Osaka Central Advanced Mathematical Institute (OCAMI)
Osaka Metropolitan University
MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics

OCAMI Reports Vol. 7 (2023)
doi: 10.24544/omu.20240219-001

Seminar on Abelian Functions 2023
Organized by
Takanori Ayano

December 4, 2023 and January 9, 2024.

Abstract
We report on the seminar on Abelian functions held via Zoom on December 4,2023 and January 9, 2024.

2020 Mathematics Subject Classiﬁcation.
14H40, 14H42, 14H70, 14H81, 14K25, 30F10, 32G20, 33E05.

Key words and Phrases.
theta function, sigma function, Riemann surface,degeneration of an algebraic curve, Sato Grassmannian, Abelian integrals© 2023 OCAMI.

OCAMI. Seminar on Abelian Functions 2023. OCAMI Reports. Vol. 7, Osaka Central Advanced
Mathematical Institute, Osaka Metropolitan University. 2023, 37 pp. doi: 10.24544/omu.20240219-001

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OCAMI Reports Vol. 7 (2023)

Preface
The theory of elliptic functions was one of the main subjects of research in the 19thcentury and the concrete theory of elliptic functions was constructed. It is applied to manyﬁelds in mathematics and science. As the technology advances, there are movements toapply Abelian functions, which are generalizations of elliptic functions to many variables,
to many ﬁelds in mathematics and science. The elliptic sigma function σ(u) and theelliptic function ℘(u), which are deﬁned and studied by Weierstrass, are generalized tothe multivariable sigma functions and the Abelian functions associated with hyperelliptic
curves by Klein and Baker about 100 years ago. The hyperelliptic sigma functions andthe hyperelliptic Abelian functions are generalized and extensively studied for the last
three decades.
The purpose of this seminar is to learn the latest ﬁndings about Abelian functions
and related ﬁelds from lectures by experts and discuss them. The 1st seminar on Abelianfunctions was held via Zoom on December 4, 2023. In this seminar, 62 people registeredfor participation. The 2nd seminar on Abelian functions was held via Zoom on January
9, 2024. In this seminar, 45 people registered for participation. There were lively discussions among participants. We are grateful to the participants of the seminar for theircontribution. In this report, we record the slides of the lectures.
The website of this seminar is as follows:
https://sites.google.com/view/abelianfunction/%E3%83%9B%E3%83%BC%E3%83%A0
This seminar was supported by Osaka Central Advanced Mathematical Institute (MEXT
Promotion of Distinctive Joint Research Center Program JPMXP0723833165), Osaka
Metropolitan University.

Seminar on Abelian Functions 2023

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Organizers
Takanori Ayano
Department of Mathematics, Graduate School of Science, Osaka Metropolitan University,
3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
Email address: ayano@omu.ac.jp

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OCAMI Reports Vol. 7 (2023)

Contents
Atsushi Nakayashiki
Introduction to Sato Grassmannian and theta functions. . . . . . . . . .

1Takashi Ichikawa
Abelian integrals and solitons . . . . . . . . . . . . . . . . . . . . . . . . . 22

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OCAMI Reports Vol. 7 (2023).

「第 2 回 アーベル関数論セミナー」.

Abelian integrals and solitons

supported by 大阪公立大学数学研究所「数学・理論物理の協働・共創...
Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp)
Saga University

January 9, 2024

Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

supported
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Seminar on Abelian Functions 2023

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Contents

アブストラクト：Riemann 面の Schottky 一意化表示を用いて Abel 微分と積分の明示公式を導き，トロピカル曲線に付随したソリトンの構成と計算に応用する。

§1. Aim of this seminar§2. Abelian diﬀerentials and period integrals§3. Explicit variational formulas§4. Calculation of universal periods§5. Tropical curves and master functions of solitons§6. Sketch of the proof§7. Supplementary comments§8. Related results
References

Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

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OCAMI Reports Vol. 7 (2023)§1. Aim of this seminar§2–4. We review of the theory of Abelian integrals on Riemann surfaces containing:Deﬁnition and basic facts of Abelian diﬀerentials and (period) integrals.

