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Harmonic analysis and the Riemann-Roch theorem. (English. Russian original) Zbl 1350.14015

Dokl. Math. 84, No. 3, 826-829 (2011); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 441, No. 4, 444-448 (2011).
Let \(X\) be a smooth projective surface over a finite field \(k\). In this note, the authors apply their two-dimensional Poisson formulas on certain adelic spaces related to \(X\) [Izv. Math. 72, No. 5, 915–976 (2008); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 72, No. 5, 77–140 (2008; Zbl 1222.11137); Izv. Math. 75, No. 4, 749–814 (2011); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 75, No. 4, 91–164 (2011; Zbl 1232.11122)] to deduce the Riemann-Roch theorem in the following form: for any divisor \(C\) and any nonzero rational 2-form \(\omega\) on \(X\), \[ h^0(C)-h^1(C)+h^0((\omega)-C)=h^0(0)-h^1(0)+h^0((\omega))-(C,(\omega)-C)/2. \] This argument extends the proof of the Riemann-Roch theorem for a smooth projective curve \(D\) over \(k\) based on the Poisson summation formula for the discrete subgroup \(k(D)\) of the adèle space \(\mathbb{A}_D\) [A. N. Parshin, in: Invitation to higher local fields. Extended version of talks given at the conference on higher local fields, Münster, Germany, August 29 – September 5, 1999. Coventry: Geometry and Topology Publications, 199–213 (2000; Zbl 1008.11060)].

MSC:

14C40 Riemann-Roch theorems
11R56 Adèle rings and groups
14G15 Finite ground fields in algebraic geometry
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
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References:

[1] D. V. Osipov, Mat. Sb. 196(10), 111–136 (2005); Sb. Math. 196 (10), 1503-1527 (2005).
[2] D. Osipov, Int. J. Math. 18(3), 269–279 (2007). · Zbl 1126.14004
[3] D. V. Osipov and A. N. Parshin, Izvestiya RAN, Ser. Mat. 72(5), pp. 77–140 2008; Izvestiya: Mathematics 72 (5), 915–976 (2008).
[4] D. V. Osipov and A. N. Parshin, Izvestiya RAN, Ser. Mat. 75(4), 91–164 (2011); Izvestiya: Mathematics 75 (4), 749–814 (2011).
[5] A. N. Parshin, Journal für die reine and angewandte Mathematik, Band 341, 174–192 (1983). · Zbl 0518.14013
[6] A. N. Parshin, ”Higher-Dimensional Local Fields and L-Functions,” in Invitation to Higher Local Fields (Münster, 1999), Geom. Topol. Monogr., 3 (Geom. Topol. Publ., Coventry, 2000), pp. 199-213. · Zbl 1008.11060
[7] J.-P. Serre, Groupes algebriques et corps de classes, Publications de l’institut de math-ematique de l’universite de Nancago (Hermann, Paris, 1959), Vol. VII.
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