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

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  10 January 2022

PENG-JIE WONG Open the ORCID record for PENG-JIE WONG [Opens in a new window]
PENG-JIE WONG*
Affiliation:
National Center for Theoretical Sciences, No. 1, Sec. 4, Roosevelt Rd., Taipei City, Taiwan
*
e-mail: pengjie.wong@ncts.tw
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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 $L\in \mathcal {C}$.

Keywords

Stark’s conjectureBrauer–Siegel conjectureclass numbersasymptotically exact families

MSC classification

Primary: 11R29: Class numbers, class groups, discriminants
Secondary: 11R42: Zeta functions and $L$-functions of number fields 11R47: Other analytic theory

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 2 , October 2022 , pp. 288 - 300
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author is currently an NCTS postdoctoral fellow.

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