Modular curves and Fermat's theorem

Nekovář, Jan

doi:10.21136/mb.1994.126199

Math. Bohem.

1994

DOI: 10.21136/mb.1994.126199

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Modular curves and Fermat's theorem

Jan Nekovář¹

Abstract: Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

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Cited by 3 publications

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Arakelov motivic cohomology I

Holmstrom¹,

Scholbach²

2015

J. Algebraic Geom.

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This paper introduces a new cohomology theory for schemes of finite type over an arithmetic ring. The main motivation for this Arakelovtheoretic version of motivic cohomology is the conjecture on special values of L-functions and zeta functions formulated by the second author [Sch13]. Taking advantage of the six functors formalism in motivic stable homotopy theory, we establish a number of formal properties, including pullbacks for arbitrary morphisms, pushforwards for projective morphisms between regular schemes, localization sequences, h-descent. We round off the picture with a purity result and a higher arithmetic Riemann-Roch theorem.In the second part of the paper, we relate Arakelov motivic cohomology to classical constructions such as arithmetic K and Chow groups and the height pairing.

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Arakelov motivic cohomology I

Holmstrom¹,

Scholbach²

2015

J. Algebraic Geom.

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Polylogarithms, regulators, and Arakelov motivic complexes

Goncharov

2004

J. Amer. Math. Soc.

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We construct an explicit regulator map from the weight n n Bloch higher Chow group complex to the weight n n Deligne complex of a regular projective complex algebraic variety X X . We define the weight n n Arakelov motivic complex as the cone of this map shifted by one. Its last cohomology group is (a version of) the Arakelov Chow group defined by H. Gillet and C. Soulé. We relate the Grassmannian n n -logarithms to the geometry of the symmetric space S L n ( C ) / S U ( n ) SL_n(\mathcal {C})/SU(n) . For n = 2 n=2 we recover Lobachevsky’s formula expressing the volume of an ideal geodesic simplex in the hyperbolic space via the dilogarithm. Using the relationship with symmetric spaces we construct the Borel regulator on K 2 n − 1 ( C ) K_{2n-1}(\mathcal {C}) via the Grassmannian n n -logarithms. We study the Chow dilogarithm and prove a reciprocity law which strengthens Suslin’s reciprocity law for Milnor’s group K 3 M K^M_3 on curves. Our note,“Chow polylogarithms and regulators”, can serve as an introduction to this paper.

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On unipotent radicals of motivic Galois groups

Eskandari

Murty

2023

Alg. Number Th.

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Let T be a neutral Tannakian category over a field of characteristic zero with unit object 1, and equipped with a filtration W • similar to the weight filtration on mixed motives. Let M be an object of T , and u(M) ⊂ W −1 Hom(M, M) the Lie algebra of the kernel of the natural surjection from the fundamental group of M to the fundamental group of Gr W M. A result of Deligne gives a characterization of u(M) in terms of the extensions 0 → W p M → M → M/W p M → 0: it states that u(M) is the smallest subobject of W −1 Hom(M, M) such that the sum of the aforementioned extensions, considered as extensions of 1 by W −1 Hom(M, M), is the pushforward of an extension of 1 by u(M). We study each of the abovementioned extensions individually in relation to u(M). Among other things, we obtain a refinement of Deligne's result, where we give a sufficient condition for when an individual extension 0 → W p M → M → M/W p M → 0 is the pushforward of an extension of 1 by u(M). In the second half of the paper, we give an application to mixed motives whose unipotent radical of the motivic Galois group is as large as possible (i.e., with u(M) = W −1 Hom(M, M)). Using Grothendieck's formalism of extensions panachées we prove a classification result for such motives. Specializing to the category of mixed Tate motives we obtain a classification result for 3-dimensional mixed Tate motives over ‫ޑ‬ with three weights and large unipotent radicals.

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Modular curves and Fermat's theorem

Abstract: Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Cited by 3 publications

References 10 publications

Arakelov motivic cohomology I

Arakelov motivic cohomology I

Polylogarithms, regulators, and Arakelov motivic complexes

On unipotent radicals of motivic Galois groups

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