Harmonic analysis and the Riemann-Roch theorem

Abstract

This paper is a continuation of papers: arXiv:0707.1766 [math.AG] and arXiv:0912.1577 [math.AG]. Using the two-dimensional Poisson formulas from these papers and two-dimensional adelic theory we obtain the Riemann-Roch formula on a projective smooth algebraic surface over a finite field.Comment: 7 pages; to appear in Doklady Mathematic

On the stability property in positional differential games

“…Examples of con flict control systems (1) for which such families can be constructed were given in [11]. For we define (6) which is called the stability defect of the path W* at the time . The function on , which characterizes the instability of W*, is referred to as the instability spectrum of W*.…”

“…Vari ous formulations of this problem, its generalizations, and approaches to its solution can be found in [2][3][4][5][6][7]. In this paper, some basic constructions are presented concerning an extension of stability, which is a central concept in the theory of positional differential games.…”

Geometric Adeles and the Riemann—Roch Theorem for 1-Cycles on Surfaces

“…The paper [P1] posed the problem to find an adelic proof of the Riemann-Roch theorem for divisors on projective smooth irreducible surfaces. A sketch of line of thought to solve this problem was recently announced in [OP2] without proofs of several key statements. In addition to the older foundational papers [P1], [B1], [P2], the incomplete argument in [OP2] essentially relies on lengthy texts [OP1].…”

“…See [P1], [P2], [Y], [HY1], [HY2], [F4, §27- §29], [M1], [M2] for more detail about basic properties of d ω and its generalizations. There is an omission in [P1] and [OP2]: the trace maps maps Tr x,z do not appear there in the definition of the map d ω at singular points x ∈ y; however, it is straightforward to extend the arguments on [P1] to this general situation.…”

Пересечения Адельных Групп На Поверхностях

“…Позже это использовалось в работе [3] при доказательстве теоремы Римана-Роха для поверхностей адель-ными методами. В нашей статье дается положительный ответ на поставленный выше вопрос в случае, если основное поле несчетно (теорема 1), и приводится Работа выполнена при поддержке Российского фонда фундаментальных исследований (гранты № 11-01-00145 и № 13-01-12420 офи_м2), Программы Президента РФ поддержки ведущих научных школ (грант № НШ-5139.2012.1), Лаборатории алгебраической геометрии НИУ ВШЭ, Правительством РФ (договор 11.G34.31.0023).…”