On number fields with power-free discriminant

König, Joachim

doi:10.1007/s11856-020-1962-7

Cited by 2 publications

(3 citation statements)

References 18 publications

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“…Theorem 5.4, together with the obvious fact that every A n -discriminant is a square, then yields that Conjecture 5.5 is true for G = A n . The same holds for G = S n , with a similar, but easier construction (see, e.g., Lemma 5.1 in [11]).…”

Section: Strengthenings Of Theorem 21mentioning

confidence: 84%

“…This shows that "most" of the specializations E t0 /Q above have group G. The fact that such a set (i.e., positive density inside Z, up to fractional linear transformation) of specialization values t 0 yields asymptotically at least B α distinct extensions E t0 /Q with discriminant of absolute value ≤ B with some constant α > 0 has been used several times, e.g. in [4, Theorem 1.1] or [11,Theorem 4.2]. The constant α here depends only on the branch point number and ramification indices of E/Q(T ), and in particular not on p.…”

mentioning

confidence: 99%

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On the mod-$p$ distribution of discriminants of $G$-extensions

König¹

2019

Preprint

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This paper was motivated by a recent paper by Krumm and Pollack ([16]) investigating modulo-p behaviour of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the Malle conjecture: our results show in a precise sense the non-existence of "local obstructions" to such heuristics, in many cases essentially only under the assumption that G occurs as the Galois group of a Galois cover defined over Q. This complements and generalizes a similar result in the direction of the Malle conjecture by Dèbes ([4]).

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Section: Strengthenings Of Theorem 21mentioning

confidence: 84%

mentioning

confidence: 99%

On the mod-$p$ distribution of discriminants of $G$-extensions

König¹

2019

Preprint

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“…Infinitely many tamely ramified extensions L/Q with alternating Galois group A n for n ≥ 3 with all inertia groups generated by 3-cycles are realized in [17] using Mestre's construction [21] and building on the specialization method in Section 4.3.…”

Section: A Reduction and First Examplesmentioning

confidence: 99%

Unramified extensions over low degree number fields

König

Neftin

Sonn

2020

Journal of Number Theory

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For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply that, for every finite group G, there exists a quadratic number field admitting an unramified G-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when Q is replaced with a function field k(t) where k is an ample field.On the other hand, It is well known that every finite group G appears as a Galois group of an unramified extension over some number field. This is obtained by realizing G as a Galois group of a tame (tamely ramified) extension L 0 /K over some number field, and finding a (not necessarily Galois) number field M which is disjoint from L 0 and satisfies: "for every prime P of M, the ramification index of P over its restriction p to K is divisible by the ramification index of p in L 0 ". Abhyankar's lemma then implies that L := L 0 M is an unramified extension of M with Gal(L/M) ∼ = G. Moreover, the resulting extension L/M is tamely defined

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On number fields with power-free discriminant

Cited by 2 publications

References 18 publications

On the mod-$p$ distribution of discriminants of $G$-extensions

On the mod-$p$ distribution of discriminants of $G$-extensions

Unramified extensions over low degree number fields

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