[Paper Review] PEL Modulispaces without $\mathbb C$-valued points

Oliver Bueltel|arXiv (Cornell University)|Aug 29, 2008

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper provides new moduli interpretations for fibers of Shimura varieties at odd prime characteristics, demonstrating that every bounded symmetric domain admits quotients by arithmetic groups whose models have good reduction at prime divisors of an odd prime p. The key contribution is establishing the existence of such good reduction models for all bounded symmetric domains in odd prime characteristics.

ABSTRACT

We give several new moduli interpretations of the fibers of certain Shimura varieties over several prime numbers. As a corollary we obtain that for every prescribed odd prime characteristic $p$ every bounded symmetric domain possesses quotients by arithmetic groups whose models have good reduction at a prime divisor of $p$.

Motivation & Objective

• To understand the structure of Shimura variety fibers at odd prime characteristics.
• To provide new moduli interpretations for these fibers using arithmetic groups.
• To establish the existence of good reduction models for quotients of bounded symmetric domains at prime divisors of an odd prime p.

Proposed method

• Analyzing the fibers of Shimura varieties over prime numbers using arithmetic group actions.
• Employing moduli-theoretic interpretations to describe the geometry of these fibers.
• Utilizing the theory of bounded symmetric domains and their arithmetic quotients.
• Establishing good reduction at prime divisors of an odd prime p through moduli constructions.
• Leveraging known results on good reduction and arithmetic group quotients in the context of Shimura varieties.
• Connecting moduli spaces of abelian varieties with level structures to the fibers at p.

Experimental results

Research questions

• RQ1 Do all bounded symmetric domains admit arithmetic quotients with good reduction at prime divisors of an odd prime p?
• RQ2 Can the fibers of Shimura varieties at odd primes be given new moduli interpretations?
• RQ3 What is the relationship between moduli spaces of abelian varieties and good reduction in odd prime characteristics?
• RQ4 How do arithmetic groups act on bounded symmetric domains to yield models with good reduction?
• RQ5 What conditions ensure that a quotient of a bounded symmetric domain has good reduction at a prime dividing p?

Key findings

• For every odd prime p, every bounded symmetric domain admits a quotient by an arithmetic group whose model has good reduction at a prime divisor of p.
• The fibers of certain Shimura varieties over odd primes admit new moduli interpretations via arithmetic group quotients.
• Good reduction is achieved at prime divisors of p for arithmetic quotients of bounded symmetric domains.
• The construction relies on moduli-theoretic descriptions of Shimura variety fibers at odd primes.
• The results extend the scope of good reduction phenomena to all bounded symmetric domains in odd prime characteristics.
• The paper establishes a general existence result for good reduction models in the specified setting.