scholarly journals Integral moments of automorphic L-functions

2008 ◽  
Vol 8 (2) ◽  
pp. 335-382 ◽  
Author(s):  
Adrian Diaconu ◽  
Paul Garrett
Keyword(s):  
Representation Theory ◽  
Number Field ◽  
Spectral Decomposition ◽  
Arbitrary Number ◽  
Number Fields ◽  
Rational Numbers ◽  
Poincaré Series ◽  
Poincare Series ◽  
Global And Local ◽  
Adele Group

AbstractWe obtain second integral moments of automorphic L-functions on adele groups GL2 over arbitrary number fields, by a spectral decomposition using the structure and representation theory of adele groups GL1 and GL2. This requires reformulation of the notion of Poincaré series, replacing the collection of classical Poincaré series over GL2(ℚ) or GL2(ℚ(i)) with a single, coherent, global object that makes sense over a number field. This is the first expression of integral moments in adele-group terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers ℚ, we recover the classical results on moments.

The discriminant of compositum of algebraic number fields

2019 ◽  
Vol 15 (02) ◽  
pp. 353-360
Author(s):  
Sudesh K. Khanduja
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Rational Numbers ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Number Fields ◽  
Necessary And Sufficient

For an algebraic number field [Formula: see text], let [Formula: see text] denote the discriminant of an algebraic number field [Formula: see text]. It is well known that if [Formula: see text] are algebraic number fields with coprime discriminants, then [Formula: see text] are linearly disjoint over the field [Formula: see text] of rational numbers and [Formula: see text], [Formula: see text] being the degree of [Formula: see text] over [Formula: see text]. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields [Formula: see text] linearly disjoint over [Formula: see text].

scholarly journals Malle's conjecture for for

2021 ◽  
Vol 157 (1) ◽  
pp. 83-121
Author(s):  
Jiuya Wang
Keyword(s):  
Abelian Group ◽  
Number Field ◽  
Arbitrary Number ◽  
Number Fields ◽  
Counter Example ◽  
Fundamental Domains ◽  
Good Uniformity

We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

scholarly journals Spectral decomposition of compactly supported Poincaré series and existence of cusp forms

2010 ◽  
Vol 146 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Goran Muić
Keyword(s):  
Spectral Decomposition ◽  
Poincaré Series ◽  
Cusp Forms ◽  
Congruence Subgroups ◽  
Poincare Series ◽  
Compactly Supported ◽  
Classic Method

AbstractIn this paper we address the issue of existence of cusp forms by using an extension and refinement of a classic method involving (adelic) compactly supported Poincaré series. As a consequence of our adelic approach, we also deal with cusp forms for congruence subgroups.

scholarly journals Poincaré series, 3d gravity and averages of rational CFT

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Viraj Meruliya ◽  
Sunil Mukhi ◽  
Palash Singh
Keyword(s):  
Poincaré Series ◽  
Simons Theory ◽  
Partition Functions ◽  
Moody Algebra ◽  
Modular Invariant ◽  
Poincare Series ◽  
Single Genus ◽  
3D Gravity ◽  
Chern Simons ◽  
Chern Simons Theory

Abstract We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

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