Abstract versus classical algebraic geometry

Weil, André

doi:10.1007/978-1-4757-1705-1_71

Cited by 16 publications

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“…If k is infinite, then k-points are Zariski dense on G (3,5), and so there is a k-point on ~G(3, 5)S, and hence on hY. Following (11~ we may end the proof in the finite field case by refering to a general theorem of Weil [12] that a smooth projective rational surface defined over a finite field k always has a k-point (see also [7, 23.1]). However, a simple general argument is available, which I owe to J.-L. Colliot-Thelene:…”

Section: Introductionmentioning

confidence: 99%

On a theorem of Enriques - Swinnerton-Dyer

Skorobogatov¹

1993

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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Section: Introductionmentioning

confidence: 99%

On a theorem of Enriques - Swinnerton-Dyer

Skorobogatov¹

1993

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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“…Furthering this idea, A. Weil then proved, in [1946][1947][1948], that this same version of (RH) holds for algebraic curves of arbitrary genus and for abelian varieties in [40]. (See also [38], [39] and [41].) Below we present a sketch of some of the ideas contained in a modern proof of these results, which are based on Weil's ideas, and will motivate the work contained in this paper.…”

Section: The Weil Conjecturesmentioning

confidence: 99%

Towards a Fractal Cohomology: Spectra of Polya–Hilbert Operators, Regularized Determinants and Riemann Zeros

Cobler

Lapidus

2017

Exploring the Riemann Zeta Function

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Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil in the case of curves and eventually, by Deligne in the case of varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to the properties of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis has continued to this day. In this paper we study the properties of the derivative operator D = d dz on a particular family of weighted Bergman spaces of entire functions on C. The operator D can be naturally viewed as the 'infinitesimal shift of the complex plane' since it generates the group of translations of C. Furthermore, this operator is meant to be the replacement for the Frobenius operator in the general case and is used to construct an operator associated to any given meromorphic function. With this construction, we show that for a wide class of meromorphic functions, the function can be recovered by using a regularized determinant involving the operator constructed from the meromorphic function. This is illustrated in some important special cases: rational functions, zeta functions of algebraic curves (or, more generally, varieties) over finite fields, the Riemann zeta function, and cul- minating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. This shows that all of the information about the given meromorphic function is encoded into the special operator we constructed. Our construction is motivated in part by work of Herichi and the second author on the infinitesimal shift of the real line (instead of the complex plane) and the associated spectral operator, as well as by earlier work and conjectures of Deninger on the role of cohomology in analytic number theory, and a conjectural 'fractal cohomology theory' envisioned in work of the second author and of Lapidus and van Frankenhuijsen on complex fractal dimensions.

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“…Let N(S 3 ) = Pic (S 3 ⊗ F q ) and F * : N(S 3 ) → N(S 3 ) be the action of Frobenius on the Picard group. Weil proved that the following formula for the number of F q rational points on a cubic surface [Wei56];…”

Section: Degree 6 Maps For Cubic Surfacesmentioning

confidence: 99%

Degree of Unirationality for del Pezzo Surfaces over Finite Fields

Knecht¹

2015

J. Théor. Nombres Bordeaux

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Abstract. We address the question of the degree of unirational parameterizations of degree four and degree three del Pezzo surfaces. Specifically we show that degree four del Pezzo surfaces over finite fields admit degree two parameterizations and minimal cubic surfaces admit parameterizations of degree 6. It is an open question whether or not minimal cubic surfaces over finite fields can admit degree 3 or 4 parameterizations.

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Abstract versus classical algebraic geometry

Cited by 16 publications

References 0 publications

On a theorem of Enriques - Swinnerton-Dyer

On a theorem of Enriques - Swinnerton-Dyer

Towards a Fractal Cohomology: Spectra of Polya–Hilbert Operators, Regularized Determinants and Riemann Zeros

Degree of Unirationality for del Pezzo Surfaces over Finite Fields

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