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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the generators of the first homology with compact supports of the Weierstrass family in characteristic zero
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by Goro C. Kato PDF
Trans. Amer. Math. Soc. 278 (1983), 361-368 Request permission

Abstract:

Let ${{\mathbf {W}}_{\mathbf {Q}}} = \operatorname {Proj}({\mathbf {Q}}[{{\text {g}}_2},{g_3},X,Y,Z]/({\text {homogeneous ideal generated by }} - {Y^2}Z + 4 {X^3} - {g_2} X{Z^2} - {g_3} {Z^3}))$. This is said to be the Weierstrass Family over the field ${\mathbf {Q}}$. Then the first homology with compact supports of the Weierstrass Family is computed explicitly, i.e., it is generated by ${\{ {C^{ - k}} dX \wedge \;dY\}_{k \geqslant 1}}$ and ${\{ X{C^{ - k}}dX \wedge dY\}_{k \geqslant 1}}$ over the ring ${\mathbf {Q}}[{g_{2}},{g_3}]$, where $C$ is a polynomial ${Y^2} - 4{X^3} + {g_2}X + {g_3}$. When one tensors the homology of the Weierstrass Family with ${\Delta ^{ - 1}} {\mathbf {Q}}[{g_2},{g_3}]$, being localized at the discriminant $\Delta = g_2^3 - 27g_3^2$, over ${\mathbf {Q}}[{{\text {g}}_2},{g_3}]$, the first homology is generated by ${C^{ - 1}}dX\; \wedge \;dY$ and $X{C^{ - 1}}dX \wedge dY$. One also obtains the first homologies with compact supports of singular fibres over $\wp = ({g_2} = {g_{3}} = 0)$ and $\wp = ({g_2} = 3,{g_{3}} = 1)$ as corollaries.
References
  • Goro C. Kato and Saul Lubkin, Zeta matrices of elliptic curves, J. Number Theory 15 (1982), no. 3, 318–330. MR 680536, DOI 10.1016/0022-314X(82)90036-1
  • G. C. Kato, Lifted $p$-adic homology with compact supports of Weierstrass family and zeta matrices (in preparation). —, Bounded Witt vector cohomology of elliptic curves (in preparation).
  • Saul Lubkin, Cohomology of completions, Notas de Matemática [Mathematical Notes], vol. 71, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 589714
  • —, A $p$-adic proof of Weil’s conjectures, Ann. of Math. (2) 87 (1968), 105-255.
  • Saul Lubkin, Finite generation of lifted $p$-adic homology with compact supports. Generalization of the Weil conjectures to singular, noncomplete algebraic varieties, J. Number Theory 11 (1979), no. 3, S. Chowla Anniversary Issue, 412–464. MR 544265, DOI 10.1016/0022-314X(79)90010-6
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 278 (1983), 361-368
  • MSC: Primary 14K15; Secondary 11D25, 11G05, 14F30, 14G10
  • DOI: https://doi.org/10.1090/S0002-9947-1983-0697080-X
  • MathSciNet review: 697080