[Paper Review] Strong approximation for a family of norm varieties

Yang Cao, Dasheng Wei+1 more|arXiv (Cornell University)|Mar 29, 2018

Algebraic Geometry and Number Theory32 references3 citations

TL;DR

This paper establishes strong approximation with Brauer–Manin obstruction for norm equations $ N_{L/k}(x) = extstyleigotimes_{i=1}^n p_i(t) $ over number fields, using a torus action to overcome Brauer group obstructions on fibers. It proves the result unconditionally when $ L $ embeds into $ k[t]/(p_i(t)) $, and conditionally under Schinzel’s hypothesis when $ L/k $ is cyclic.

ABSTRACT

We study strong approximation of the equation N_{L/k}(x) = \prod_{i=1}^n p_i(t) where L/k is a finite extension of number fields and p_i(t)'s are distinct irreducible polynomials over k. We prove this equation satisfies strong approximation with Brauer-Manin obstruction when L can be imbedded in k[t]/(p_i(t)) over k for all 1\leq i\leq n. Under Schinzel's hypothesis, we prove that the same result is true without assuming that L can be imbedded in k[t]/(p_i(t)) for all 1\leq i\leq n when L/k is cyclic.

Motivation & Objective

• To establish strong approximation for norm equations $ N_{L/k}(x) = extstyleigotimes_{i=1}^n p_i(t) $ over number fields.
• To overcome the difficulty that Brauer groups of special fibers may have infinite torsion, a key obstruction in fibration methods.
• To extend conditional results under Schinzel’s hypothesis to unconditional results via torus actions and descent techniques.
• To generalize prior results on weak approximation and strong approximation for Châtelet surfaces and norm varieties.
• To provide a framework for strong approximation off a finite set of places using the geometry of torsors and Brauer–Grothendieck obstruction.

Proposed method

• Uses the action of the torus $ ext{Res}_{L/k}^1(bG_m) $ to construct a finite subgroup of the Brauer group of the generic fiber.
• Applies this action to test rational points on fibers via restriction of Brauer classes to special fibers.
• Employs the fibration method with a fibration $ f: W o bP^1 $, where $ W $ is a torsor under $ ext{ker}( ho) $, a product of Weil restrictions.
• Relies on the Hochschild–Serre spectral sequence to compute $ ext{Br}_1(W) = ext{Br}(k) $ and $ ext{Pic}(W_{ar{k}}) = 0 $, ensuring trivial Picard group.
• Uses the fact that $ ar{k}[W]^ imes = ar{k}^ imes $ to control units and simplify the Brauer group computation.
• Applies Conjecture 9.1 and 9.2 from Harpaz–Wittenberg [31] to deduce strong approximation off a finite set of places.

Experimental results

Research questions

• RQ1 Can strong approximation be established for norm varieties $ N_{L/k}(x) = extstyleigotimes_{i=1}^n p_i(t) $ with Brauer–Manin obstruction?
• RQ2 How can the infinite Brauer group obstruction on fibers (D1) be overcome in fibration-based approaches?
• RQ3 Under what conditions on $ L/k $ and the polynomials $ p_i(t) $ does strong approximation hold unconditionally?
• RQ4 Can Schinzel’s hypothesis be replaced by a geometric condition to remove the conditional assumption?
• RQ5 What role does the torus action on the universal torsor play in controlling rational points across fibers?

Key findings

• The equation $ N_{L/k}(x) = extstyleigotimes_{i=1}^n p_i(t) $ satisfies strong approximation with Brauer–Manin obstruction when $ L $ embeds into $ k[t]/(p_i(t)) $ over $ k $ for all $ i $.
• Under Schinzel’s hypothesis, strong approximation holds unconditionally for cyclic extensions $ L/k $, even without the embedding assumption.
• The Brauer group of the variety $ W $ satisfies $ ext{Br}_1(W) = ext{Br}(k) $, and $ ext{Pic}(W_{ar{k}}) = 0 $, which simplifies the obstruction theory.
• The unit group $ ar{k}[W]^ imes = ar{k}^ imes $, ensuring no nontrivial units complicate the Brauer–Grothendieck obstruction.
• The fibration $ f: W o bP^1 $ induces a torsor under $ ext{ker}( ho) $, and rational points on fibers are controlled via the torus action.
• Corollary 5.6 shows that if Conjecture 9.1 [31] holds, then strong approximation holds off a finite, uniformly bounded set $ S_0 $ independent of $ S $.