Duality for the de Rham cohomology of an abelian scheme

Robert F. Coleman

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 5, page 1379-1393
  • ISSN: 0373-0956

Abstract

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In this paper the equality is established of three different pairings between the first de Rham cohomology group of an abelian scheme over a base flat over ℤ and that of its dual. These pairings have appeared and been used either explicitly or implicitly in the literature.In the last section we deduce a generalization to arbitrary characteristic of Serre’s formula for the Poincaré pairing on the first de Rham cohomology group of a curve over a field of characteristic zero.

How to cite

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Coleman, Robert F.. "Duality for the de Rham cohomology of an abelian scheme." Annales de l'institut Fourier 48.5 (1998): 1379-1393. <http://eudml.org/doc/75323>.

@article{Coleman1998,
abstract = {In this paper the equality is established of three different pairings between the first de Rham cohomology group of an abelian scheme over a base flat over $\{\Bbb Z\}$ and that of its dual. These pairings have appeared and been used either explicitly or implicitly in the literature.In the last section we deduce a generalization to arbitrary characteristic of Serre’s formula for the Poincaré pairing on the first de Rham cohomology group of a curve over a field of characteristic zero.},
author = {Coleman, Robert F.},
journal = {Annales de l'institut Fourier},
keywords = {abelian scheme; Poincaré pairing; first de Rham cohomology group; Serre's formula},
language = {eng},
number = {5},
pages = {1379-1393},
publisher = {Association des Annales de l'Institut Fourier},
title = {Duality for the de Rham cohomology of an abelian scheme},
url = {http://eudml.org/doc/75323},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Coleman, Robert F.
TI - Duality for the de Rham cohomology of an abelian scheme
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 5
SP - 1379
EP - 1393
AB - In this paper the equality is established of three different pairings between the first de Rham cohomology group of an abelian scheme over a base flat over ${\Bbb Z}$ and that of its dual. These pairings have appeared and been used either explicitly or implicitly in the literature.In the last section we deduce a generalization to arbitrary characteristic of Serre’s formula for the Poincaré pairing on the first de Rham cohomology group of a curve over a field of characteristic zero.
LA - eng
KW - abelian scheme; Poincaré pairing; first de Rham cohomology group; Serre's formula
UR - http://eudml.org/doc/75323
ER -

References

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  1. [BBM] P. BERTHELOT, L. BREEN and W. MESSING, Théorie de Dieudonné cristalline, II, SLN 930 (1982). Zbl0516.14015MR85k:14023
  2. [C1] R. COLEMAN, Hodge-Tate periods and p-adic abelian integrals, Invent. Math., 78 (1984), 351-374. Zbl0572.14024MR87f:11040
  3. [C2] R. COLEMAN, The universal vectorial bi-extension and p-adic heights, Invent. Math., 103 (1991), 631-650. Zbl0763.14009MR92k:14021
  4. [D] P. DELIGNE, Théorie de Hodge, III, IHES Publ. Math., 44 (1974). Zbl0237.14003MR58 #16653b
  5. [SGAI] A. GROTHENDIECK, Revêtements étales et groupe fondamental, (SGA I), SLN 224 (1971). Zbl0234.14002
  6. [H] R. HARTSHORNE, On the de Rham cohomology of algebraic varieties, IHES Publ. Math., 45 (1976). Zbl0326.14004
  7. [MM] B. MAZUR and W. MESSING, Universal extensions and one-dimensional crystalline cohomology, SLN 370 (1974). Zbl0301.14016MR51 #10350
  8. [NO] P. NORMAN and F. OORT, Moduli of abelian varieties, Ann. of Math., 112 (1980), 412-439. Zbl0483.14010MR82h:14026

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