Quadratic Chabauty and Rational Points II: Generalised Height Functions on Selmer Varieties

Balakrishnan, Jennifer S.; Dogra, Netan

doi:10.1093/imrn/rnz362

Cited by 27 publications

(34 citation statements)

References 40 publications

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“…On the one hand, the gauge-theoretical perspective has the potential to be applicable to a very broad class of phenomena encompassing many of the central problems of current day number theory [38]. On the other, unlike Faltings's proof, which is widely regarded as ineffective, the gauge-theory proof conjecturally leads to a computational method for actually finding rational solutions [37], a theme that is currently under active investigation [8,9,22,23]. It should be remarked that the map A that associates gauge fields to points has been well-known since the 1950s when the variety V is an elliptic curve, an abelian variety, or generally, a commutative algebraic group.…”

Section: Diophantine Geometry and Gauge Theorymentioning

confidence: 99%

“…But then, instead of a moduli space of torsors, we will be dealing with a gerbe. 9 We will not give the precise definitions in terms of fibre functors. A good general introduction is the book of Szamuely [59], while the algebraic group realisation we will use below is given a careful discussion in [24].…”

Section: Homotopy and Gauge Fieldsmentioning

confidence: 99%

“…The last object U DR is the De Rham fundamental group [24] endowed with a Hodge flitration F i , which can be computed explicitly in such a way that the map A DR is also described explicitly in terms of p-adic iterated integrals. From this point of view, computing the E-L equation is the main tool for finding the points V (Q) [35,36,8,9].…”

Section: Non-abelian Gauge Fields and Diophantine Geometrymentioning

confidence: 99%

“…This implies that the image is a constructible set for the Zariski topology, which therefore admits a finite polynomial description. There is an increasing collection of examples for which the image has a precise enough description for the rational points to be computed completely [8,9,22,23]. A spectacular recent result [10] carries this out for the modular curve…”

Section: Non-abelian Gauge Fields and Diophantine Geometrymentioning

confidence: 99%

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Arithmetic gauge theory: A brief introduction

Kim

2019

Topology and Physics

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Much of arithmetic geometry is concerned with the study of principal bundles. They occur prominently in the arithmetic of elliptic curves and, more recently, in the study of the Diophantine geometry of curves of higher genus. In particular, the geometry of moduli spaces of principal bundles appears to be closely related to an effective version of Faltings's theorem on finiteness of rational points on curves of genus at least 2. The study of arithmetic principal bundles includes the study of Galois representations, the structures linking motives to automorphic forms according to the Langlands programme. In this article, we give a brief introduction to the arithmetic geometry of principal bundles with emphasis on some elementary analogies between arithmetic moduli spaces and the constructions of quantum field theory. For the most part, it can be read as an attempt to explain standard constructions of arithmetic geometry using the language of physics, albeit employed in an amateurish and ad hoc manner.

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Section: Diophantine Geometry and Gauge Theorymentioning

confidence: 99%

Section: Homotopy and Gauge Fieldsmentioning

confidence: 99%

Section: Non-abelian Gauge Fields and Diophantine Geometrymentioning

confidence: 99%

Section: Non-abelian Gauge Fields and Diophantine Geometrymentioning

confidence: 99%

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Arithmetic gauge theory: A brief introduction

Kim

2019

Topology and Physics

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“…In a roughly chronological order, on the thrice punctured projective line P 1 Q − {0, 1, ∞} [22] it has been verified for the Z S -points and for the S-integers of a totally real field [21]; if X is a once punctured elliptic curve with CM over Q then n ≥ #S + r + 1 is sufficient to show the finiteness of X(Z S ) for some r depending on E [25] and if all the Tamagawa numbers are 1 and X has rank 1 then n = 2 is sufficient [24]; if X is a complete hyperbolic curve with CM Jacobian [15]; and if X is a complete hyperbolic curve and a solvable cover of P 1 Q (and hence any smooth superelliptic curve over Q with genus at least 2 [20]. In [4], [8] Balakrishnan and Dogra have made an explicit application of non-abelian Chabauty when n = 2 -what they refer to as 'Quadratic Chabauty' -to p-adic heights on elliptic and hyperelliptic curves. They extended their methods in [7] to propose an effective Chabauty-Kim theorem which provides bounds of the type produced by Coleman under certain hypothesis even when r = g. Recently with Müller, Tuitman and Vonk they demonstrated an application of the Chabauty-Kim method in [9] to a nonhyperelliptic curve and used the method to complete the classification of non-CM elliptic curves over Q with split Cartan level structure.…”