Their computable variational formulas by the Schottky uniformization theory.
§3–8. Combining the above formulas with tropical geometry, we study the program:
(∗)

Here

Quasi-periodic solutions to soliton equations ⇒ All(?) soliton solutions.
quasi-periodic solution: given by Krichever,...using general Riemann theta functions.
soliton equation: KP, Toda lattice (and Universal?) hierarchies containing()
3 u − 6u ∂ u ↔ shallow water waves
KP equation: 3∂y2 u = ∂x 4∂t u − ∂xx⊂ KP hierarchy ∼ Sato Grassmannian (Nakayashiki’s talk), Characterizing Jacobians.
un−1 −unToda lattice: d2− eun −un+1 ↔ nonlinearly interacting chainst un = e⊂ Toda hierarchy ∼ 2D gravity, String theory, Enumerative geometry,...

(∗): taking limit under rational degeneration of Riemann surfaces.
soliton: expressed by compositions of rational, exponential, logarithm functions.
We also discuss their calculation, regularity and nonrational degeneration.
Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

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Seminar on Abelian Functions 2023

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§2. Abelian diﬀerentials and period integrals
Set up.
R: Riemann surface:{compact 1-dim. complex mfd.
proper smooth algebraic curve/C}, genus g = ♯(holes).

Abelian diﬀerential (of the 1st kind): meromorphic (holomorphic) 1-form on RRiemann-Roch⇒dim{hol. diﬀ. on R} = − deg(OR ) − 1 + g + dim H 0 (OR ) = g.

{ai , bi }1≤i≤g : symp. basis of H1 (R); (ai ,aj ) = (bi ,bj ) = 0, (ai ,bj ) = δij∫⇒ ∃1 {wi }1≤i≤g : holomorphic abelian diﬀ. on R normalized i.e., a wj = δiji()∫⇒ Pij = b wj
: period matrix of R (⇒ Pij = Pji , Im(Pij ) > 0).
ii,jAbel’s theorem and Riemann theta function.

Abel∑∑ ∫p
Jacobian of R ∼= Cg /(Zg + Zg (Pij )); k (pk − qk ) 7→ ( k qkk wi )i / ∼.
( √())∑
ΘR (z) := u∈Zg exp π −1 u(Pij )uT + 2zuT = ΘR (z + v) (v ∈ Zg ).

Question. Can we calculate exact values of Pij ?

Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

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OCAMI Reports Vol. 7 (2023)§3. Explicit variational formulas of diﬀerentials and periods
Schottky uniformization. Γ: hyperbolic subgroup of P GL2 (C), free of rank g⇒ RΓ = ΩΓ /Γ: genus g Riemann surface; ΩΓ = P1 \(limit set of Γ) (Indra’s Pearls).
n→∞Explicit formula of wi ,Pij (Schottky, Manin-Drinfeld). Γ = ⟨γi ⟩, γi±n (a) −→ α±i .

Under bi ↔ γi and convergence conditions, the diﬀerentials wi and integrals Pij satisfy(∫)∑√(γ(αi )−γ(α−i ))dz
Riemann
γi (a) ∑dγ(ζ)
2π −1wi=dz a=γ∈Γ/⟨γi ⟩ (z−γ(αi ))(z−γ(α−i )) ,γ∈Γ γ(ζ)−z( √)∫ γ (a)∏√exp(2π −1Pij ) = exp 2π −1 a i
wj = γ∈⟨γi ⟩\Γ/⟨γj ⟩ ψij (γ),{multiplier of γi(i = j, γ ∈ ⟨γi ⟩),(αi −γ(αj ))(α−i −γ(α−j ))where a ∈ ΩΓ , ψij (γ) =(otherwise).
(α−γ(αj ))(αi −γ(α))−i−jComputable variational formula. ∆ = (V = {vertices}, E = {edges}): graph⇒ ∃R∆ : family of Riemann surfaces, genus = dim H1 (∆)∪
1 dual R∆ : deformation by ye (e ∈ E) of singular curve v∈V Pv (= P ) ↔ ∆,∃s.t.
R∆ : Schottky
uniformized by Γ∆ = Im(π1 (∆) ,→ P GL2 (C)),
√
exp(2π −1Pij ): computable Laurent series Uij in ye /Z[xh(∈±E) ,...].
Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

supportedJanuaryby 9,
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5 / 12Seminar on Abelian Functions 202327§4. Calculation of universal periods Uij
Example 1. ∆ = {single vertex, g loops}.

γ (z)−x−i(z)−xi
x±i , yi : variables (i = 1,..., g), Γ∆ = ⟨γi ⟩; γiz−x= yi iz−xi⇒

1−iR∆ : universal deformation of P /(xi = x−i )i by y1 ,..., yg .

