Connected components of Hurwitz schemes and Malle's conjecture

Turkelli, Seyfi

doi:10.1016/j.jnt.2015.03.005

Cited by 21 publications

(21 citation statements)

References 16 publications

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“…The application to the 3-part of the class group of quadratic fields is a special case of our application. More generally, the principle also makes correct predictions with infinitely many local conditions for quadratic, S 3 cubic, S 4 quartic, S 5 quintic fields [Bha14], and abelian [FLN15] fields with local conditions at infinitely many primes such that for sufficiently large primes all unramified and minimally ramified local extensions are allowed (in the abelian case excepting local conditions that are never satisfied; see also [BST13,TT13] [Tür15] both suggest ways to correct Malle's conjecture, and in particular suggest that it should still hold if we do not count extensions with nontrivial subfields that are subfields of certain cyclotomic fields. First, we note that the only admissible subgroup of C S 2 is D ⊂ C S 2 (with C → C × C as a → (a, −a)), so have not above applied the Malle-Bhargava principle in the cases of known counter-examples.…”

Section: The Malle-bhargava Principle Motivation For Conjecture 11mentioning

confidence: 99%

Nonabelian Cohen–Lenstra moments

Wood¹,

Wood²

2019

Duke Math. J.

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In this paper we give a conjecture for the average number of unramified Gextensions of a quadratic field for any finite group G. The Cohen-Lenstra heuristics are the specialization of our conjecture to the case that G is abelian of odd order. We prove a theorem towards the function field analog of our conjecture, and give additional motivations for the conjecture including the construction of a lifting invariant for the unramified Gextensions that takes the same number of values as the predicted average and an argument using the Malle-Bhargava principle. We note that for even |G|, corrections for the roots of unity in Q are required, which can not be seen when G is abelian.

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Section: The Malle-bhargava Principle Motivation For Conjecture 11mentioning

confidence: 99%

Nonabelian Cohen–Lenstra moments

Wood¹,

Wood²

2019

Duke Math. J.

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“…Unfortunately, the conjecture is not true in this form, as Klüners [Klü05a] provided a counter-example for G " C3 ≀ C2 Ă S6 for which bpK, Gq is too small. There have been proposed corrections for bpK, Gq by Türkelli [Tür15], but the 1{apGq exponent is still widely believed to be correct. This leads to the weak form of Malle's conjecture, which will be the major focus of study of this paper:…”

Section: Introductionmentioning

confidence: 99%

The weak form of Malle’s conjecture and solvable groups

Alberts

2020

Res. number theory

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For a fixed finite solvable group G and number field K, we prove an upper bound for the number of G-extensions L{K with restricted local behavior (at infinitely many places) and invpL{Kq ă X for a general invariant "inv". When the invariant is given by the discriminant for a transitive embedding of a nilpotent group G Ă Sn, this realizes the upper bound given in the weak form of Malle's conjecture. For other solvable groups, the upper bound depends on the size of torsion of the class group of number fields with fixed degree. In particular, the bounds we prove realize the upper bound given in the weak form of Malle's conjecture for the transitive embedding of a solvable group G Ă Sn if we assume that for each finite abelian group A the average size of class group torsion |HompClpLq, Aq| is smaller than X ǫ as L{K varies over certain families of extensions with invpL{Kq ă X.

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“…Klüners [Klü05a] shows that the conjecture does not hold for C 3 C 2 number fields over Q in its S 6 representation, where Malle's conjecture predicts a smaller power for ln X in the main term. See [Klü05a] and [Tur08] for suggestions on how to fix the conjecture. And by relaxing the precise description of the power for ln X, the weak form of Malle's conjecture states that for arbitrary given small > 0, the distribution satisfies C 1 X 1/a(G) ≤ N k (G, X) ≤ C 2 ( )X 1/a(G)+ when X is large enough.…”

Section: J Wangmentioning

confidence: 99%

Malle's conjecture for for

Wang¹

2021

Compositio Math.

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We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method, we prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6, and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ representation is the first counter-example of Malle's conjecture given by Klüners. We also prove new local uniformity results for ramified $S_5$ quintic extensions over arbitrary number fields by adapting Bhargava's geometric sieve and averaging over fundamental domains of the parametrization space.

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Connected components of Hurwitz schemes and Malle's conjecture

Cited by 21 publications

References 16 publications

Nonabelian Cohen–Lenstra moments

Nonabelian Cohen–Lenstra moments

The weak form of Malle’s conjecture and solvable groups

Malle's conjecture for for

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