Companion forms and Kodaira-Spencer theory
type: Item
instance of: scholarly article
title: Companion forms and Kodaira-Spencer theory
zbMATH Open document ID: 0770.11024
DOI: 10.1007/BF01231333
author: Robert F. Coleman
author: José Felipe Voloch
published in: Inventiones Mathematicae
publication date: 1993-05-16T00:00:00Z
full work available at URL: https://eudml.org/doc/144051
review text: Let \(N\) be a positive integer and let \(p>2\) be a prime not dividing \(N\). With a normalized eigenform \(f=\sum a_ nq^ n\) on \(X_ 1(N)\) modulo \(p\) of weight \(k\) (and supposed to have nebentypus \(\varepsilon)\) one can associate a representation \(\rho_ f:G=\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\to GL_ 2(E)\), where \(E\) is a finite field of characteristic \(p\). An eigenform \(g=\sum b_ nq^ n\) modulo \(p\) of weight \(k'=p+1-k\) on \(X_ 1(N)\) such that \(na_ n=n^ kb_ n\) is called a companion form of \(f\). A conjecture of Serre says that the existence of such a companion form is equivalent to tame ramification of \(\rho_ f\) above \(p\). The conjecture was proved by \textit{B. H. Gross} [Duke Math. J. 61, 445-517 (1990; Zbl 0743.11030)] in `most cases'. In the underlying paper the assumption Gross had to make is removed, and the following result is proved: For an ordinary cuspidal eigenform \(f\) on \(X_ 1(N)\) modulo \(p\) of weight \(k\) such that \(2<k\leq p\), the representation \(\rho_ f\) is tamely ramified above \(p\) if and only if \(f\) has a companion form. The results and ideas of Gross (loc. cit.) pervade the proof of the main theorem above, but to circumvent Gross's need for an extra assumption, the Kodaira-Spencer pairing is applied as a useful tool to compute the logarithmic derivative of the Serre-Tate pairings between the Tate module of the reduction of a family of ordinary \(p\)-divisible groups \(G\) over a complete local \(p\)-ring \(R\) and that of its dual \(^ tG\). Here a \(p\)- divisible group \(G\) is said to be ordinary if the dual of the connected subgroup of its special fiber is étale. An explicit general formula for the Kodaira-Spencer pairing can be derived for a semi-stable curve over a one-parameter infinitesimal deformation of a point. Regarding \(X_ 1(pN)\) as a family over \(\text{Spec}(\mathbb{Z}_ p[\zeta_ p])\) with base \(\text{Spec}(\mathbb{Z}_ p)\), this can be applied to modular forms. It leads to a formula…
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14L05
zbMATH DE Number: 167395
zbMATH Keywords: \(p\)-divisible group
zbMATH Keywords: modular elliptic curves
zbMATH Keywords: Galois representation
zbMATH Keywords: companion form
zbMATH Keywords: tame ramification
zbMATH Keywords: cuspidal eigenform
zbMATH Keywords: Kodaira-Spencer pairing
zbMATH Keywords: Serre-Tate pairing
zbMATH Keywords: Tate module of the Jacobian
zbMATH Keywords: de Rham cohomology of the Igusa curve
reviewed by: W. W. J. Hulsbergen
Wikidata QID: Q56813948
MaRDI profile type: Publication
cites work: Q5749421
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OpenAlex ID: W2064846883
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description: scientific article; zbMATH DE number 167395