Two cases of Stark's conjecture

1985 ◽  
Vol 272 (3) ◽  
pp. 349-359 ◽  
Author(s):  
Jonathan W. Sands
Keyword(s):  
Stark's Conjecture

ON STARK’S CLASS NUMBER CONJECTURE AND THE GENERALISED BRAUER–SIEGEL CONJECTURE

2022 ◽  
pp. 1-13
Author(s):  
PENG-JIE WONG
Keyword(s):  
Class Number ◽  
Number Fields ◽  
Galois Closure ◽  
Stark's Conjecture

Abstract Stark conjectured that for any $h\in \Bbb {N}$ , there are only finitely many CM-fields with class number h. Let $\mathcal {C}$ be the class of number fields L for which L has an almost normal subfield K such that $L/K$ has solvable Galois closure. We prove Stark’s conjecture for $L\in \mathcal {C}$ of degree greater than or equal to 6. Moreover, we show that the generalised Brauer–Siegel conjecture is true for asymptotically good towers of number fields $L\in \mathcal {C}$ and asymptotically bad families of $L\in \mathcal {C}$ .

scholarly journals Stark's conjecture over complex cubic number fields

2003 ◽  
Vol 73 (247) ◽  
pp. 1525-1547 ◽  
Author(s):  
David S. Dummit ◽  
Brett A. Tangedal ◽  
Paul B. van Wamelen
Keyword(s):  
Number Fields ◽  
Stark's Conjecture ◽  
Cubic Number Fields

scholarly journals Stark's conjecture in multi-quadratic extensions, revisited

2003 ◽  
Vol 15 (1) ◽  
pp. 83-97 ◽  
Author(s):  
David S. Dummit ◽  
Jonathan W. Sands ◽  
Brett Tangedal
Keyword(s):  
Quadratic Extensions ◽  
Stark's Conjecture

Stark's Conjecture and New Stickelberger Phenomena

2006 ◽  
Vol 58 (2) ◽  
pp. 419-448 ◽  
Author(s):  
Victor P. Snaith
Keyword(s):  
Number Fields ◽  
Supporting Evidence ◽  
Etale Cohomology ◽  
Galois Action ◽  
Rings Of Integers ◽  
Stark's Conjecture ◽  
Higher Dimensional ◽  
Cyclotomic Extensions ◽  
Étale Cohomology ◽  
Totally Real

AbstractWe introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraicK–groups of rings of integers in number fields. Our conjecture ismotivic in the sense that it involves the (transcendental) Borel regulator as well as being related tol–adic étale cohomology. In addition, the conjecture generalises the wellknown Coates–Sinnott conjecture. For example, for a totally real extension whenr= –2,–4,–6, … the Coates–Sinnott conjecturemerely predicts that zero annihilatesK–2rof the ring ofS–integers while our conjecture predicts a non-trivial annihilator. By way of supporting evidence, we prove the corresponding (conjecturally equivalent) conjecture for the Galois action on the étale cohomology of the cyclotomic extensions of the rationals.

Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

2016 ◽  
Vol 152 (6) ◽  
pp. 1159-1197
Author(s):  
Yingkun Li
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Hilbert Modular Forms ◽  
Algebraic Numbers ◽  
Modular Functions ◽  
Maass Forms ◽  
Stark's Conjecture ◽  
Hilbert Modular ◽  
Harmonic Maass Forms ◽  
The Individual

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

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