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Mass formulas for local Galois representations (with an appendix by Daniel
 Gulotta), International Mathematics Research Notices
"... Bhargava has given a formula, derived from a formula of Serre, computing a certain count of extensions of a local field, weighted by conductor and by number of automorphisms. We interpret this result as a counting formula for permutation representations of the absolute Galois group of the local fiel ..."
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Bhargava has given a formula, derived from a formula of Serre, computing a certain count of extensions of a local field, weighted by conductor and by number of automorphisms. We interpret this result as a counting formula for permutation representations of the absolute Galois group of the local field, then speculate on variants of this formula in which the role of the symmetric group is played by other groups. We prove an analogue of Bhargava’s formula for representations into a Weyl group in the Bn series, which suggests a possible link with integration on padic groups. We also obtain positive results in some other cases, and some negative results in residual characteristic 2. 1
Counting discriminants of number fields
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX 18 (2006), 573–593
, 2006
"... For each transitive permutation group G on n letters with n ≤ 4, we give without proof results, conjectures, and numerical computations on discriminants of number fields L of degree n over Q such that the Galois group of the Galois closure of L is isomorphic to G. ..."
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For each transitive permutation group G on n letters with n ≤ 4, we give without proof results, conjectures, and numerical computations on discriminants of number fields L of degree n over Q such that the Galois group of the Galois closure of L is isomorphic to G.
Connected components of Hurwitz schemes and Malle’s conjecture
, 2008
"... Let Z(X) be the number of degreed extensions of Fq(t) with bounded discriminant and some specified Galois group. The problem of computing Z(X) can be related to a problem of counting Fqrational points on certain Hurwitz spaces. Ellenberg and Venkatesh used this idea to develop a heuristic for the ..."
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Let Z(X) be the number of degreed extensions of Fq(t) with bounded discriminant and some specified Galois group. The problem of computing Z(X) can be related to a problem of counting Fqrational points on certain Hurwitz spaces. Ellenberg and Venkatesh used this idea to develop a heuristic for the asymptotic behavior of Z ′ (X), the number ofgeometrically connected extensions, and showed that this agrees with the conjectures of Malle for function fields. We extend EllenbergVenkatesh’s argument to handle the more complicated case of covers of P 1 which may not be geometrically connected, and show that the resulting heuristic suggests a natural modification to Malle’s conjecture which avoids the counterexamples, due to Klüners, to the original conjecture.
ASYMPTOTICS OF NUMBER FIELDS AND THE COHEN–LENSTRA HEURISTICS
, 2005
"... Abstract. We study the asymptotics conjecture of Malle for dihedral groups Dℓ of order 2ℓ, where ℓ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those ..."
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Abstract. We study the asymptotics conjecture of Malle for dihedral groups Dℓ of order 2ℓ, where ℓ is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen–Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds. Dedicated to Michael Pohst on the occasion of his 60th birthday 1.
Counting cyclic quartic extensions of a number field
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
, 2005
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ON NUMBER FIELDS WITH NONTRIVIAL SUBFIELDS
"... Abstract. What is the probability for a number field of composite degree d to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability i ..."
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Abstract. What is the probability for a number field of composite degree d to have a nontrivial subfield? As the reader might expect the answer heavily depends on the interpretation of probability. We show that if the fields are enumerated by the smallest height of their generators the probability is zero, at least if d> 6. This is in contrast to what one expects when the fields are enumerated by the discriminant. The main result of this article is an estimate for the number of algebraic numbers of degree d = en and bounded height which generate a field that contains an unspecified subfield of degree e. If n> max{e 2 + e, 10} we get the correct asymptotics as the height tends to infinity. 1. Introduction and
appendix by Daniel Gulotta)
, 2007
"... Mass formulas for local Galois representations (with an ..."