Harmonic analysis and the Riemann-Roch theorem

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

Harmonic analysis on local fields and adelic spaces I

We develop a harmonic analysis on objects of some category $C_2$ of infinite-dimensional filtered vector spaces over a finite field. It includes two-dimensional local fields and adelic spaces of algebraic surfaces defined over a finite field. The main result is the theory of the Fourier transform on these objects and two-dimensional Poisson formulas.Comment: 69 pages; corrected typos and inserted some changes into the last sectio

Quantum Oscillations of the Critical Current of Asymmetric Aluminum Loops in Magnetic Field

The periodical dependencies in magnetic field of the asymmetry of the current-voltage curves of asymmetric aluminum loop are investigated experimentally at different temperatures below the transition into the superconducting state T < Tc. The obtained periodical dependencies of the critical current on magnetic field allow to explain the quantum oscillations of the dc voltage as consequence of the rectification of the external ac current and to calculate the persistent current at different values of magnetic flux inside the loop and temperatures.Comment: 2 pages, 2 figures, will be published in the Proceedings of the 24th International Conference on Low Temperature Physic