The motivic fundamental group of P1∖{0,1,∞} and the theorem of Siegel

Kim, Minhyong

doi:10.1007/s00222-004-0433-9

Cited by 130 publications

(174 citation statements)

References 12 publications

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“…the image consists exactly of the crystalline torsors (exactly as in the proof of [23], section 2, lemma 6). By passing to the limit over n, we see that the set of isomorphisms classes of crystalline torsors for U et is represented by a…”

Section: §2étale Realizationsmentioning

confidence: 95%

“…We warn the reader at this point that this subvariety was denoted H 1 f in the reference [23]. In this paper, we will use the notation H That is, we are requiring the existence of a G v -invariant point in T B DR (R ⊗ B DR ).…”

Section: §2étale Realizationsmentioning

confidence: 99%

“…However, since the reference [23] dealt explicitly only with H g , we sketch the modification necessary to represent the crystalline torsors. For each n, consider the functor on Q p -algebras The proof is verbatim the same as [23], section 1, proposition 3 and will be therefore omitted.…”

mentioning

confidence: 99%

“…For each n, consider the functor on Q p -algebras The proof is verbatim the same as [23], section 1, proposition 3 and will be therefore omitted. And then, as in the few paragraphs preceding that proposition, we have an exact sequence from which we get the connecting homomorphism…”

mentioning

confidence: 99%

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The Unipotent Albanese Map and Selmer Varieties for Curves

Kim¹

2009

Publ. Res. Inst. Math. Sci.

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187

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We study the unipotent Albanese map that associates the torsor of paths for p-adic fundamental groups to a point on a hyperbolic curve. It is shown that the map is very transcendental in nature, while standard conjectures about the structure of mixed motives provide control over the image of the map. As a consequence, conjectures of 'Birch and Swinnerton-Dyer type' are connected to finiteness theorems of Faltings-Siegel type.

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Section: §2étale Realizationsmentioning

confidence: 95%

Section: §2étale Realizationsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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The Unipotent Albanese Map and Selmer Varieties for Curves

Kim¹

2009

Publ. Res. Inst. Math. Sci.

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187

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“…Although this is not the first 'algebro-geometric' proof of finiteness of Z S -solutions to the unit equation (see the important work of M. Kim [11], where the result is stated in terms of Z S -points of P 1 − {0, 1, ∞} and it is attributed to Siegel), our method is effective and gives explicit constants.…”

Section: Here Z × S Denotes the Group Of Units Of The Ring Z S Of Ramentioning

confidence: 99%

Modular forms and effective Diophantine approximation

Murty

Pasten

2013

Journal of Number Theory

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After the work of G. Frey, it is known that an appropriate bound for the Faltings height of elliptic curves in terms of the conductor (Frey's height conjecture) would give a version of the ABC conjecture. In this paper we prove a partial result towards Frey's height conjecture which applies to all elliptic curves over Q, not only Frey curves. Our bound is completely effective and the technique is based in the theory of modular forms. As a consequence, we prove effective explicit bounds towards the ABC conjecture of similar strength to what can be obtained by linear forms in logarithms, without using the latter technique. The main application is a new effective proof of the finiteness of solutions to the S-unit equation (that is, S-integral points of P 1 − {0, 1, ∞}), with a completely explicit and effective bound, without using any variant of Baker's theory or the Thue-Bombieri method.

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Remark on fundamental groups and effective Diophantine methods for hyperbolic curves

Kim

2011

Number Theory, Analysis and Geometry

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Dedicated to the memory of Serge LangIn a few earlier papers ([8], [9], [10]) attention was called to the striking parallel between the ideas surrounding the well-known conjecture of Birch and Swinnerton-Dyer for elliptic curves, and the mysterious section conjecture of Grothendieck [6] that concerns hyperbolic curves. We wish to explain here some preliminary ideas for 'effective non-abelian descent' on hyperbolic curves equipped with at least one rational point. We again follow in an obvious manner the method of descent on elliptic curves and, therefore, rely on conjectures. In fact, the main point is to substitute the section conjecture for the finiteness of the Shafarevich group. That is to say, the input of the section conjecture is of the form section conjecture ⇒ termination of descent At a number of different lectures on the topic of fundamental groups and Diophantine geometry, the question was raised about the role of surjectivity in the section conjecture as far as Diophantine applications are concerned. This implication is intended as something of a reply.To start the descent, on the other hand, requires the use of p-adic Hodge theory and the unipotent Albanese map. In this process, in general, one another conjecture is unfortunately needed. It could be, for example, the Bloch-Kato conjecture on surjectivity of the p-adic Chern class map that has been referred to in [9]. In other words, via the construction of Selmer varieties and Albanese maps, one deduces an implication Bloch-Kato conjecture ⇒ beginning of descentThe main caveat here arises from lack of actual knowledge of computational issues on the part of the author. To avoid misleading anyone about what is being achieved here, we have in the following section separated out the questionable portions as hypotheses [H] and [H']. That is to say, the objects that mediate this process, namely Galois cohomology groups/varieties and maps between them, seem in principle to be computable. But even to the algorithmically illiterate perspective, it is obvious that actual computation would be daunting to the point of impossibility given the technology of the present day. Nevertheless, it is perhaps not entirely devoid of value to point out one direction of investigation in effective methods, in the hope that even incompetent strategies may eventually be improved through the focussing of sharper skills obviously available in the community. Hence, the present paper.One point of some theoretical interest concerns the comparison with 'effective Mordell conjectures' in the usual sense where upper bounds for heights are proposed. If we fix a point b on the curve and measure heights with respect to the corresponding divisor, the height of another point measures the distance from b at all places. So an upper bound for the height corresponds to a lower bound for the distance from b at all places. On the other hand, what the p-adic Hodge theory provides (in principle) is a lower bound for the p-adic distance between all pairs of points at one place. This lower bound is ...

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The motivic fundamental group of P1∖{0,1,∞} and the theorem of Siegel

Cited by 130 publications

References 12 publications

The Unipotent Albanese Map and Selmer Varieties for Curves

The Unipotent Albanese Map and Selmer Varieties for Curves

Modular forms and effective Diophantine approximation

Remark on fundamental groups and effective Diophantine methods for hyperbolic curves

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