Proceedings of the Japan Academy, Series A, Mathematical Sciences

On indivisibility of relative class numbers of totally imaginary quadratic extensions and these relative Iwasawa invariants

Yuuki Takai

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Abstract

In this paper, we announce some results on indivisibility of relative class numbers of CM quadratic extensions $K/F$ of a fixed totally real number field $F$ which is Galois over $\mathbf{Q}$ and on vanishing of these relative Iwasawa $\lambda_{p}$-, $\mu_{p}$-invariants. In particular, we give a lower bound of the number of such CM extensions $K/F$ with bounded (norm of) relative discriminants. To prove them, we use Hilbert modular forms of half-integral weight.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 90, Number 2 (2014), 33-36.

Dates
First available in Project Euclid: 30 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.pja/1391091380

Digital Object Identifier
doi:10.3792/pjaa.90.33

Mathematical Reviews number (MathSciNet)
MR3161543

Zentralblatt MATH identifier
1286.11067

Subjects
Primary: 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] 11R29: Class numbers, class groups, discriminants
Secondary: 11F37: Forms of half-integer weight; nonholomorphic modular forms 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 11R23: Iwasawa theory

Keywords
Relative class numbers relative Iwasawa invariants Hilbert modular forms of half-integral weight Sturm’s theorem

Citation

Takai, Yuuki. On indivisibility of relative class numbers of totally imaginary quadratic extensions and these relative Iwasawa invariants. Proc. Japan Acad. Ser. A Math. Sci. 90 (2014), no. 2, 33--36. doi:10.3792/pjaa.90.33. https://projecteuclid.org/euclid.pja/1391091380


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References

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