A note on ray class fields of global fields

The notion of a ray class field, which is fundamental in Takagi’s class field theory, has no immediate analogon in the function field case. The reason for this lies in the lacking of a distinguished maximal order. In this paper I overcome this difficulty by a relative version of the notion of ray class fields to be defined for every holomorphy ring of the field. The prototype for this new notion is M. Rosen’s definition of a Hilbert class field for function fields [6].

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Capitulation in Class Field Extensions of Type (p, p)

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-091-9 ◽

1980 ◽

Vol 32 (5) ◽

pp. 1229-1243 ◽

Cited By ~ 4

Author(s):

S. M. Chang ◽

R. Foote

Keyword(s):

Field Theory ◽

Abelian Group ◽

Number Field ◽

Class Field Theory ◽

Class Groups ◽

Class Field ◽

Field Extensions ◽

Class Fields ◽

Image Position ◽

Hilbert Class Field

Let K be a number field, K(1) its Hilbert class field, that is, the maximal abelian unramified extension of K, let K(2) be the Hilbert class field of K(1), and let G = Gal(K(2)/K) (alternatively, for p a prime the first and second p class fields enjoy properties analogous to those of the respective class fields discussed in this introduction; the particulars may be found surrounding Lemma 2). Since G/G’ is the largest abelian quotient of G, G/G′ = Gal (K(1))/K) and so G’ is the abelian group Gal(K(2)K(1)); moreover, class field theory provides (Artin) maps φK, φK(1) which are isomorphisms of the class groups Ck, Ck(1) onto G/G′, G′ respectively. In the remarkable paper [1] E. Artin computed the composition VG′where e is the homomorphism induced on the class groups by extending ideals of K to ideals of K(1), and he gave a formula for computing VG′, the now familiar transfer (Verlagerung) homomorphism, in terms of the group G alone (see Lemma 1).

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p-units in ray class fields of real quadratic number fields

Compositio Mathematica ◽

10.1112/s0010437x08003886 ◽

2009 ◽

Vol 145 (2) ◽

pp. 364-392

Author(s):

Hugo Chapdelaine

Keyword(s):

Field Theory ◽

Number Fields ◽

Natural Generalization ◽

Quadratic Number ◽

Quadratic Number Field ◽

Quadratic Number Fields ◽

Class Fields ◽

Ray Class Fields ◽

Real Quadratic ◽

Narrow Ring

AbstractLet K be a real quadratic number field and let p be a prime number which is inert in K. We denote the completion of K at the place p by Kp. We propose a p-adic construction of special elements in Kp× and formulate the conjecture that they should be p-units lying in narrow ray class fields of K. The truth of this conjecture would entail an explicit class field theory for real quadratic number fields. This construction can be viewed as a natural generalization of a construction obtained by Darmon and Dasgupta who proposed a p-adic construction of p-units lying in narrow ring class fields of K.

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GENUS FIELDS OF CYCLIC l-EXTENSIONS OF RATIONAL FUNCTION FIELDS

International Journal of Number Theory ◽

10.1142/s1793042113500243 ◽

2013 ◽

Vol 09 (05) ◽

pp. 1249-1262 ◽

Cited By ~ 2

Author(s):

VÍCTOR BAUTISTA-ANCONA ◽

MARTHA RZEDOWSKI-CALDERÓN ◽

GABRIEL VILLA-SALVADOR

Keyword(s):

Prime Number ◽

Rational Function ◽

Function Field ◽

Function Fields ◽

Cyclic Extension ◽

Class Field ◽

Dirichlet Characters ◽

Genus Field ◽

Hilbert Class Field

We give a construction of genus fields for Kummer cyclic l-extensions of rational congruence function fields, l a prime number. First we find this genus field for a field contained in a cyclotomic function field using Leopoldt's construction by means of Dirichlet characters and the Hilbert class field defined by Rosen. The general case follows from this. This generalizes the result obtained by Peng for a cyclic extension of degree l.

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The p-adic Kummer–Leopoldt constant: Normalized p-adic regulator

International Journal of Number Theory ◽

10.1142/s1793042118500203 ◽

2018 ◽

Vol 14 (02) ◽

pp. 329-337 ◽

Cited By ~ 3

Author(s):

Georges Gras

Keyword(s):

