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On the generators of the first homology with compact supports of the Weierstrass family in characteristic zero


Author: Goro C. Kato
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
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Abstract: Let $ {{\mathbf{W}}_{\mathbf{Q}}} = \operatorname{Proj}({\mathbf{Q}}[{{\text{g}}_2},... ...eous 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 [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0697080-X
Keywords: Lifted $ p$-adic homology with compact supports, Weierstrass Family, elliptic curves
Article copyright: © Copyright 1983 American Mathematical Society

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