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A Note on Manin's Theorem of the Kernel

The purpose of this note is to show that Manin's original statement of the theorem of the kernel is an easy consequence of Deligne's theory of differential equations with regular singularities and Deligne's Hodge theory. Since this note is primarily of historical interest, some discussion about the history may be beneficial to the reader, and I stand to be corrected for any inaccuracy. More than a quarter of a century ago Manin in [M] proved Mordell's conjecture for function fields over C. His major tool was what is now commonly called the Gauss-Manin connection, introduced by Manin. A major step is a statement called the theorem of the kernel, an equivalent form of which will be recalled later. Recently, Robert Coleman found a mistake in elementary linear algebra in the proof of the theorem of kernel in [M], which unfortunately invalidated the whole proof. Coleman however proved a weaker version of the theorem of kernel, and deduced Mordell's conjecture over function fields following Manin's original ideas, and he used analogue of Siegel's theorem on integral points over function fields over C. In this note, it will be shown that Manin was right after all, and it is possible to make a local correction using Deligne's theorems. It may be of interest to note that our proof uses global monodromy of the Gauss-Manin connection systematically instead of the local monodromy as in [M]. I learned of Manin's beautiful ideas from lectures of Professor Coleman on Jan. 7-8, 1989 at the Tucson conference on the arithmetic of curves. This note is a response to the stimulating question he raised at Tucson, and I thank him very much. I would also like to thank Carlos Simpson for pleasant discussions and a pertinent remark. I strongly recommend Manin's original paper [M] for readers interested in Manin's circle of ideas about Mordel's conjecture for function fields (over C). For a modern treatment of Manin's proof using algebraic DeRham cohomology, the reader is referred to [C].