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$K_2$ of families of elliptic curves over non-Abelian cubic and quartic fields

François Brunault, Rob de Jeu, Hang Liu, Fernando Rodriguez Villegas

Abstract

We give two constructions of families of elliptic curves over cubic or quartic fields with three, respectively four, `integral' elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements linearly independent. For their integrality, we prove a new criterion that does not ignore any torsion. We also verify Beilinson's conjecture numerically for just over 90 of the curves.

$K_2$ of families of elliptic curves over non-Abelian cubic and quartic fields

Abstract

We give two constructions of families of elliptic curves over cubic or quartic fields with three, respectively four, `integral' elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements linearly independent. For their integrality, we prove a new criterion that does not ignore any torsion. We also verify Beilinson's conjecture numerically for just over 90 of the curves.
Paper Structure (14 sections, 11 theorems, 86 equations, 12 tables)

This paper contains 14 sections, 11 theorems, 86 equations, 12 tables.

Table of Contents

  1. Introduction
  2. Beilinson's conjecture
  3. A new integrality criterion
  4. K2 of families of elliptic curves with 7, 8 and 10-torsion points
  5. Bloch's construction of elements in K2T
  6. Families of elliptic curves with torsion
  7. Integrality of the elements
  8. The main result
  9. Linear independence of the elements
  10. The elliptic dilogarithm
  11. Complex conjugation on the universal elliptic curve
  12. End of the proof of Theorem \ref{['thm:independence']}
  13. K2T of families of elliptic curves over cubic fields
  14. Numerical results

Key Result

Theorem 1.1

Let $f_a(X)$ be one the following polynomials Let $F = \mathbb Q(t)$ be the cubic field generated by a root $t$ of $f_a$. Consider the elliptic curve $E$ over $F$ defined by with $f = t^3-t^2$ and $g = t^2-t$. Then $S_1$, $S_2$ and $S_3$ are in $K_2^T(E)_{\textup{int}}$, and they are $\mathbb Z$-linearly independent for $|a| \gg 0$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Remark 3.3
  • Example 3.4
  • Remark 3.5
  • Remark 4.1
  • Remark 4.6
  • ...and 30 more