Nonsolvable number fields ramified only at 3 and 5

For p = 3 and p = 5, we exhibit a finite nonsolvable extension of Q which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computations with Hilbert modular forms. The study of Galois number fields with prescribed ramification remains a central question in number theory. Class field theory, a triumph of early twentieth century algebraic number theory, provides a satisfactory way to understand solvable extensions of a number field. To investigate nonsolvable extensions, the use of the modern techniques of arithmetic geometry is essential. Implicit in his work on algebraic modular forms on groups of higher rank, Gross [Gro98] proposed the following conjecture. Conjecture. For any prime p, there exists a nonsolvable Galois number field ramified only at p. Serre [Ser73, Ser97] has answered this conjecture in the affirmative for primes p> 11 in his analy-sis of Galois representations associated to classical cusp forms, exhibiting an irreducible (continuous, odd) representation Gal(Q/Q) → GL2(Fp) ramified only at p. Conversely, it is a consequence of the proof of Serre’s conjecture by Khare and Wintenberger [KhWi09] together with standard level low