Galois module structure of unramified covers

Pappas, G.

doi:10.1007/s00208-007-0183-2

Cited by 4 publications

(6 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If π is tamely ramified, then the complex RΓ(X, F ) is "perfect", i.e. isomorphic to a bounded complex (P • ) of finitely generated projective Z[G]-modules ( [6], [7], [31]). This observation goes back to E. Noether when X is the spectrum of the ring of integers of a number field; in this general set-up it is due to T. Chinburg.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, it allowed us to show that, under this hypothesis, gcd(2, #G) • χ(X, F ) = 0 if d = 1 ( [30]). Some general results were obtained in [31] when d > 1 but the problem appears to be quite hard. In fact, as is explained in loc.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, as is explained in loc. cit., it is plausible that the statement that χ(X, F ) = 0 for G abelian and all X of dimension < #G is equivalent to the truth of Vandiver's conjecture for all prime divisors of the order #G. In [31], [10], it was conjectured that for all G, there are integers N (that depends only on d) and δ (that depends only on #G), such that gcd(N, #G) δ • χ(X, F ) = 0; this was shown for G with abelian Sylows.…”

Section: Introductionmentioning

confidence: 99%

“…When d > 0, some progress towards calculating χ(X, F ) for general F was achieved after the introduction of the technique of cubic structures ( [30], [31], [10]; the paper [8] only dealt with the deRham complex). This technique is very effective when all the Sylow subgroups of G are abelian.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Adams operations and Galois structure

Pappas

2015

Algebra Number Theory

Self Cite

View full text Add to dashboard Cite

We present a new method for determining the Galois module structure of the cohomology of coherent sheaves on varieties over the integers with a tame action of a finite group. This uses a novel Adams-Riemann-Roch type theorem obtained by combining the Kunneth formula with localization in equivariant K-theory and classical results about cyclotomic fields. As an application, we show two conjectures of Chinburg-Pappas-Taylor, in the case of curves.Comment: 30 pp, few additional change

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Adams operations and Galois structure

Pappas

2015

Algebra Number Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…(Note that G-torsors X → Y with Y as in Prop. 6.4 can be constructed for any group G as in [50,Appendix] by using the theorems of Moret-Bailly and Rumely on the existence of integral points.) 6.c.…”

Section: Elementary Structures and The Second Chern Classmentioning

confidence: 99%

Higher adeles and non-abelian Riemann–Roch

Chinburg

Pappas

Taylor

2015

Advances in Mathematics

View full text Add to dashboard Cite

Abstract. We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P 1 . Our set-up provides an extension of several aspects of the classical Fröhlich theory of the Galois module structure of rings of integers of number fields to arithmetic surfaces. ContentsIntroduction §1. Beilinson-Parshin adeles on a surface §2. Equivariant adelic Chow groups §3. Lattices, determinant functors and determinant theories §4. Pushdown maps and reciprocity laws §5. Transition matrices and the first Chern class §6. Elementary structures and the second Chern class §7. Equivariant Euler characteristics and the Riemann-Roch theorem §8. The proof of the theorem; reduction to the case of P 1 Z §9. The proof of the theorem; bundles over P 1 Z §10. Appendix: Adelic Riemann-Roch for general bundles when G = 1 References

show abstract

Cubic structures, equivariant Euler characteristics and lattices of modular forms

2009

Self Cite

View full text Add to dashboard Cite

We use the theory of cubic structures to give a fixed point Riemann-Roch formula for the equivariant Euler characteristics of coherent sheaves on projective flat schemes over ‫ޚ‬ with a tame action of a finite abelian group. This formula supports a conjecture concerning the extent to which such equivariant Euler characteristics may be determined from the restriction of the sheaf to an infinitesimal neighborhood of the fixed point locus. Our results are applied to study the module structure of modular forms having Fourier coefficients in a ring of algebraic integers, as well as the action of diamond Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of Jacobians of modular curves.

show abstract

Galois module structure of unramified covers

Cited by 4 publications

References 26 publications

Adams operations and Galois structure

Adams operations and Galois structure

Higher adeles and non-abelian Riemann–Roch

Cubic structures, equivariant Euler characteristics and lattices of modular forms

Contact Info

Product

Resources

About