Rob Benedetto (Amherst)
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We will interpret the $abc$ conjecture as a statement about four points in the projective line. For function fields of characteristic zero, where the $abc$ conjecture is a theorem, we will consider generalizations corresponding to $N$ points in the projective line. Analogous conjectures can then be made for number fields.
If time permits, we will also consider applications of such conjectures to arithmetic problems, such as Morton and Silverman's conjecture that there is a uniform bound for the number of rational preperiodic points of a morphism defined over a number field. |