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 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 for the leading term of the logarithmic derivative of the Serre-Tate pairing for the ordinary factor \(G\) of the Tate module of the Jacobian of \(X_ 1(pN)\) cut out by the natural action of \((\mathbb{Z}/p\mathbb{Z})^*\). The formula shows that the vanishing of the leading term of \(ds/s\) is equivalent to the vanishing of a certain class \(h\) in the de Rham cohomology of the Igusa curve, and this amounts to the existence of a companion form of the modular form \(f\). The last argument necessary for the proof of the theorem now comes from the observation that the vanishing of the leading term is equivalent to the tameness of the ramification of the restriction of \(\rho_ f\) to a decomposition group at \(p\).