Essential dimension of isogenies

TL;DR

This work develops a birational-geometric framework to bound the essential dimension of isogenies between abelian varieties. The authors prove a sharp lower bound ed(α) ≥ dim A − dim B + (p−1)/p · rank_p(ker(α) ∩ B) for each prime p, yielding the incompressibility of multiplication maps m_A (for m ≥ 2) and a coprime-to-(dim A)! formula ed(α) = min_B(dim A − dim B + rank(ker(α) ∩ B)). The approach combines descent via Galois actions, Albanese/MRC fibrations, and sharp bounds for abelian p-group actions on rationally connected varieties, along with equivariant intersection theory to control characteristic classes. The results illuminate the geometry of rational images of abelian varieties, constrain group actions on RC varieties, and connect essential dimension to Cremona-type questions, with several sharp examples confirming tightness. Overall, the paper advances understanding of how birational geometry governs the complexity of isogenies and related morphisms.

Abstract

We give a lower bound for the essential dimension of isogenies of complex abelian varieties. The bound is sharp in many cases. In particular, the multiplication-by-$m$ map is incompressible for every $m\geq 2$, confirming a conjecture of Brosnan.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2
• Corollary 3
• Proposition 4
• proof : First proof of Proposition \ref{['prop:MRC quotient abelian']}
• proof : Second proof of Proposition \ref{['prop:MRC quotient abelian']}
• Corollary 7
• proof
• Proposition 8
• proof