Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

Ellenberg, Jordan S.; Venkatesh, Akshay; Westerland, Craig

doi:10.4007/annals.2016.183.3.1

Cited by 118 publications

(219 citation statements)

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“…In contrast to Theorem A, the étale cohomology groups H ié t (Simp m n ; Q ℓ ) do not stabilize when charK > 0-a divergence from other comparable stability results (see e.g [EVW15], and Farb-Wolfson's work on configuration spaces see [FW15]). Indeed, the moduli space of polynomials f ∈ F p [x] of degree n that are unramified as self-maps of A 1 F p is nonempty if and only if n is a prime power.…”

contrasting

confidence: 80%

Cohomology of the space of polynomial maps on A1 with prescribed ramification

Banerjee¹

2020

Advances in Mathematics

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contrasting

confidence: 80%

Cohomology of the space of polynomial maps on A1 with prescribed ramification

Banerjee¹

2020

Advances in Mathematics

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“…The theme of homology (cohomology-) stabilization is indeed a very general one, which has been recently revived through the work of several authors, also in other contexts ( see for instance [131,153,158,203]). …”

Section: Stabilization Results For the Homology Of Moduli Spaces Of Cmentioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)'s, through Teichmüller theory. The main trhust of the paper is to show how in the case of K(H,1)'s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lönne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an Communicated by Efim Zelmanov.The present work took place in the realm of the DFG Forschergruppe 790 "Classification of algebraic surfaces and compact complex manifolds". Part of the article was written when the author was visiting KIAS, Seoul, as KIAS research scholar.

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“…Therefore the assumptions of Theorem 4.4 are fulfilled. Similarly, the group P Sp 4 (3).2 has a rationally rigid triple of classes of element orders (2,8,9), and the classes of elements of order 8 and 9 respectively fulfill the required assumptions. Finally, the simple group P Sp 6 (2) has a rationally rigid triple of classes of element orders (2,7,9), and the classes of elements of order 7 and 9 respectively fulfill the required assumptions.…”

Section: General Criteriamentioning

confidence: 99%

“…In the case G = P SL 2 (7), an extension of Q(t) with all inertia groups of order 2 and without universally ramified primes is deduced from [18, Proof of Theorem 3.2] by specializing some of the parameters. Finally, a P SL 3 (3)-extension of Q(t) with all inertia groups of order 2 is given in [18,Lemma 3.4].…”

Section: General Criteriamentioning

confidence: 99%

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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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Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

Cited by 118 publications

References 44 publications

Cohomology of the space of polynomial maps on A1 with prescribed ramification

Cohomology of the space of polynomial maps on A1 with prescribed ramification

Topological methods in moduli theory

Unramified extensions over low degree number fields

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