Arakelov theory

Arakelov theory   (Redirected from Arithmetic Riemann–Roch theorem) Jump to navigation Jump to search In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.

Contents 1 Background 1.1 Original definition of divisors 2 Results 3 Arithmetic Chow groups 4 The arithmetic Riemann–Roch theorem 5 See also 6 Notes 7 References 8 External links Background The main motivation behind Arakelov geometry is the fact there is a correspondence between prime ideals {displaystyle {mathfrak {p}}in {text{Spec}}(mathbb {Z} )} and finite places {displaystyle v_{p}:mathbb {Q} ^{*}to mathbb {R} } , but there also exists a place at infinity {displaystyle v_{infty }} , given by the Archimedean valuation, which doesn't have a corresponding prime ideal. Arakelov geometry gives a technique for compactifying {displaystyle {text{Spec}}(mathbb {Z} )} into a complete space {displaystyle {overline {{text{Spec}}(mathbb {Z} )}}} which has a prime lying at infinity. Arakelov's original construction studies one such theory, where a definition of divisors is constructor for a scheme {displaystyle {mathfrak {X}}} of relative dimension 1 over {displaystyle {text{Spec}}({mathcal {O}}_{K})} such that it extends to a Riemann surface {displaystyle X_{infty }={mathfrak {X}}(mathbb {C} )} for every valuation at infinity. In addition, he equips these Riemann surfaces with Hermitian metrics on holomorphic vector bundles over X(C), the complex points of {displaystyle X} . This extra Hermitian structure is applied as a substitute for the failure of the scheme Spec(Z) to be a complete variety.

Note that other techniques exist for constructing a complete space extending {displaystyle {text{Spec}}(mathbb {Z} )} , which is the basis of F1 geometry.

Original definition of divisors Let {displaystyle K} be a field, {displaystyle {mathcal {O}}_{K}} its ring of integers, and {displaystyle X} a genus {displaystyle g} curve over {displaystyle K} with a non-singular model {displaystyle {mathfrak {X}}to {text{Spec}}({mathcal {O}}_{K})} , called an arithmetic surface. Also, we let {displaystyle infty :Kto mathbb {C} } be an inclusion of fields (which is supposed to represent a place at infinity). Also, we will let {displaystyle X_{infty }} be the associated Riemann surface from the base change to {displaystyle mathbb {C} } . Using this data, we can define a c-divisor as a formal linear combination {displaystyle D=sum _{i}k_{i}C_{i}+sum _{infty }lambda _{infty }X_{infty }} where {displaystyle C_{i}} is an irreducible closed subset of {displaystyle {mathfrak {X}}} of codimension 1, {displaystyle k_{i}in mathbb {Z} } , and {displaystyle lambda _{infty }in mathbb {R} } , and the sum {displaystyle sum _{infty }} represents the sum over every real embedding of {displaystyle Kto mathbb {C} } and over one embedding for each pair of complex embeddings {displaystyle Kto mathbb {C} } . The set of c-divisors forms a group {displaystyle {text{Div}}_{c}({mathfrak {X}})} .

Results Arakelov (1974, 1975) defined an intersection theory on the arithmetic surfaces attached to smooth projective curves over number fields, with the aim of proving certain results, known in the case of function fields, in the case of number fields. Gerd Faltings (1984) extended Arakelov's work by establishing results such as a Riemann-Roch theorem, a Noether formula, a Hodge index theorem and the nonnegativity of the self-intersection of the dualizing sheaf in this context.

Arakelov theory was used by Paul Vojta (1991) to give a new proof of the Mordell conjecture, and by Gerd Faltings (1991) in his proof of Serge Lang's generalization of the Mordell conjecture.

Pierre Deligne (1987) developed a more general framework to define the intersection pairing defined on an arithmetic surface over the spectrum of a ring of integers by Arakelov.

Arakelov's theory was generalized by Henri Gillet and Christophe Soulé to higher dimensions. That is, Gillet and Soulé defined an intersection pairing on an arithmetic variety. One of the main results of Gillet and Soulé is the arithmetic Riemann–Roch theorem of Gillet & Soulé (1992), an extension of the Grothendieck–Riemann–Roch theorem to arithmetic varieties. For this one defines arithmetic Chow groups CHp(X) of an arithmetic variety X, and defines Chern classes for Hermitian vector bundles over X taking values in the arithmetic Chow groups. The arithmetic Riemann–Roch theorem then describes how the Chern class behaves under pushforward of vector bundles under a proper map of arithmetic varieties. A complete proof of this theorem was only published recently by Gillet, Rössler and Soulé.

Arakelov's intersection theory for arithmetic surfaces was developed further by Jean-Benoît Bost (1999). The theory of Bost is based on the use of Green functions which, up to logarithmic singularities, belong to the Sobolev space {displaystyle L_{1}^{2}} . In this context, Bost obtains an arithmetic Hodge index theorem and uses this to obtain Lefschetz theorems for arithmetic surfaces.

