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

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 $L\in \mathcal {C}$...

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Bibliographic Details
Published inBulletin of the Australian Mathematical Society Vol. 106; no. 2; pp. 288 - 300
Main Author WONG, PENG-JIE
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.10.2022
Subjects
Asymptotic properties
Hypotheses
Number theory
11R29
Brauer–Siegel conjecture
asymptotically exact families
11R42
Stark’s conjecture
class numbers
11R47
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ISSN0004-9727
1755-1633
DOI10.1017/S0004972721001076

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Summary: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 $L\in \mathcal {C}$ .
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ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972721001076