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A proof of Sugawara's conjecture on Hasse-Weber ray class invariants

Patrick Morton

TL;DR

This work proves Sugawara's conjecture that, for any imaginary quadratic field K and a nontrivial modulus \\mathfrak{f}, the ray class field \\textsf{K}_{\\mathfrak{f}} is generated over K by a single tau-invariant from Weber/Hasse’s \\tau-function. The author advances a case-by-case, CM-based approach using CM elliptic curves in Legendre, Deuring, and Tate normal forms to produce explicit invariants \\tau(\\mathfrak{k}^*) associated with ray classes, establishes their distinctness, and proves irreducibility of ray-class polynomials T_{\\mathfrak{m}}(t, j(\\mathfrak{k})). In many CM settings, the tau-invariant already generates the Hilbert class field, and in all analyzed moduli (including (2), (4), primes dividing 2,3,5, and combinations such as wp_2 wp_3, wp_3 wp_5, wp_3 wp_3' wp_5, and beyond), Sugawara’s claim holds. The paper completes several previously known special cases and extends the generation results to a broad suite of moduli, providing a unified arithmetic-algebraic framework for ray class generation via tau-values. Overall, it solidifies the role of CM-elliptic curves and explicit tau-values as universal generators of ray class fields in the imaginary-quadratic setting, with substantial consequences for the irreducibility of ray-class polynomials and explicit class-field computations.

Abstract

In this paper a proof is given of Sugawara's conjecture from 1936, that the ray class field of conductor $\mathfrak{f}$ over an imaginary quadratic field $K$ is generated over $K$ by a single primitive $\mathfrak{f}$-division value of the $τ$-function, first defined by Weber and then modified by Hasse in his 1927 paper giving a new foundation of complex multiplication.

A proof of Sugawara's conjecture on Hasse-Weber ray class invariants

TL;DR

This work proves Sugawara's conjecture that, for any imaginary quadratic field K and a nontrivial modulus \\mathfrak{f}, the ray class field \\textsf{K}_{\\mathfrak{f}} is generated over K by a single tau-invariant from Weber/Hasse’s \\tau-function. The author advances a case-by-case, CM-based approach using CM elliptic curves in Legendre, Deuring, and Tate normal forms to produce explicit invariants \\tau(\\mathfrak{k}^*) associated with ray classes, establishes their distinctness, and proves irreducibility of ray-class polynomials T_{\\mathfrak{m}}(t, j(\\mathfrak{k})). In many CM settings, the tau-invariant already generates the Hilbert class field, and in all analyzed moduli (including (2), (4), primes dividing 2,3,5, and combinations such as wp_2 wp_3, wp_3 wp_5, wp_3 wp_3' wp_5, and beyond), Sugawara’s claim holds. The paper completes several previously known special cases and extends the generation results to a broad suite of moduli, providing a unified arithmetic-algebraic framework for ray class generation via tau-values. Overall, it solidifies the role of CM-elliptic curves and explicit tau-values as universal generators of ray class fields in the imaginary-quadratic setting, with substantial consequences for the irreducibility of ray-class polynomials and explicit class-field computations.

Abstract

In this paper a proof is given of Sugawara's conjecture from 1936, that the ray class field of conductor over an imaginary quadratic field is generated over by a single primitive -division value of the -function, first defined by Weber and then modified by Hasse in his 1927 paper giving a new foundation of complex multiplication.
Paper Structure (14 sections, 34 theorems, 447 equations)

This paper contains 14 sections, 34 theorems, 447 equations.

Table of Contents

  1. Introduction.
  2. The case $\frak{m} = (2)$.
  3. The case $\frak{m} = (4)$.
  4. The case $\mathfrak{m} = \wp_2^3$.
  5. The case $\frak{m} = (3)$.
  6. The case $\mathfrak{m} \mid 9$.
  7. The case $\mathfrak{m} = \wp_3'^2$.
  8. The case $\mathfrak{m} = \wp_3 \wp_3'^2$.
  9. The case $\mathfrak{m} = (2)\wp_3 \wp_3'$.
  10. The case $\mathfrak{m} = \wp_5$.
  11. The case $\mathfrak{m} = \wp_3 \wp_5'$.
  12. The case $\mathfrak{m} = \wp_3 \wp_3' \wp_5$.
  13. Proof of the conjecture when $(5) = \wp_5^2$.
  14. $E_{12}$ and the case $\mathfrak{m} = \wp_2^2 \wp_3$.

Key Result

Theorem 1

If the Legendre normal form $E_2(a)$ has complex multiplication by the ring of integers in the quadratic field $K = \mathbb{Q}(\sqrt{d_K})$, where $d_K \equiv 5$ (mod $8$), then the parameter $a$ generates the ray class field $\textsf{K}_{(2)}$ of conductor $\mathfrak{m} = (2)$ over $K$. Also, $\tex

Theorems & Definitions (54)

  • Theorem 1
  • Conjecture 1
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • ...and 44 more