Arithmetic Chow groups An arithmetic cycle of codimension p is a pair (Z, g) where Z ∈ Zp(X) is a p-cycle on X and g is a Green current for Z, a higher-dimensional generalization of a Green function. The arithmetic Chow group {displaystyle {widehat {mathrm {CH} }}_{p}(X)} of codimension p is the quotient of this group by the subgroup generated by certain "trivial" cycles.[1] The arithmetic Riemann–Roch theorem The usual Grothendieck–Riemann–Roch theorem describes how the Chern character ch behaves under pushforward of sheaves, and states that ch(f*(E))= f*(ch(E)TdX/Y), where f is a proper morphism from X to Y and E is a vector bundle over f. The arithmetic Riemann–Roch theorem is similar, except that the Todd class gets multiplied by a certain power series. The arithmetic Riemann–Roch theorem states {displaystyle {hat {mathrm {ch} }}(f_{*}([E]))=f_{*}({hat {mathrm {ch} }}(E){widehat {mathrm {Td} }}^{R}(T_{X/Y}))} where X and Y are regular projective arithmetic schemes. f is a smooth proper map from X to Y E is an arithmetic vector bundle over X. {displaystyle {hat {mathrm {ch} }}} is the arithmetic Chern character. TX/Y is the relative tangent bundle {displaystyle {hat {mathrm {Td} }}} is the arithmetic Todd class {displaystyle {hat {mathrm {Td} }}^{R}(E)} is {displaystyle {hat {mathrm {Td} }}(E)(1-epsilon (R(E)))} R(X) is the additive characteristic class associated to the formal power series {displaystyle sum _{m>0 atop m{text{ odd}}}{frac {X^{m}}{m!}}left[2zeta '(-m)+zeta (-m)left({1 over 1}+{1 over 2}+cdots +{1 over m}right)right].} See also Hodge–Arakelov theory Hodge theory P-adic Hodge theory Adelic group Notes ^ Manin & Panchishkin (2008) pp.400–401 References Arakelov, Suren J. (1974), "Intersection theory of divisors on an arithmetic surface", Math. USSR Izv., 8 (6): 1167–1180, doi:10.1070/IM1974v008n06ABEH002141, Zbl 0355.14002 Arakelov, Suren J. (1975), "Theory of intersections on an arithmetic surface", Proc. Internat. Congr. Mathematicians Vancouver, vol. 1, Amer. Math. Soc., pp. 405–408, Zbl 0351.14003 Bost, Jean-Benoît (1999), "Potential theory and Lefschetz theorems for arithmetic surfaces" (PDF), Annales Scientifiques de l'École Normale Supérieure, Série 4, 32 (2): 241–312, doi:10.1016/s0012-9593(99)80015-9, ISSN 0012-9593, Zbl 0931.14014 Deligne, P. (1987), "Le déterminant de la cohomologie", Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) [The determinant of the cohomology], Contemporary Mathematics, vol. 67, Providence, RI: American Mathematical Society, pp. 93–177, doi:10.1090/conm/067/902592, MR 0902592 Faltings, Gerd (1984), "Calculus on Arithmetic Surfaces", Annals of Mathematics, Second Series, 119 (2): 387–424, doi:10.2307/2007043, JSTOR 2007043 Faltings, Gerd (1991), "Diophantine Approximation on Abelian Varieties", Annals of Mathematics, Second Series, 133 (3): 549–576, doi:10.2307/2944319, JSTOR 2944319 Faltings, Gerd (1992), Lectures on the arithmetic Riemann–Roch theorem, Annals of Mathematics Studies, vol. 127, Princeton, NJ: Princeton University Press, doi:10.1515/9781400882472, ISBN 0-691-08771-7, MR 1158661 Gillet, Henri; Soulé, Christophe (1992), "An arithmetic Riemann–Roch Theorem", Inventiones Mathematicae, 110: 473–543, doi:10.1007/BF01231343 Kawaguchi, Shu; Moriwaki, Atsushi; Yamaki, Kazuhiko (2002), "Introduction to Arakelov geometry", Algebraic geometry in East Asia (Kyoto, 2001), River Edge, NJ: World Sci. Publ., pp. 1–74, doi:10.1142/9789812705105_0001, ISBN 978-981-238-265-8, MR 2030448 Lang, Serge (1988), Introduction to Arakelov theory, New York: Springer-Verlag, doi:10.1007/978-1-4612-1031-3, ISBN 0-387-96793-1, MR 0969124, Zbl 0667.14001 Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002. Soulé, Christophe (2001) [1994], "Arakelov theory", Encyclopedia of Mathematics, EMS Press Soulé, C.; with the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer (1992), Lectures on Arakelov geometry, Cambridge Studies in Advanced Mathematics, vol. 33, Cambridge: Cambridge University Press, pp. viii+177, doi:10.1017/CBO9780511623950, ISBN 0-521-41669-8, MR 1208731 Vojta, Paul (1991), "Siegel's Theorem in the Compact Case", Annals of Mathematics, Annals of Mathematics, Vol. 133, No. 3, 133 (3): 509–548, doi:10.2307/2944318, JSTOR 2944318 External links Original paper Arakelov geometry preprint archive Categories: Algebraic geometryDiophantine geometry

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