Furthermore,
γi (z) − γi (w) = (z−x(xi −x−i )2 (z−w)yii)= (z−x O(y−i −yi (z−xi ))(w−x−i −yi (w−xi ))−i )(w−x−i )⇒⇒(x −x)(γ(x )−γ(x))

j−i−jψij (γ) = 1 + (xi −γ(x= 1 + O(yk ) if γ ̸∈ ⟨γi ⟩ · ⟨γj ⟩j ))(xi −γ(x−j ))−i(i = j), yi + O(yk yl )(x−x)(x−x)
Uij =ij−i−j+ O(yk ) (i ̸= j).
(x−i − xj )(xi − x−j )Summary. ∃ Computable theory of Abelian integrals/Z[xh ,...]((ye )) containing/⟨⟩
Abel’s theorem: Jacobian of R∆ ∼= Ggm (Uij )1≤i≤g | 1 ≤ j ≤ g .

Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

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OCAMI Reports Vol. 7 (2023)§5. Tropical curves and master functions of solitons
Tropical curve (of weights 0): graph ∆ = (V , E) with length function l : E → R>0 genus:= dim H1 (∆),{±l(e) (e = ±e′ ),⇒⊗2′s.t. ⟨e, e ⟩ = period matrix: bilinear form on H1 (∆)
0(e ̸= ±e′ ).
Theorem (Proof of Agostini-Fevola-Mandelshtam-Sturmfels’ conjecture (2021)).
C: tropical curve of genus g with (sym. pos.-def.) period matrix BC⇒∃{Rs }s>0 : family of Riemann surfaces of genus g with symp. H1 -basis s.t.
√(1) (Period matrices of Rs ) × (2π −1) = log s · BC + ∃ B0 as s ↓ 0.

(2) DC,α := {u ∈ Zg ; |u − α|BC :(minimal }: Delaunay
finite set for α ∈ Rg)∑uB0 uTz∈Cg⇒ MC,α (z) =exp+ zuT : master τ -function of solitons.
2u∈DC,αCorollary. u(x, y, t) = 2∂x2 log MC,α (xr1 + yr2 + tr3 + c): KP soliton{B0 , ri : explicitly represented by the above xh(∈±E) ,;c: any constant vector.
Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

supported
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7 / 12Seminar on Abelian Functions 202329§6. Sketch of the proof
Proof of (1). [ , ] : H1 (∆)⇒ye =sl(e)⇒{±ye (e = ±e′ ),Zys.t.
[e,e]=ee∈E0(e ̸= ±e′ )∏[γ ,γ ]eUij = e∈E ye i j (Uij (0) + O(ye′ ))⊗2→⊕′(1), where B0 = (log Uij (0))i,j : explicit(?).

Proof of (2). (z ∈ Cg , α ∈ Rg ).

( √())
π −1 uPs uT + 2zuT ; Ps : period matrix of Rs ,{}ΘC (α) := maxu∈Zg αBC uT − 12 uBC uT : tropical theta function for C
ΘRs (z) =∑u∈Zg exp⇒ MC,α (z) = lims→0 sΘC (α) · ΘRs (z − (log s)αBC ) satisﬁes (2).
Example 2. C = {single vertex, g loops with length 1} ⇒ BC = Ig .
γ (z)−x⇒(z)−xiΓ∆ = ⟨γi ⟩1≤i≤g ; γiz−x= yi iz−x −i : as in Example 1i−i(∏
( (x −x )(x −x ) )ui uj ∏)∑gi
j−i−jzi uiMC,α (z) = u∈{0,1}gei<ji=1(x−xj )(xi −x)−i−j: gives the g soliton for α = (1/2,..., 1/2).

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Abelian integrals and solitons

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30OCAMI Reports Vol. 7 (2023)§7. Supplementary comments
defRegularity of MC,α ⇔ MC,α (z) > 0 (z ∈ Rg ) ⇒ ∂x2 log MC,α : real, nonsingular.
def

M-curve = real alg. curve with max. number (= genus + 1) of components⇒⇒∃

Rs : M-curves as plane ﬁgures obtained by R∆ taking xh ∈ R, ye = ±sl(e)∫√2π −1 b wj ∈ R ∴ B0 = (log |Uij (0)|)i,j : explicit and MC,α : regular.
i

Master function for general tropical curve.

C = (V , E, l): tropical curve with weight function w : V → Z≥0 ,⇒Rv : Riemann surface of genus w(v), wv,i : normalized Abelian diﬀ. on Rv
()
(1)
∫∑
TT ∏(z)+(w)
expuBu+zu
Θv,i i0Rv
v,i iu∈DC,αv∈V2u|Rv(): master function of quasi-periodic solitons z ∈ Ch1 (∆) , (zv,i )i ∈ Cw(v) .

