Sigma function associated with a hyperelliptic curve with two points at infinity

This paper investigates the properties of Baker functions on the Jacobian of a hyperelliptic curve with two points at infinity by constructing an associated entire function, proving its power series expansion is algebraically determined by the curve's defining equation and a branch point, and expressing it in terms of the Riemann theta function.

The Gist

The Big Picture: Mapping a Strange Landscape

Imagine you are an explorer trying to map a very strange, multi-dimensional landscape called a Hyperelliptic Curve. In mathematics, these curves are like complex, twisted shapes that exist in higher dimensions.

For a long time, mathematicians had a very good map for a specific type of these curves: ones that have one "exit" or "infinity" point (like a road that ends at a single horizon). They had a special tool called a Sigma Function that acted like a perfect GPS. It could tell you exactly where you were, how the terrain twisted, and even predict how waves would move across this landscape (solving famous physics equations like the KdV and KP equations).

The Problem:
There is another type of curve that has two "exits" or points at infinity (imagine a road that splits and goes off into two different horizons). For these "two-point" curves, the old GPS (the Sigma function) didn't work perfectly. The famous mathematician H.F. Baker, back in the early 1900s, built some basic tools (called Baker functions) to navigate this terrain, but he didn't have the "master key" (the Sigma function) to unlock the full potential of these tools. He knew how to walk the path, but he didn't have the map that explained why the path looked the way it did.

What This Paper Does: Building the Master Key

The authors, Takanori Ayano and Victor Buchstaber, have finally built that missing master key. They constructed a new, complete function (which they also call a Sigma function, but specifically for the "two-point" curves) that acts as the ultimate guide.

Here is how they did it, broken down into simple steps:

1. The "Translation" Trick

Imagine the "two-point" curve is a foreign country with a language no one speaks. The "one-point" curve is a country where everyone speaks a language the authors already know.

The authors found a clever way to translate the "two-point" curve into the "one-point" curve. They built a bridge (a mathematical transformation) that takes the messy "two-point" world and folds it into the familiar "one-point" world.

• The Metaphor: It's like taking a crumpled piece of paper with two holes in it and flattening it out so it looks like a standard sheet with one hole. Once it's flattened, they can use their old, trusted GPS (the one-point Sigma function) to navigate it.

2. The "Magic Formula" (The Baker Function Connection)

Once they had the translation, they realized that the "Baker functions" (the basic tools Baker built 100 years ago) were actually just the second derivative (the rate of change of the rate of change) of their new, big Sigma function.

• The Metaphor: Think of the Baker functions as the ripples on a pond. The authors discovered that if you know the shape of the entire pond (the new Sigma function), you can mathematically calculate exactly how every single ripple moves. They proved that the ripples aren't random; they are perfectly determined by the shape of the pond itself.

3. The "Recipe" (Power Series)

One of the most exciting discoveries is that this new Sigma function is built from a very specific "recipe."

• The Metaphor: Imagine you are baking a cake. Usually, a cake recipe might depend on the brand of flour you buy or the humidity in the kitchen. But the authors proved that this specific "mathematical cake" depends only on the ingredients listed in the curve's original equation.
• If you give them the equation of the curve (the list of ingredients), they can write down the exact recipe for the Sigma function without needing any outside information. It is purely algebraic. This is huge because it means the function is "native" to the curve; it doesn't rely on arbitrary choices made by the mathematician.

4. The "Repeating Pattern" (Quasi-Periodicity)

The authors also showed how this function behaves when you move around the landscape.

• The Metaphor: Imagine walking on a tiled floor. If you take a step of a certain size, you land on a tile that looks exactly like the one you started on. But with this function, it's like walking on a floor where the tiles look the same, but the colors are slightly shifted (multiplied by a factor). This "shifted repetition" is called quasi-periodicity. They figured out the exact rule for how the colors shift, which is crucial for understanding the function's behavior over long distances.

Why Does This Matter?

1. Completing the Puzzle: For over a century, there was a gap in the theory of these curves. This paper fills that gap, connecting the "one-point" and "two-point" worlds.
2. Physics Applications: These curves aren't just abstract math; they describe real-world phenomena like water waves, light pulses in fibers, and quantum mechanics. By having a better map (the new Sigma function), physicists can solve complex equations (like the KP equation mentioned in the paper) more accurately.
3. Universality: The fact that the function is determined only by the curve's equation means it is a fundamental property of the shape itself, not just a mathematical trick.

Summary

In short, the authors took a difficult, two-exit mathematical landscape that was hard to navigate. They built a bridge to a familiar one-exit landscape, used that to construct a new "Master Map" (the Sigma function), and proved that this map is built entirely from the blueprint of the landscape itself. They also showed how this map connects to the old tools (Baker functions) and how it repeats itself in a predictable, beautiful pattern. This allows mathematicians and physicists to finally understand and predict the behavior of these complex shapes with total precision.

Technical Summary

1. Problem Statement

The paper addresses a specific gap in the theory of multidimensional sigma functions associated with algebraic curves.

• Context: Historically, the theory of sigma functions (generalizing the Weierstrass $\sigma$-function) has been well-developed for hyperelliptic curves with one point at infinity (Weierstrass curves). These functions are holomorphic on the Jacobian variety, and their power series expansions are determined solely by the coefficients of the curve's defining equation.
• The Gap: While H.F. Baker (1923) constructed basic meromorphic functions (now called Baker functions, denoted $P_{i,j}$) for hyperelliptic curves with two points at infinity, he did not define an associated sigma function. Consequently, there was no known entire function whose second logarithmic derivatives yield these Baker functions, nor was there a proof that such a function's power series expansion depends algebraically only on the curve's coefficients.
• Objective: The authors aim to construct a sigma function $H(v)$ for hyperelliptic curves with two points at infinity, prove its properties (specifically its power series expansion and quasi-periodicity), and establish its relationship with the Baker functions and the Riemann theta function.

