Hurwitz Number Fields David P. Roberts University of Minnesota, Morris. Context coming from mass formulas. Sketch of definitions and key properties. A full Hurwitz number field with Galois group A 5 and discriminant d = 56 5 0. Results and open problems
. Context coming from mass formulas. The mass of an Q-algebra F is by definition µ(f ) = Aut(F ). Examples: F = R : µ(r r C s ) = r!s! s F = Q p : µ(q p f ) = f The mass of a class of Q-algebras is the sum of the masses of one representing algebra for each isomorphism type. For example, the class of all unramified Q p -algebras of degree n has total mass, as in the case n = : µ(q p )+µ(q p Q p )+µ(q p Q p Q p ) = + + 6 =. More generally, for any tame partition τ n the class of Q p -algebras with ramification partition τ has total mass. Total mass µ n (Q v ) of Q v -algebras of degree n: v\n 5 6 7 8 9.66.7...05.0.0 5 6 0 5 80 57 7 9 9 8 99 7 00 5 5 7 55 8 0
Let NF n (d) be the number of degree n number fields of absolute discriminant d, which are full in the sense that G {A n, S n }. Some average cardinalities NF n (d) : 00 00 00-000 -000-000 -000 Limit 5.000.00.00.006.50 [B].08.0.05.056.5 [B].5.77.8.97.77 [DH] ζ().607.6.60.6.608 ζ() A natural local-global heuristic for n is that on average NF n ( p p cp ) µ n(r) p µ n (Q p, p cp ). Here µ n (Q p, p c p) is the total mass of Q p -algebras of degree n and discriminant p c p. For n =,, and 5, this heuristic is exactly right in the limit of large d.
What about the vertical rather than horizontal direction? Let NF n (P) be the set of full fields of degree n ramified within a given finite set P of primes. The heuristic gives e.g. the following predictions for NF n ({,, 5}). 0 5 0 0 0 5 50 00 50 n In fact, for any fixed P, the heuristic says that NF n (P) is eventually zero. However with Venkatesh we expect that Hurwitz number fields form an enormous exception to the mass heuristic: Conjecture. Suppose P contains the set of primes dividing a finite nonabelian simple group. Then lim sup NF n (P) =.
. Sketch of definitions and key properties. A Hurwitz parameter is a triple h = (G, C, ν) where G is a finite centerless group, C = (C,..., C r ) is a list of conjugacy classes, ν = (ν,..., ν r ) is a list of positive integers, The quotient elements [C i ] generate G ab and satisfy [C i ] ν i =. A Hurwitz parameter h determines an unramified covering of complex algebraic varieties: π h : Hur h Conf ν. Here the cover Hur h is a Hurwitz variety parameterizing certain covers of the complex projective line P of type h. The base is the variety whose points are tuples (D,..., D r ) of disjoint divisors D i in P, with D i consisting of ν i distinct points. The map π h sends a cover to its branch locus.
Let G h be the set of tuples (g,,..., g,ν,..., g r,,..., g r,νr ) C ν Cν r r which generate G and have product. Then G acts G h by conjugation and the fiber Hur h,u above any base point u can be identified with F h = G h G. A canonical approximation to the degree n h = F h is ri= C i ν i n h = G G. When enough sufficiently different C i are present, in fact n h = n h. The fundamental group π (Conf ν, u) can be identified with a classical braid group Br ν. Under suitable hypotheses most crucially that G is close to being a nonabelian simple group the action of Br ν on Hur h,u is full, in the sense of having image all of A n or S n.
If all the C i are rational, then the map π h canonically descends to a map of varieties over Q. Fibers Hur h,u above rational points u Conf ν (Q) Conf ν are the root-sets of Hurwitz number algebras F h,u. If π h is full, then F h,u is full for generic u, by the Hilbert irreducibility theorem. The cover π h has good reduction outside of P G, the set of primes dividing G. Let Z[P] be the set of rational numbers having denominator divisible only by primes in P. Then for u Conf ν (Z[P]) the algebra can have bad reduction only at primes in P G P. The sets Conf ν (Z[P G ]) can be arbitrarily big in the way needed by the conjecture. So either the conjecture is true or specialization to u Conf ν (Z[P G ]) behaves in an extremely nongeneric way.
. A full Hurwitz number field with Galois group A 5 and discriminant d = 56 5 0. Take h = (G, C, ν) = (S 5, (, 5), (, )), u = (D, D ) = ({, 0,, }, { }). The definition requires us to look at g(z) = z 5 + z + bz + cz + dz + e with critical values {, 0,, }. Explicitly, we need to find solutions (b, c, d, e, w) C 5 to Res z (g(z) t, g (z)) = w(t + )t(t )(t ). Finding these solutions takes one second. Exactly 5 different e work. They are the roots of an irreducible degree 5 polynomial f(x). The Hurwitz number field F h,u = Q[x]f(x) has discriminant d = 56 5 0. Fixing h but varying u Conf, (Z[{,, 5}]) gives more than ten thousand different F h,u. All have d = ± a b 5 c and Galois group in {A, S, A 5, S 5 }.
The intuitive reason that a Hurwitz number field F h,u = Q[x]f(x) is special is that each root of a defining polynomial f(x) is not just a complex number. Rather behind this complex number is a delicate geometric situation: the unique covering of P t with a prescribed topology. For the five real e, the coverings P z P t are as follows (drawn in the real z-t plane and also superimposed). In general, there are n = [F h,u : Q] different geometric objects, with their arithmetic coordinated by F h,u.
To get images for all 5 different e, we draw in the t-plane. plane are 5 Then its preimages in the z
F B C E I G I J J E H A C B H B E D D D A C G F A Actions of the standard generators σ (vertical arrows), σ (symbols), and σ (horizontal arrows) of the braid group Br, are given above. 5
. Results and Open Problems. A. Explicit equations for the covers Hur h Conf ν have been worked out in a broad range of situations: exotic groups like SL (), G (), Sp (), M,... ; constrained ramification and large degrees like {,, 5} and n = 00; certain sequences with G = S d with d and n both increasing without bound. Further understanding geometry would allow computations in new regimes (higher genus..., more ramification points...)
B. Tame ramification is completely understood: given (h, u) and a tame prime p, the partition of n measuring p-adic ramification in F h,u is given by a universal braid group formula. Wild ramification experimentally is subject to strong upper bounds. For example, for degree 5 fields of discriminant ± a b 5 c, the locally allowed maxima are (a, b, c) = (0, 6, 7). The largest exponents occurring in the h = (S 5, (, 5), (, )) family are (79, 57, 5). In larger degree, the bounds are much stronger. Open problem: get formulas for wild ramification in terms of (h, u) and establish the upper bounds. C. Specialization has been observed to be near generic to start with and become more generic in higher degrees. Open problem: control specialization enough to prove the expected lim sup NF n (P) =!