Quantum Statistical Mechanics, L-Series and Anabelian Geometry I: Partition Functions

Cornelissen, Gunther; Marcolli, Matilde

doi:10.1007/978-3-319-05254-0_4

Cited by 22 publications

(39 citation statements)

References 30 publications

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“…Basic Galois theory (as in proposition 8.1 in ref. 4) then shows that it also maps the group of the maximal abelian extension of K unramified outside m to that of L ramified outside φðmÞ; i.e., the inertia group of v in K ab is mapped to that of of φðvÞ in L ab . If we combine this with the map π v ↦π φðvÞ , we find the desired map Φ v : K * v → L * φðvÞ that automatically respects the order functions.…”

Section: Proof Of Theoremmentioning

confidence: 98%

“…4, section 7.3, but some differences arise from the geometric situation. The rough idea is as follows: A priori Φ is only a homeomorphism (between Cantor sets).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In ref. 4, one finds the analogue of these theorems for number fields. The way to deduce Theorem 1 from Theorem 2 is identical to that in ref.…”

mentioning

confidence: 93%

“…One may then show, as in ref. 4, section 6, that degree preservation for this system [and ergodicity with regard to the natural measure induced from Haar measure, as established in ref . 7] implies full topological conjugacy.…”

mentioning

confidence: 93%

“…The way to deduce Theorem 1 from Theorem 2 is identical to that in ref. 4 and is sketched in the next section; the part about restricting to geometric characters is proved in the final section. As is explained in the middle two sections of this paper, the proof of Theorem 2 in the case of function fields will be somewhat different from that in number fields and will rely on Theorem 5 below, which might be of independent interest, and which we deduce from the proof of Uchida's anabelian theorem for global function fields in ref.…”

mentioning

confidence: 99%

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Curves, dynamical systems, and weighted point counting

Cornelissen

2013

Proc. Natl. Acad. Sci. U.S.A.

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Suppose X is a (smooth projective irreducible algebraic) curve over a finite field k. Counting the number of points on X over all finite field extensions of k will not determine the curve uniquely. Actually, a famous theorem of Tate implies that two such curves over k have the same zeta function (i.e., the same number of points over all extensions of k) if and only if their corresponding Jacobians are isogenous. We remedy this situation by showing that if, instead of just the zeta function, all Dirichlet L-series of the two curves are equal via an isomorphism of their Dirichlet character groups, then the curves are isomorphic up to "Frobenius twists", i.e., up to automorphisms of the ground field. Because L-series count points on a curve in a "weighted" way, we see that weighted point counting determines a curve. In a sense, the result solves the analogue of the isospectrality problem for curves over finite fields (also know as the "arithmetic equivalence problem"): It states that a curve is determined by "spectral" data, namely, eigenvalues of the Frobenius operator of k acting on the cohomology groups of all ℓ-adic sheaves corresponding to Dirichlet characters. The method of proof is to show that this is equivalent to the respective class field theories of the curves being isomorphic as dynamical systems, in a sense that we make precise.L et X denote a smooth projective curve over a finite field k = F q of characteristic p. Its zeta function encodes precisely the sequence of integers given by the number of points of X over finite extensions F q n of k. A celebrated theorem of Tate (ref. 1, theorem 1, compare ref.2) implies that an equality of the zeta functions of two curves over k (and so equality of their number of points over all extensions of k) is equivalent to the curves having isogenous Jacobians over the field k. Because there are plenty of nonisomorphic curves with isogenous Jacobians, this shows that the k-isomorphism type of a curve over a finite field k is not determined solely by point counting over field extensions. For a very explicit example, consider the curves of genus two given byIt is known that these curves are not (even geometrically) isomorphic, but have the same zeta function, and furthermore their Jacobian is absolutely simple (3).The aim of the current paper is to show that if this counting is extended from using only the zeta function to abelian L-series, then this does determine the curve uniquely, as follows: Theorem 1. Let X, Y denote two smooth projective curves over a finite field k. Assume that there is an isomorphism ψ : G ab kðXÞ → ∼ G ab kðY Þ of Galois groups of the maximal abelian extensions of their respective function fields, such that the corresponding L-series match,for all characters χ ∈ G ab∨ kðXÞ . Then X is isomorphic to a Frobenius twist Y σ of Y over k, where σ is an automorphism of the ground field k.For the conclusion to hold, it suffices that the L-series of geometric characters match (i.e., characters that factor through a finite geometric extension of the fu...

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Section: Proof Of Theoremmentioning

confidence: 98%

“…4, section 7.3, but some differences arise from the geometric situation. The rough idea is as follows: A priori Φ is only a homeomorphism (between Cantor sets).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In ref. 4, one finds the analogue of these theorems for number fields. The way to deduce Theorem 1 from Theorem 2 is identical to that in ref.…”

mentioning

confidence: 93%

mentioning

confidence: 93%

mentioning

confidence: 99%

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Curves, dynamical systems, and weighted point counting

Cornelissen

2013

Proc. Natl. Acad. Sci. U.S.A.

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Reconstructing global fields from dynamics in the abelianized Galois group

Cornelissen

Marcolli

et al. 2019

Sel. Math. New Ser.

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We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to field isomorphism, relating it to anabelian geometry.Date: June 15, 2017 (version 1.0). This paper supersedes [5], of which it is the second (and final) part, dealing with the dynamical systems aspects of the theory. The first part [6] dealt with physics aspects related to partition functions, and [8] is a companion paper, only containing number theoretical results. Part of this work was done whilst the first two authors enjoyed the hospitality of the University of Warwick (special thanks to Richard Sharp for making it possible).

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