Integral differentials of elliptic function fields
Let \(K\) be an arithmetic function field of transcendence degree \(r\) over a number field \(k\) and denote by \(A\) the ring of integers of \(k\). In his fundamental work ``Geometria aritmetica'' published in [Ann. Mat. Pura Appl., IV. Ser. 45 (1958; Zbl 0142.18101)], which laid the foundations for an algebraic differential calculus on arithmetic varieties, \textit{E. Kähler} has introduced the so-called module of integral differential forms of \(K\) with respect to the different. This graded \(A\)-module \(D({K\over\vartheta}|A)\) has recently been reconsidered by the authors of the paper under review who gave a geometric-cohomological interpretation of it in terms of a proper and normal model scheme \(X\) of the function field \(K\) [Abh. Math. Semin. Univ. Hamb. 73, 297--310 (2003; Zbl 1045.14009)]. On the other hand, as a variation of Kähler's original differential module, J.-B. Bost has introduced the graded ring \(D_s(K):= \bigcap_{R\in V_s} \Omega_{R/A}\), where \(V_s\) denotes the set of all discrete valuation rings with quotient field \(K\) which are smooth over \(A\) [cf., \textit{J.-B. Bost}, A neglected aspect of Kähler's work on arithmetic geometry birational invariants of algebraic varieties over number fields; in: Erich Kähler, Mathematical Works (R. Berndt and O. Riemenschneider (eds.), De Gruyter Verlag, Berlin, 854--869 (2003; Zbl 1039.01013)]. In the present note, the authors supplement these recent developments by a comparison theorem in the special case of elliptic function fields. More precisely, let \(K\) be the arithmetic function field of an elliptic curve \(E\) defined over the number field \(k\) and having a \(k\)-rational point. Using a Weierstrass model for \(E\), the authors construct a module \(W(E/k)\) describing, in terms of differentials, the reciprocal of the Weierstrass class of \(E\). Then their main theorem establishes the equalities \[ D\biggl({K\over\vartheta} \Bigl| A\biggr)= W(E/k)= D_s(K) \] for elliptic function fields \(K\). This comparison theorem implies that, for elliptic function fields, the Kähler-Bost module is \(A\)-free if and only if the elliptic curve \(E\) can be defined by a globally minimal Weierstrass polynomial.
scientific article; zbMATH DE number 2139437
scientific article