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Infinitely many primes of basic reduction for some abelian fourfolds

Wanlin Li, Elena Mantovan, Rachel Pries, Yunqing Tang

Abstract

If $E$ is an elliptic curve, defined over $\mathbb{Q}$ or a number field having at least one real embedding, then Elkies proved that $E$ has supersingular reduction at infinitely many primes $p$. Baba and Granath extended this result to certain curves $C$ of genus $2$ with field of moduli $\mathbb{Q}$, under a condition on the endomorphism ring of the Jacobian. In this paper, we extend these results to certain curves of genus $4$ having an automorphism of order $5$, proving that the Jacobians of these curves have basic reduction (as defined by Kottwitz) for infinitely many primes $p$. To do this, we study the complex uniformization of the Deligne--Mostow Shimura variety $\mathrm{Sh}$ associated with the one dimensional family of these curves. By analyzing the real points on $\mathrm{Sh}$, we compute three geodesics in the upper half plane that are edges of a fundamental triangle for the action of the unitary similitude group. Using representations of quadratic forms, we determine the points on $\mathrm{Sh}$ which represent curves whose Jacobians have complex multiplication by certain quadratic extensions of the cyclotomic field $\mathbb{Q}(ζ_5)$. We conclude by studying the equidistribution of these points and the reduction of these CM cycles on the Shimura variety.

Infinitely many primes of basic reduction for some abelian fourfolds

Abstract

If is an elliptic curve, defined over or a number field having at least one real embedding, then Elkies proved that has supersingular reduction at infinitely many primes . Baba and Granath extended this result to certain curves of genus with field of moduli , under a condition on the endomorphism ring of the Jacobian. In this paper, we extend these results to certain curves of genus having an automorphism of order , proving that the Jacobians of these curves have basic reduction (as defined by Kottwitz) for infinitely many primes . To do this, we study the complex uniformization of the Deligne--Mostow Shimura variety associated with the one dimensional family of these curves. By analyzing the real points on , we compute three geodesics in the upper half plane that are edges of a fundamental triangle for the action of the unitary similitude group. Using representations of quadratic forms, we determine the points on which represent curves whose Jacobians have complex multiplication by certain quadratic extensions of the cyclotomic field . We conclude by studying the equidistribution of these points and the reduction of these CM cycles on the Shimura variety.

Paper Structure

This paper contains 61 sections, 71 theorems, 95 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Infinitely many primes of supersingular reduction
  3. An approach using moduli spaces and complex multiplication
  4. Review of proof of Elkies
  5. Strategy of the proof
  6. Related work
  7. Table of Contents
  8. Moduli of cyclic covers and the Shimura variety
  9. Description of curves and signature types
  10. Curves in the family with extra automorphisms
  11. Two curves in the family with extra automorphisms
  12. No other curves in the family have extra automorphisms
  13. A singular curve in the family
  14. The Klein $J$-function and field of definition
  15. The Deligne-Mostow Shimura variety
  16. ...and 46 more sections

Key Result

Theorem 1.1

Suppose $C_t$ is a smooth projective genus $4$ curve with an affine equation of the form Assume that the reduction of $C_t$ at $5$ is singular. Suppose that $J(t):=(t^2 -t+1)^3/t^2(t-1)^2$ is in ${\mathbb Q} \cap (-\infty, 27/4)$. Then $\mathrm{Jac}(C_t)$ has basic reduction at infinitely many primes.

Figures (4)

  • Figure 1: The hyperbolic triangle with vertices $\tilde{P}$, $\tilde{Q}$, and $\tilde{R}$
  • Figure 2: Analytic position of points on the geodesic $G_{Q,P}$
  • Figure 3: Schematic of the arch $\overset{\frown}{PQR}$, with modification in Case (A)
  • Figure 4: Schematic of the arch $\overset{\frown}{PQR}$, with modification in Case (B)

Theorems & Definitions (152)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.4
  • ...and 142 more