. J. Number Theory 13 (2017), no. 6, 1443–1455.
  * [Kur02] M. Kurihara, _On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I_ , Invent. Math. 149 (2002), 195–224.
  * [MT87] B. Mazur and J. Tate, _Refined conjectures of the “Birch and Swinnerton-Dyer type”_ , Duke Math. J. 54 (1987), no. 2, 711–750.
  * [Pol03] R. Pollack, _On the $p$-adic $L$-function of a modular form at a supersingular prime_, Duke Math. J. 118 (2003), no. 3, 523–558.
  * [PW11] R. Pollack and T. Weston, _On anticyclotomic $\mu$-invariants of modular forms_, Compos. Math. 147 (2011), 1353–1381.
  * [Shi] R. Shii, _On non-trivial $\Lambda$-submodules with finite index of the plus/minus Selmer group over anticyclotomic $\mathbb{Z}_{p}$-extension at inert primes_, preprint, arXiv:2308.16384.
  * [Vat02] V. Vatsal, _Uniform distribution of Heegner points_ , Invent. Math. 148 (2002), no. 1, 1–48.
  * [Vig80] M.-F. Vignéras, _Arithmétique des algèbres de quaternions_ , Lecture Notes in Math., vol. 800, Springer, 1980.