Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Prym varieties that are not isomorphic to Jacobian

Yuri G. Zarhin

TL;DR

This work constructs Prym varieties from ramified double covers of curves with a prime-order automorphism to obtain abelian varieties carrying a canonical ${\mathbb Z}[\zeta_p]$-action that are not isomorphic to Jacobians. By combining Galois theory of carefully chosen odd polynomials with Prym theory, the paper proves End$(\mathrm{Prym}(C_{f,p}))\cong {\mathbb Z}[\zeta_p]$ and describes the $\delta_p$-spectrum on $\Omega^1(\mathrm{Prym}(C_{f,p}))$, demonstrating absolute simplicity under suitable hypotheses (notably ${\rm Gal}(f)\supseteq {\mathbb W}({\mathbb D}_m)$ and $r$ even). It further shows that the Prym varieties are not only non-Jacobians but remain so even when polarizations are ignored, providing new explicit examples in the landscape of abelian varieties with extra endomorphisms. The Appendix supplies explicit Galois-realizable polynomial families (e.g., $h(x)=x^{2m}-x^2-1$) to realize these Prym varieties concretely with End$\cong {\mathbb Z}[\zeta_p]$.

Abstract

We study Prym varieties of ramified (at precisely two points) double covers of smooth irreducible complex projectives curves that admit an automorphism of prime order $p>2$. Using Galois theory, we give an explicit constructions of Prym varieties that are not isomorphic to jacobians (even if one ignores the polarizations).

Prym varieties that are not isomorphic to Jacobian

TL;DR

This work constructs Prym varieties from ramified double covers of curves with a prime-order automorphism to obtain abelian varieties carrying a canonical -action that are not isomorphic to Jacobians. By combining Galois theory of carefully chosen odd polynomials with Prym theory, the paper proves End and describes the -spectrum on , demonstrating absolute simplicity under suitable hypotheses (notably and even). It further shows that the Prym varieties are not only non-Jacobians but remain so even when polarizations are ignored, providing new explicit examples in the landscape of abelian varieties with extra endomorphisms. The Appendix supplies explicit Galois-realizable polynomial families (e.g., ) to realize these Prym varieties concretely with End.

Abstract

We study Prym varieties of ramified (at precisely two points) double covers of smooth irreducible complex projectives curves that admit an automorphism of prime order . Using Galois theory, we give an explicit constructions of Prym varieties that are not isomorphic to jacobians (even if one ignores the polarizations).
Paper Structure (4 sections, 9 theorems, 145 equations)

This paper contains 4 sections, 9 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Galois groups of odd polynomials and permutation modules
  3. Cyclic covers, jacobians and prymians
  4. Appendix: Galois properties of certain polynomials

Key Result

Theorem 1.1

Suppose that $r\ge 2$ is an integer and let us put Let $f(x)\in {\mathbb C}[x]$ be an odd polynomial of degree $n=2m+1$ without repeated roots. Then:

Theorems & Definitions (26)

  • Theorem 1.1
  • Remark 1.2
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Example 2.5: Example 2.5 of ZarhinI
  • Lemma 2.8
  • proof
  • Remark 3.4
  • ...and 16 more