Tame actions of group schemes: integrals and slices

Chinburg, Ted; Erez, Boas; Pappas, G.; Taylor, Martin J.

doi:10.1215/s0012-7094-96-08212-5

Cited by 32 publications

(48 citation statements)

References 40 publications

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“…X H is tame (see [CEPT96]), the cokernel Ꮿ and kernel of Tr are coherent and supported on the subset of X H over which the cover has wild ramification. By [Maz77,II.2] this is (at most) a finite set of points in characteristics 2 and 3.…”

Section: Galois Structure Of Modular Formsmentioning

confidence: 99%

Cubic structures, equivariant Euler characteristics and lattices of modular forms

2009

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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.

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Section: Galois Structure Of Modular Formsmentioning

confidence: 99%

Cubic structures, equivariant Euler characteristics and lattices of modular forms

2009

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“…Suppose that G is a finite, flat commutative group scheme over Y of exponent N , and let G D denote the Cartier dual of G. Let π : X → Y be a G-torsor, and write π 0 : G → Y for the trivial G-torsor. Then O X is an O G -comodule, and so it is also an O G D -module (see [12]). As an O G D -module, O X is locally free of rank one, and it therefore gives a line bundle…”

Section: Introductionmentioning

confidence: 99%

On arithmetic class invariants

Agboola

Pappas²

2001

Math Ann

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“…For a detailed account of the formation of Euler characteristics (without metrics) associated to a tame action, the reader is referred to [CEPT4]. Let F • denote a bounded complex of coherent G-X sheaves.…”

Section: A Preliminary Resultsmentioning

confidence: 99%

-constants and equivariant Arakelov–Euler characteristics

Chinburg

Pappas

Taylor

2002

Annales Scientifiques de l’École Normale Supérieure

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characteristic for a bounded complex of hermitian G-bundles. Roughly speaking, our main results show that the Arakelov-Euler characteristic of the (logarithmic) de Rham complex of X determines certain ε-constants which are associated to the L-functions of the Artin motives obtained from X and the symplectic representations of G.Let us now describe these results in more detail. Firstly we need to say a little more about arithmetic classgroups. Let R G (resp. R s G ) be the group of virtual characters (resp. virutal symplectic characters) of G. In Section 4 we obtain a quotient. Now let S denote a finite set of primes which includes those primes where X has non-smooth reduction. Let Ω 1 X /Z log X red S / log S denote the sheaf of degree one relative logarithmic differentials of X with respect to the morphism X , X red S → (Spec (Z) , S) of schemes with log-structures.This sheaf is locally free if (as we now assume) each special fibre of X is a divisor with strictly normal crossings and the multiplicities of the irreducible components of each fibre are prime to the residue characteristic. The logarithmic de Rham complex Ω • X /Z (log X red S / log S) is defined to be the complexWe view each term of this complex as carrying the hermitian metric given by the corresponding exterior power of h D . In Theorem 7.1 we completely describe the image c s of the loga-). Here we limit ourselves to stating the result for characters of degree zero, since this result is particularly striking (see Theorem 7.1 for the result for characters of arbitrary degree):) is a rational class and for any virtual symplectic character ψ of degree zeroWhilst this result is of interest in itself, our principal concern rests with the sheaf of differentials Ω 1 X /Z . One can associate to Ω 1 X /Z and the metric h D a class Ω in the arithmetic Grothendieck group K 0 (X ), which is a λ-ring by [GS1, §7]. In [CPT2] we show d i=0 χ(λ i (Ω)) = −log|ε(X )|.

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Tame actions of group schemes: integrals and slices

Cited by 32 publications

References 40 publications

Cubic structures, equivariant Euler characteristics and lattices of modular forms

Cubic structures, equivariant Euler characteristics and lattices of modular forms

On arithmetic class invariants

-constants and equivariant Arakelov–Euler characteristics

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