Néron Models and Formal Groups

Фалтингс, Герд

doi:10.1007/s00032-008-0086-z

Cited by 3 publications

(2 citation statements)

References 8 publications

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“…The purpose of this paper is to show that the theory of generalized entropies can be mathematically interpreted, and widely extended, by means of an approach based on formal group theory [12]. Since the seminal paper by Bochner [11], formal groups have been intensively investigated in the last decades, because they play a prominent role in several branches of mathematics, especially algebraic topology [13]- [16], combinatorics [17], the theory of elliptic curves [18], arithmetic and analytic number theory [19]- [23].…”

Section: Introductionmentioning

confidence: 99%

Formal Groups and $Z$-Entropies

Tempesta

2015

Preprint

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We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the Z-entropies. Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi.A crucial aspect is that every Z-entropy is composable [1]. This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required, to describe the composition process in terms of a group law.The composability axiom, introduced as a generalization of the fourth Shannon-Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the Z-entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts.From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.

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Section: Introductionmentioning

confidence: 99%

Formal Groups and $Z$-Entropies

Tempesta

2015

Preprint

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“…The theory of formal groups [9], [16] has been intensively investigated in the last decades, due to its relevance in many branches of mathematics, especially algebraic topology [10], [7], [25], [15], the theory of elliptic curves [29], and arithmetic number theory [1], [3], [31], [33].…”

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confidence: 99%

The Lazard formal group, universal congruences and special values of zeta functions

Tempesta¹

2015

Trans. Amer. Math. Soc.

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A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist-Meurman-type congruences for the universal Bernoulli polynomials that are related with the Lazard universal formal group [31]- [33]. Their role in the theory of L-genera for multiplicative sequences is illustrated. As an application, sequences of integer numbers are constructed. New congruences are also obtained, useful to compute special values of a new class of Riemann-Hurwitz-type zeta functions.

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Formal groups and Z -entropies

Tempesta

2016

Proc. R. Soc. A.

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We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the . Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi. A crucial aspect is that every-entropy is (Tempesta 2016, 180-197. (doi:10.1016/j.aop.2015.08.013)). This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon-Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the -entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.

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Néron Models and Formal Groups

Abstract: Abstract. We show that formal groups can be used to simplify the construction of Néron models. Also we give a new proof of the stable reduction theorem for abelian varieties.

Cited by 3 publications

References 8 publications

Formal Groups and $Z$-Entropies

Formal Groups and $Z$-Entropies

The Lazard formal group, universal congruences and special values of zeta functions

Formal groups and Z -entropies

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