Nonlinear descent on moduli of local systems

Whang, Junho Peter

doi:10.1007/s11856-020-2085-x

Cited by 9 publications

(13 citation statements)

References 42 publications

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“…This was proved for t = −2 in [Sil90], and using Markoff descent this extends to all t = −2. The geometric descent in [Wha17] is carried out over these rings and also yields this finiteness. As explained in Section 3.1 this effective finiteness together with the general lifting procedures, yields an effective and feasible means to decide all of (E 1 ), (E 2 ) and (E 3 ).…”

Section: The Casementioning

confidence: 99%

“…This shows that the key diophantne equation that is at the heart of (1.0.1) is the cubic affine surface M k . For k = 4 these surfaces are nonsingular log K3's (see [CTWX20], [Wha17]) and their diophantine analysis over D is delicate. If D = Z, one can exploit the "Markoff" non-linear group of polynomial morphisims of A 3 which preserve the surfaces M k , to give a diophantine theory of these surfaces (see [GS17]).…”

Section: Introductionmentioning

confidence: 99%

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Commutators in $\mathrm{SL}_2$ and Markoff surfaces I

Ghosh,

Meiri,

Sarnak

2021

Preprint

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We show that the commutator equation over SL 2 (Z) satisfies a profinite local to global principle, while it can fail with infinitely many exceptions for SL 2 (Z[ 1 p ]). The source of the failure is a reciprocity obstruction to the Hasse Principle for cubic Markoff surfaces. Contents 1. Introduction 2. Local to global principle for the commutator word in SL 2 (Z) 2.1. Proof of Proposition 2.5 2.2. The ring Z[G m,n /G ′ m,n ] and the module G ′ m,n /G ′′ m,n . 2.3. Trace computations 2.4. Proof of Proposition 2.6 3. Lifting solutions 3.1. General D 3.1.1. An explicit analysis 3.2. The case D = Z 3.2.1. An algorithmn for (E 2 (Z)) 3.3. D = Z[ √ −d], Bianchi groups 4. Universal domains 5. Hasse failures 5.1. Hasse failures for Markoff surfaces 5.1.1. Markoff surfaces and ternary quadratic forms, SL 2 (Z)-revisited 5.1.2. Some Hasse failures revisited 5.1.3. Hasse failures over S-integers 5.2. Hasse failures for commutators Appendix A. Preliminaries B. Local analysis References

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Section: The Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Commutators in $\mathrm{SL}_2$ and Markoff surfaces I

Ghosh,

Meiri,

Sarnak

2021

Preprint

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“…We achieve this by considering points fixed by various subsurfaces of Σ. Below, we recall some elementary facts about SL 2 (C), and reproduce some topological notions considered in [22] to conveniently keep track of subsurfaces of Σ. Definition 1.…”

Section: Rank Two Local Systems On Surfacesmentioning

confidence: 99%

The rank two $p$-curvature conjecture on generic curves

Patel¹,

Shankar²,

Whang³

2018

Preprint

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We prove the p-curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological arguments. Contents 1. Introduction 1 2. Rank two local systems on surfaces 5 3. Good reduction at the special fiber 9 4. Constancy of monodromy 16 5. Proof of Theorem 1.4 20 6. Complements of hypersurfaces in projective space 24 References 25

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“…This shows that the key diophantne equation that is at the heart of (1.0.1) is the cubic affine surface M k . For k = 4 these surfaces are nonsingular log K3's (see [7], [27]) and their diophantine analysis over D is delicate. If D = Z, one can exploit the "Markoff" non-linear group of polynomial morphisims of A 3 which preserve the surfaces M k , to give a diophantine theory of these surfaces (see [9]).…”

Section: Introductionmentioning

confidence: 99%

“…This was proved for t = −2 in [25], and using Markoff descent this extends to all t = −2. The geometric descent in [27] is carried out over these rings and also yields this finiteness. As explained in Section 3.1 this effective finiteness together with the general lifting procedures, yields an effective and feasible means to decide all of (E 1 ), (E 2 ) and (E 3 ).…”

mentioning

confidence: 99%