Summary.

Singular curve with tropical dual graph ↔ Degenerating Riemann surfaces⇒ Master function of solitons reﬂecting the degeneration rates (:l(e):)e∈E .

Arithmetic of Abelian diﬀerentials and integrals ⇒ Regularity of master functions.
Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

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§8. Related results (Are there direct connections with our results?)
Line KP soliton. Kodama and Williams studied the regularity,... of line KP solitons:()∑∏∏ κ x+κ2 y+κ3 t[n]ii ;I ∈∆I (A)(κil − κim )e i(A ∈ Grk,n , κi ),kIm<li∈Iby the theory of positive Grassmannians, and Nakayashiki, Abenda-Grinevich and Kodama
showed that line KP solitons are obtained as limits of quasi-periodic KP solutions.
Sato Grassmannian approach. By the analysis of the KP hierarchy using SatoGrassmannian, Nakayashiki and others constructed KP solutions of mixed type withsolitons and quasi-periodic functions by degenerations of hyperelliptic curves:
y 2 = f (x); f (x): polynomial of x without multiple root
or trigonal curves. Further, Nakayashiki gave KP solutions on elliptic background using
the action of vertex operators, and also there were works of Li-Zhang and Kakei.
Tropical theta and ultra-discretization. Tropical theta functions of tropical curves wereintroduced by Mikhalkin and Zharkov for studying tropical Jacobians, and used toconstruct solutions to the ultra-discrete Toda lattice and box-ball system by Inoue, Iwaoand Takenawa.

Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

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OCAMI Reports Vol. 7 (2023)

References (1)S. Abenda, P. G. Grinevich: Rational degeneration of M-curves, totally positive
Grassmannians and KP2-solitons. Commun. Math. Phys. 361 (2018) 1029–1081
D. Agostini, C. Fevola, Y. Mandelshtam, B. Sturmfels: KP solitons from tropicallimits. J. Symb. Comput. 114 (2023) 282–301
J. Bernatska, V. Enolski, A. Nakayashiki: Sato Grassmannian and degenerate sigma
function. Commun. Math. Phys. 374 (2020) 627–660
T. Ichikawa: Periods of tropical curves and associated KP solutions. Commun.
Math. Phys. 402 (2023) 1707–1723
T. Ichikawa: Families of KP solutions associated with tropical curves having
nontrivial weights. arXiv:2312.06998
R. Inoue, S. Iwao: Tropical spectral curves, Fay’s trisecant identity, and generalized
ultradiscrete Toda lattice. in: New Trends in Quantum Integrable Systems, World
Scientiﬁc, pp. 101–116 (2010), arXiv:1003.0057
S. Kakei: Solutions of the KP hierarchy with an elliptic background. Preprint (2023)
Y. Kodama: KP solitons and the Riemann theta functions. arXiv:2308.06902
Y. Kodama, L. Williams: KP solitons and total positivity for the Grassmannian.
Invent. Math. 198 (2014) 637–699
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Abelian integrals and solitons

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References (2)I. Krichever, A. Zabrodin: Quasi-periodic solutions of the universal hierarchy.
arXiv:2308.12187
X. Li, D.-J. Zhang: Elliptic soliton solutions: τ functions, vertex operators andbilinear identities. J. Nonlin. Sci. 32 (2022), Article number 70
A. Nakayashiki: Degeneration of trigonal curves and solutions of the KP hierarchy.
Nonlinearity 31 (2018) 3567–3590
A. Nakayashiki: On reducible degeneration of hyperelliptic curves and soliton
solutions. SIGMA 15 (2019). https://doi.org/10.3842/SIGMA.2019.009
A. Nakayashiki: One step degeneration of trigonal curves and mixing of solitons andquasi-periodic solutions of the KP equation. Geometric Methods in Physics XXXVIII(2020) 163–186, arXiv:1911.06524
A. Nakayashiki: Vertex operators of the KP hierarchy and singular algebraic curves.
arXiv:2309.08850
A. Zabrodin: Quasi-periodic solutions to hierarchies of nonlinear integrable equations
and bilinear relations. arXiv:2304.05108

Takashi Ichikawa (ichikawn@cc.saga-u.ac.jp) (Saga University)「第 2 回 アーベル関数論セミナー」

Abelian integrals and solitons

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