[Paper Review] Local points on P-adically uniformized Shimura varieties

Bruce W. Jordan, Ron Livné+1 more|ArXiv.org|Apr 26, 2002

Advanced Algebra and Geometry13 references4 citations

TL;DR

This paper determines over which local fields certain $p$-adically uniformized Shimura varieties have rational points, using $p$-adic uniformization and class field theory. It proves that for specific Shimura curves, the Jacobians are even in the sense of [PS], establishing a key arithmetic property via analysis of deficient primes and Hilbert symbols over number fields.

ABSTRACT

Using the p-adic uniformization of Shimura varieties we determine, for some of them, over which local fields they have rational points. Using this we show in some new curve cases that the jacobians are even in the sense of Poonen and Stoll.

Motivation & Objective

• To determine over which local fields certain $p$-adically uniformized Shimura varieties have rational points.
• To extend the theory of local points beyond the case of Shimura curves with good reduction, particularly for bad reduction cases.
• To investigate the arithmetic of Jacobians of Shimura curves, specifically whether they are even in the sense of [PS].
• To analyze the interplay between $p$-adic uniformization, class field theory, and the existence of rational points on Shimura varieties.
• To establish conditions under which the number of deficient primes (infinite and finite) is even, using Hilbert symbols and local class field theory.

Proposed method

• Uses $p$-adic uniformization of Shimura varieties via Drinfel'd's theory and the results of Čerednik and Rapoport–Zink.
• Applies the theory of $p$-adic uniformization to reduce the existence of local points to the solubility of certain quaternion algebras over local fields.
• Employs class field theory and the Hilbert symbol $(-1, - heta)_{v}$ to analyze splitting behavior of quaternion algebras at places of $F$.
• Analyzes the action of the group $W \simeq (\mathbb{Z}/2\mathbb{Z})^r$ on the curve $X$, where $r$ is the number of finite primes in the discriminant of the quaternion algebra.
• Uses the Riemann–Hurwitz formula to relate the genus of $X$ to the genus of the quotient $X/W$, with correction terms depending on fixed points of $W$.
• Applies genus theory and the product formula for Hilbert symbols to determine when the number of deficient primes is even, particularly at places of residue characteristic 2 and at infinite places.

Experimental results

Research questions

• RQ1 For which local fields do $p$-adically uniformized Shimura varieties have rational points?
• RQ2 Under what conditions is the number of deficient infinite and finite primes of a Shimura curve even?
• RQ3 When is the Jacobian of a Shimura curve even in the sense of [PS]?
• RQ4 How does the splitting behavior of the quaternion algebra $B(-1, - heta)$ at places of $F$ relate to the parity of deficient primes?
• RQ5 What role does the Hilbert symbol $(-1, - heta)_v$ play in determining the existence of local points and the evenness of the Jacobian?

Key findings

• The number of infinite deficient primes of a Shimura variety $X$ is even if and only if $(-1, - heta)_{F_{rak{P}}}=1$, where $ heta$ is a totally positive generator of the prime ideal $rak{P}^{k_+}$.
• The number of finite relevant deficient primes is even if and only if $(-1, - heta)_{F_{rak{Q}}}=1$ for all places $rak{Q}$ of residue characteristic 2.
• For $m \equiv 5 \pmod{8}$, the number of relevant deficient primes is even, implying the Jacobian is even.
• If $m \equiv 1 \pmod{8}$ and $p$ is a rational prime inert in $F$ with $p \equiv 1 \pmod{4}$, then the number of relevant deficient primes is odd.
• When the discriminant of the quaternion algebra has exactly one finite prime (i.e., $r=1$), the genus of the curve $X$ is odd if and only if the number of deficient primes is odd, and the Jacobian is even if the number of deficient primes is even.
• The theorem concludes that for $r \geq 3$, the genus $g(X)$ is odd, so the Jacobian $\operatorname{Jac}(X)$ is even; for $r=1$, the evenness of the Jacobian is established via explicit analysis of Hilbert symbols at places above 2 and at infinite places.