On distribution of supersingular primes of abelian varieties and K3 surfaces

TL;DR

The paper studies the distribution of supersingular primes for non-CM abelian varieties and K3 surfaces over number fields. By relating supersingular reductions to the associated ℓ-adic Galois representations, it analyzes the algebraic monodromy groups G_ℓ and employs an effective Chebotarev density theorem to bound the counting function π_{SS_X}(x) of supersingular primes. It shows that the natural density of supersingular primes is zero in the non-CM case and provides explicit upper bounds that depend on dim G_ℓ and rk G_ℓ, with improvements under GRH and when G_ℓ is connected; it also demonstrates the convergence of the sum ∑_{v∈SS_X} 1/#F_v. These results extend Lang-Trotter-type questions from elliptic curves to general non-CM abelian varieties and K3 surfaces using ℓ-adic monodromy techniques and effective density theorems.

Abstract

Let X be an abelian variety or a K3 surface defined over a number field K. We prove that the density of the supersingular primes of X is zero if X is non-CM. By applying an effective Chebotarev density theorem of Serre, we obtain asymptotic upper bounds of the counting function for these supersingular primes.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• Remark 2.4
• Corollary 2.5
• proof