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Letter to Don Zagier. (English) Zbl 0728.14009

Arithmetic algebraic geometry, Proc. Conf., Texel/Neth. 1989, Prog. Math. 89, 285-292 (1991).
[For the entire collection see Zbl 0711.00011.]
For a fibration \(V\to B\) of relative dimension 1 over a curve B, one has a (so-called VBMY-) inequality, giving an upper bound for \(\omega^ 2_{V/B}\). It is more or less obtained as a consequence of \(h^ 1(\Omega^ 1_ V)\geq 0\). Special attention goes to the case of a finite ground field and to the analogue over a number field instead of B. This is compared with the situation of relative dimension 0, where the functional equations of zeta-functions are used.

MSC:

14D10 Arithmetic ground fields (finite, local, global) and families or fibrations
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties
14H25 Arithmetic ground fields for curves

Citations:

Zbl 0711.00011