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Kleinian hyperelliptic funtions of weight 2 associated with curves of genus 2

Matvey Smirnov

Abstract

We introduce a new collection of special functions associated to a complex curve of genus 2 similar to Kleinian hyperelliptic $σ$-function. These functions are related to weight 2 $θ$-functions in the same fashion as $σ$-function is related to the classical $θ$-function. A key feature of the introduced functions is the fact that they are well-defined for genus 2 algebraic curves without any restrictions (in particular it is not needed to assume that the curve has a Weierstrass point at infinity).

Kleinian hyperelliptic funtions of weight 2 associated with curves of genus 2

Abstract

We introduce a new collection of special functions associated to a complex curve of genus 2 similar to Kleinian hyperelliptic -function. These functions are related to weight 2 -functions in the same fashion as -function is related to the classical -function. A key feature of the introduced functions is the fact that they are well-defined for genus 2 algebraic curves without any restrictions (in particular it is not needed to assume that the curve has a Weierstrass point at infinity).
Paper Structure (9 sections, 15 theorems, 60 equations)

This paper contains 9 sections, 15 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Meromorphic functions associated with curves of genus 2
  3. Notation
  4. Meromorfic functions on .
  5. Kleinian functions of weight 2
  6. Periods of certain meromorphic forms of second kind.
  7. Kleinian functions of weight 2.
  8. Relation to the Kleinian sigma function of genus 2.
  9. Taylor expansions of Kleinian functions of weight 2

Key Result

Proposition 2.1

The following statements hold.

Theorems & Definitions (33)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • proof
  • ...and 23 more