Dihedral and cyclic extensions with large class numbers

Cho, Peter; Kim, Henry H.

doi:10.5802/jtnb.812

Cited by 4 publications

(3 citation statements)

References 19 publications

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“…We claim that (θ 1 + t) 5 /t and (θ 2 1 + 5t) 5 /t 2 are two independent units in K t . Since f (x, t) = x 5 + tx 4 + 15tx 3 + 15t 2 x 2 + 50t 2 x + 50t 3 + t, = −(θ 4 1 + 15θ 3 1 + 15tθ 2 1 + 50tθ 1 + 50t 2 + 1).…”

Section: Remark 52mentioning

confidence: 99%

“…Let p | (27t 4 + 1), p > 3. Then p t and f (x, t) ≡ (x + t)(x − 3t) 3 mod p. The vertices of the Newton polygon with respect to x − 3t are (0,0), (1,0), (4, i) with i = 1, 2. By Cohen [5, p. 315…”

Section: Proposition 81 the Regulator R Kt Satisfies R Ktmentioning

confidence: 99%

“…For example, Daileda [7] showed that for a pure cubic extension given by f (x, t) = x 3 − t, L (1, ρ) log log |d K |. In a separate paper [3], we study dihedral and cyclic extensions with large class numbers. In this case, the representation ρ is no longer attached to a cuspidal automorphic representation.…”

Section: Introductionmentioning

confidence: 99%

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Application of the Strong Artin Conjecture to the Class Number Problem

Cho¹,

Kim²

2013

Can. j. math.

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Abstract. We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group A 4 , S 4 and S 5 . We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying zero density result of Kowalski-Michel, we choose subfamilies of L-functions which are zero free close to 1. For these subfamilies, the L-functions have the extremal value at s = 1, and by the class number formula, we obtain the extreme class numbers.

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Section: Remark 52mentioning

confidence: 99%

Section: Proposition 81 the Regulator R Kt Satisfies R Ktmentioning

confidence: 99%