2. Methodology

The authors employ a strategy of isomorphism and transformation to bridge the gap between the "two points at infinity" case and the well-understood "one point at infinity" case.

• Curve Definitions:
  • Curve $V$ (Two points at infinity): Defined by $y^2 = N(x)$, where $N(x)$ is a polynomial of degree $2g+2$. It has two points at infinity. A specific branch point $(a, 0)$ is chosen such that $N(a)=0$.
  • Curve $\tilde{C}$ (One point at infinity): Defined by $Y^2 = \tilde{M}(X)$, where $\tilde{M}(X)$ is a polynomial of degree $2g+1$.
• Isomorphism Construction:
  • The authors construct an explicit isomorphism $\zeta: V \to \tilde{C}$ using a change of variables involving parameters $s$ and $t$ (where $s^{2g+1}/t^2 = N'(a)$). This maps the curve with two points at infinity to a curve with one point at infinity.
  • This allows them to pull back the holomorphic 1-forms and the sigma function $\sigma(u)$ from $\tilde{C}$ to $V$.
• Transformation of Period Matrices:
  • They define a transformation matrix $D$ relating the period matrices of the two curves.
  • They introduce a symmetric matrix $\Omega = (n_{i,j})$ derived from the algebraic structure of the Baker functions $P_{i,j}$ and the coefficients of $N(x)$.
• Definition of the New Sigma Function:
  • The new sigma function $H(v)$ is defined on the Jacobian of $V$ ($Jac(V)$) by modifying the pulled-back sigma function $\sigma(Dv)$ with an exponential factor involving $\Omega$:
    $$H(v) = \chi \exp(v^t \Omega v) \sigma(Dv)$$
    where $\chi$ is a constant depending on the parity of $g(g+1)/2$.

3. Key Contributions and Results

A. Construction of the Entire Function $H(v)$

The paper defines $H(v)$ as an entire function on $\mathbb{C}^g$. The central result is that the second logarithmic derivatives of $H(v)$ recover the Baker functions $P_{i,j}(v)$:
$$\frac{\partial^2}{\partial v_i \partial v_j} \log H(v) = -P_{i,j}(v)$$
This establishes $H(v)$ as the true sigma function counterpart for the two-points-at-infinity case, analogous to how $\sigma(u)$ relates to the $\wp$-functions in the one-point case.

B. Algebraic Determination of Power Series

The authors prove a crucial theorem regarding the coefficients of the power series expansion of $H(v)$ around the origin:

• Result: The coefficients of the expansion are elements of the ring $\mathbb{Q}[a, \{\nu_{2i}\}]$, where $a$ is the chosen branch point and $\nu_{2i}$ are the coefficients of the defining polynomial $N(x)$.
• Significance: This confirms that $H(v)$ is determined algebraically by the curve's equation and the chosen branch point, independent of the auxiliary parameters $s$ and $t$ introduced during the isomorphism construction. This resolves the open problem of finding a sigma function for this class of curves that relies strictly on the curve's algebraic data.

C. Explicit Relations between Baker and Sigma Functions

The paper provides an explicit formula (Proposition 4.8) relating the Baker functions $P_{i,j}$ on $Jac(V)$ to the standard $\wp$-functions ($\wp_{k,l}$) on the Jacobian of the transformed curve $\tilde{C}$:
$$P_{2g+2-2i, 2g+2-2j}(v) = -n_{i,j} + \text{linear combination of } \wp_{2k-1, 2l-1}(Dv)$$
This refines previous results and clarifies the structural connection between the two types of hyperelliptic functions.

D. Quasi-Periodicity and Theta Function Representation

• Quasi-Periodicity: The authors derive the quasi-periodicity formula for $H(v)$ under the lattice of periods of $Jac(V)$. The formula involves the Riemann constant and the period matrices of the meromorphic 1-forms of the second kind on $V$.
• Theta Representation: $H(v)$ is expressed explicitly in terms of the Riemann theta function $\theta$:
  $$H(v) = \chi \varepsilon \exp\left(\frac{1}{2} v^t \kappa' (\mu')^{-1} v\right) \theta \left[ \begin{smallmatrix} \delta' \\ \delta'' \end{smallmatrix} \right] ((2\mu')^{-1}v, \tau)$$
  This connects the new sigma function directly to the classical theory of theta functions.

4. Significance and Impact

• Completion of Klein-Baker Theory: The work completes the classical program initiated by Klein and Baker by providing the missing sigma function for hyperelliptic curves with two points at infinity.
• Mathematical Physics Applications: The Baker functions $P_{i,j}$ are known to generate solutions to integrable systems, specifically the Kadomtsev-Petviashvili (KP) equation. By constructing $H(v)$, the authors provide the necessary tool to generate these solutions via the sigma function formalism, similar to how $\sigma(u)$ generates KdV/KP solutions for one-point curves.
• Grading and Homogeneity: The paper emphasizes the role of grading (weight assignments) in the construction. Unlike theta functions, which do not naturally admit grading due to normalized arguments, the constructed sigma function $H(v)$ preserves the homogeneous weight structure of the curve's coefficients. This is a fundamental distinction that allows for the algebraic determination of the series coefficients.
• Generalization: The methodology suggests a pathway for defining sigma functions for other classes of algebraic curves where the "one point at infinity" reduction is possible via birational transformations.

In summary, this paper successfully constructs the sigma function for hyperelliptic curves with two points at infinity, proves its algebraic dependence on the curve's defining equation, and links it to the Riemann theta function and integrable systems, thereby unifying the theory of hyperelliptic sigma functions across different curve configurations.