On the number of equivalence classes of binary forms of given degree and given discriminant

Details On the number of equivalence classes of binary forms of given degree and given discriminant On the number of equivalence classes of binary forms of given degree and given discriminant

2003-12-11

arxiv.org/abs/math/0312234v1

Acta Arith. 113 (2004), 363-399

Mathematics

Number Theory

45 pages, latex file

Scientific paper

We give an explicit upper bound for the number of equivalence classes of binary forms with rational integral coefficients of given degree and given discriminant, and with given splitting field. Further, we give an explicit upper bound for the number of irreducible binary forms with rational integral coefficients with given invariant order. Our bounds depend on as few parameters as possible. For instance, we show that the number of equivalence classes of irreducible binary forms with rational integral coefficients of degree r with given invariant order has an upper bound depending only on r. We have proved more general results for binary forms with coefficients in the ring of S-integers of a number field.

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