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ADDENDA: NONSOLVABLE NUMBER FIELDS RAMIFIED ONLY AT 3 AND 5

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ADDENDA:

NONSOLVABLE NUMBER FIELDS RAMIFIED ONLY AT 3 AND 5

LASSINA DEMB ´EL ´E, MATTHEW GREENBERG, AND JOHN VOIGHT

This note gives addenda for the articleNonsolvable number fields ramified only at 3 and 5 [1].

(1) Page 717, “On the assumption of the Generalized Riemann Hypothesis (GRH), the Odlyzko bounds [Mar82] imply that each of the fields we con- struct is totally imaginary”: This immediately follows from the more gen- eral fact that the representations constructed from Hilbert modular forms areodd, so that the determinant of (any) complex conjugation is equal to

−1; it thus follows that even in the projectivization, complex conjugation acts nontrivially, so the number fields so constructed are totally complex.

References

[1] Lassina Demb´el´e, Matthew Greenberg, and John Voight,Nonsolvable number fields ramified only at3and5, Compositio Math.147(2011), no. 3, 716–734.

Date: May 26, 2013.

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