Field Theory ◽

Number Field ◽

Torsion Group ◽

Class Field Theory ◽

Cyclotomic Fields ◽

Class Field ◽

Elementary Formula ◽

Ramification Theory ◽

Definition Of ◽

Theory Interpretation

The [Formula: see text]-adic Kummer–Leopoldt constant [Formula: see text] of a number field [Formula: see text] is (assuming the Leopoldt conjecture) the least integer [Formula: see text] such that for all [Formula: see text], any global unit of [Formula: see text], which is locally a [Formula: see text]th power at the [Formula: see text]-places, is necessarily the [Formula: see text]th power of a global unit of [Formula: see text]. This constant has been computed by Assim and Nguyen Quang Do using Iwasawa’s techniques, after intricate studies and calculations by many authors. We give an elementary [Formula: see text]-adic proof and an improvement of these results, then a class field theory interpretation of [Formula: see text]. We give some applications (including generalizations of Kummer’s lemma on regular [Formula: see text]th cyclotomic fields) and a natural definition of the normalized [Formula: see text]-adic regulator for any [Formula: see text] and any [Formula: see text]. This is done without analytical computations, using only class field theory and especially the properties of the so-called [Formula: see text]-torsion group [Formula: see text] of Abelian [Formula: see text]-ramification theory over [Formula: see text].

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Ramanujan and the Modular j-Invariant

Canadian Mathematical Bulletin ◽

10.4153/cmb-1999-050-1 ◽

1999 ◽

Vol 42 (4) ◽

pp. 427-440 ◽

Cited By ~ 20

Author(s):

Bruce C. Berndt ◽

Heng Huat Chan

Keyword(s):

Infinite Product ◽

Elliptic Functions ◽

Class Field ◽

New Class ◽

Class Invariant ◽

Class Invariants ◽

Definition Of ◽

Hilbert Class Field

AbstractA new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about tn by establishing new connections between themodular jinvariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn generates the Hilbert class field of . This shows that tn is a new class invariant according to H. Weber’s definition of class invariants.

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Smooth functorial field theories from B-fields and D-branes

Journal of Homotopy and Related Structures ◽

10.1007/s40062-020-00272-2 ◽

2021 ◽

Vol 16 (1) ◽

pp. 75-153

Author(s):

Severin Bunk ◽

Konrad Waldorf

Keyword(s):

Field Theory ◽

Sigma Models ◽

Lagrangian Approach ◽

Field Theories ◽

Open Strings ◽

Homotopy Invariant ◽

Target Manifold ◽

Definition Of ◽

Smooth Map

AbstractIn the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms naturally fit into a 2-dimensional, smooth open-closed functorial field theory (FFT) in the sense of Atiyah, Segal, and Stolz–Teichner. We give a detailed construction of this smooth FFT, based on the definition of a suitable smooth bordism category. In this bordism category, all manifolds are equipped with a smooth map to a spacetime target manifold. Further, the object manifolds are allowed to have boundaries; these are the endpoints of open strings stretched between D-branes. The values of our FFT are obtained from the B-field and its D-branes via transgression. Our construction generalises work of Bunke–Turner–Willerton to include open strings. At the same time, it generalises work of Moore–Segal about open-closed TQFTs to include target spaces. We provide a number of further features of our FFT: we show that it depends functorially on the B-field and the D-branes, we show that it is thin homotopy invariant, and we show that it comes equipped with a positive reflection structure in the sense of Freed–Hopkins. Finally, we describe how our construction is related to the classification of open-closed TQFTs obtained by Lauda–Pfeiffer.

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Classical black hole scattering from a worldline quantum field theory

Journal of High Energy Physics ◽

10.1007/jhep02(2021)048 ◽

2021 ◽

Vol 2021 (2) ◽

Cited By ~ 3

Author(s):

Gustav Mogull ◽

Jan Plefka ◽

Jan Steinhoff

Keyword(s):

Field Theory ◽

Quantum Field Theory ◽

Black Hole ◽

Quantum Field ◽

Expectation Values ◽

Path Integral Representation ◽

S Matrix ◽

Feynman Rules ◽

Definition Of ◽

Three Body

Abstract A precise link is derived between scalar-graviton S-matrix elements and expectation values of operators in a worldline quantum field theory (WQFT), both used to describe classical scattering of black holes. The link is formally provided by a worldline path integral representation of the graviton-dressed scalar propagator, which may be inserted into a traditional definition of the S-matrix in terms of time-ordered correlators. To calculate expectation values in the WQFT a new set of Feynman rules is introduced which treats the gravitational field hμν(x) and position $$ {x}_i^{\mu}\left({\tau}_i\right) $$ x i μ τ i of each black hole on equal footing. Using these both the 3PM three-body gravitational radiation 〈hμv(k)〉 and 2PM two-body deflection $$ \Delta {p}_i^{\mu } $$ Δ p i μ from classical black hole scattering events are obtained. The latter can also be obtained from the eikonal phase of a 2 → 2 scalar S-matrix, which we show corresponds to the free energy of the WQFT.

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