Abstract
Let be a number field, a polynomial over with , and the group of -units of , where is an appropriate finite set of places of . In this note, we prove that outside of some natural exceptional set , the prime ideals of dividing , , mostly have degree one over ; that is, the corresponding residue fields have degree one over the prime field. We also formulate a conjectural analogue of this result for rational points on an elliptic curve over a number field, and deduce our conjecture from Vojta’s conjecture. We prove this conjectural analogue in certain cases when the elliptic curve has complex multiplication.
Citation
Aaron Levin. David McKinnon. "Ideals of degree one contribute most of the height." Algebra Number Theory 6 (6) 1223 - 1238, 2012. https://doi.org/10.2140/ant.2012.6.1223
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