Patching subfields of division algebras

Harbater, David; Hartmann, Julia; Krashen, Daniel

doi:10.1090/s0002-9947-2010-05229-8

Cited by 20 publications

(35 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This problem was posed to the first author by David Harbater. We show that the result of [Harbater et al 2009] holds over these fields as well.…”

mentioning

confidence: 54%

“…direction was recently made by Harbater et al [2009], who consider a question relating Galois theory and Brauer theory over a field E: Which groups are admissible over E? That is, which finite groups occur as a Galois group of an adequate Galois extension F/E (recall that an extension F/E is called adequate if F is a maximal subfield in an E-central division algebra).…”

mentioning

confidence: 99%

“…The main theorem of [Harbater et al 2009] asserts that if E is a function field in one variable over a complete discretely valued field with an algebraically closed residue field, then a finite group G of order not divisible by char E is admissible over E if and only if each of the Sylow subgroups of G is abelian of rank at most 2 (i.e., generated by two elements).…”

mentioning

confidence: 99%

See 2 more Smart Citations

Untitled

2010

ANT

View full text Add to dashboard Cite

We introduce a notion of Hecke-monicity for functions on certain moduli spaces associated to torsors of finite groups over elliptic curves, and show that it implies strong invariance properties under linear fractional transformations. Specifically, if a weakly Hecke-monic function has algebraic integer coefficients and a pole at infinity, then it is either a holomorphic genus-zero function invariant under a congruence group or of a certain degenerate type. As a special case, we prove the same conclusion for replicable functions of finite order, which were introduced by Conway and Norton in the context of monstrous moonshine. As an application, we introduce a class of Lie algebras with group actions, and show that the characters derived from them are weakly Hecke-monic. When the Lie algebras come from chiral conformal field theory in a certain sense, then the characters form holomorphic genus-zero functions invariant under a congruence group. IntroductionWe define a holomorphic genus-zero function to be a holomorphic function f : H → ‫ރ‬ on the complex upper half-plane, with finite-order poles at cusps, such that there exists a discrete group f ⊂ SL 2 ‫)ޒ(‬ for which f is invariant under the action of f by Möbius transformations, inducing a dominant injection H/ f → ‫.ރ‬ Keywords: moonshine, replicable function, Hecke operator, generalized moonshine. This material is partly based upon work supported by the National Science Foundation under grant DMS-0354321. 650 Scott CarnahanA holomorphic genus-zero function f therefore generates the field of meromorphic functions on the quotient of H by its invariance group. In this paper, we are interested primarily in holomorphic congruence genus-zero functions, especially those f for which (N ) ⊂ f for some N > 0. These functions are often called Hauptmoduln.The theory of holomorphic genus-zero modular functions began with Jacobi's work on elliptic and modular functions in the early 1800's, but did not receive much attention until the 1970's, when Conway and Norton found numerical relationships between the Fourier coefficients of a distinguished class of these functions and the representation theory of the largest sporadic finite simple group ‫,ލ‬ called the monster. Using their own computations together with work of Thompson and McKay, they formulated the monstrous moonshine conjecture, which asserts the existence of a graded representation V = n≥−1 V n of ‫ލ‬ such that for each g ∈ ‫,ލ‬ the graded character T g (τ ) := n≥−1 Tr(g|V n )q n is a normalized holomorphic genus-zero function invariant under some congruence group 0 (N ), where the normalization indicates a q-expansion of the form q −1 + O(q). More precisely, they gave a list of holomorphic genus-zero functions f g as candidates for T g , whose first several coefficients arise from characters of the monster, and whose invariance groups f g contain some 0 (N ) [Conway and Norton 1979]. By unpublished work of Koike, the power series expansions of f g satisfy a condition known as complete replicability, given by a fa...

show abstract

“…This problem was posed to the first author by David Harbater. We show that the result of [Harbater et al 2009] holds over these fields as well.…”

mentioning

confidence: 54%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Untitled

2010

ANT

View full text Add to dashboard Cite

show abstract

“…The content of the next proposition is essentially given in [Harbater et al 2009], but for specific fields Q i , while here we present it for general fields satisfying a matrix factorization property. We note that [Harbater et al 2009, Theorem 4.1] uses the terminology of categories and equivalence of categories.…”

Section: Patching and Admissibilitymentioning

confidence: 99%

“…For a prime v of E, let ram v denote the ramification map ram v : Br E → H 1 (G E v , Q/Z ) [Saltman 1999]. Following [Harbater et al 2009], we say that an α ∈ Br E is determined by ramification with respect to a set of primes if there is a prime v ∈ for which exp(α) = exp(ram v (α)). Let D be an E-division algebra with maximal subfield L that has Galois group G = Gal(L/E).…”

Section: Patching and Admissibilitymentioning

confidence: 99%