Section: Introductionmentioning

confidence: 99%

Computation of the unipotent Albanese map on elliptic and hyperelliptic curves

Beacom

2019

Ann. Math. Québec

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We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the p-adic de Rham period map j dr n on elliptic and hyperelliptic curves over number fields via their universal unipotent connections U.Several algorithms forming part of the computation of finite level versions j dr n of the unipotent Albanese maps are presented. The computation of the logarithmic extension of U in general requires a description in terms of an open covering, and can be regarded as a simple example of computational descent theory. We also demonstrate a constructive version of a lemma of Hadian used in the computation of the Hodge filtration on U over affine elliptic and odd hyperelliptic curves.We use these algorithms to present some new examples describing the co-ordinates of some of these period maps. This description will be given in terms iterated p-adic Coleman integrals. We also consider the computation of the co-ordinates if we replace the rational basepoint with a tangential basepoint, and present some new examples here as well.Here we introduce unipotent connections and background material on the unipotent Albanese map. Throughout both sections [23] is used as a primary reference for definitions and results.Let K be a field of characteristic 0, let X be a K-scheme, and suppose we have a fixed basepoint b ∈ X(K). Let V be a vector bundle on X.Here Ω 1 X/K is the sheaf of 1-forms on X/K. Remark 2.2. We will often refer to a vector bundle V with connection ∇ simply as a connection and write such objects either as (V, ∇) or simply as V.Remark 2.3. We may extend ∇ to a covariant derivative ∇ 1 :Definition 2.4. We say that V is a flat or integrable connection if the induced morphismis the zero map. Note that if X is a curve, then any connection V is automatically flat.Given a connection (V, ∇) with V of rank n, there is a matrix Ω ∈ gl n ⊗ Ω 1 X/K called the connection matrix which determines ∇: suppose that we have a local basis e i : O X ֒→ V (1 ≤ i ≤ n). Let U ⊂ X be some trivialising neighbourhood in X. Then ∇(e i ) ∈ V ⊗ Ω 1 X/K (U ), and so there are ω ij ∈ Ω 1 X/K (U ) such thatWe let Ω := (ω ij ). We may show that in matrix notation ∇(e · f ) = e · (df + Ω · f ), and so ∇ acts locally as d + Ω. Remark 2.5. A connection (V, ∇ = d + Ω) is flat if and only if dΩ + Ω ∧ Ω = 0Definition 2.6. A morphism of connections (V, ∇) → (W, ∇ ′ ) is a morphism f : V → W of sheaves preserving the connection.Definition 2.7. A connection V is unipotent with index of unipotency less than or equal to n if there is a decreasing sequence of sub-connectionssuch that the quotients V i+1 /V i are isomorphic to a direct sum of copies of (O X , d) i.e. they are trivial.We obtain the following category on X:Definition 2.8. Let Un n (X) be defined to be category whose objects are unipotent vector bundles on X with flat connection having index of unipotency less than or equal to n with morphisms being morphisms of connections. Define Un(X) to be ∪ n≥1 Un n (X)Given so...

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Two Recent p-adic Approaches Towards the (Effective) Mordell Conjecture

Balakrishnan

Best

Bianchi

et al. 2021

Arithmetic L-Functions and Differential Geometric Methods

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Quadratic Chabauty and Rational Points II: Generalised Height Functions on Selmer Varieties

Cited by 27 publications

References 40 publications

Arithmetic gauge theory: A brief introduction

Arithmetic gauge theory: A brief introduction

Computation of the unipotent Albanese map on elliptic and hyperelliptic curves

Two Recent p-adic Approaches Towards the (Effective) Mordell Conjecture

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