For every odd integer n ≥ 3, we prove that there exist infinitely many number fields of degree n and associated Galois group S n whose class number is odd. To do so, we study the class groups of families of number fields of degree n whose rings of integers arise as the coordinate rings of the subschemes of P 1 cut out by integral binary n-ic forms. By obtaining upper bounds on the mean number of 2-torsion elements in the class groups of fields in these families, we prove that a positive proportion (tending to 1 as n tends to ∞) of such fields have trivial 2-torsion subgroup in their class groups and narrow class groups. Conditional on a tail estimate, we also prove the corresponding lower bounds and obtain the exact values of these averages, which are consistent with the heuristics of Cohen-Lenstra-Martinet-Malle and Dummit-Voight.Additionally, for any order O f of degree n arising from an integral binary n-ic form f , we compare the sizes of Cl 2 (O f ), the 2-torsion subgroup of ideal classes in O f , and I 2 (O f ), the 2-torsion subgroup of ideals in O f . For the family of orders arising from integral binary n-ic forms and contained in fields with fixed signature (r 1 , r 2 ), we prove that the mean value of the difference |Cl 2 (O f )| − 2 1−r1−r2 |I 2 (O f )| is equal to 1, generalizing a result of Bhargava and the third-named author for cubic fields. Conditional on certain tail estimates, we also prove that the mean value of |Cl 2 (O f )| − 2 1−r1−r2 |I 2 (O f )| remains 1 for certain families obtained by imposing local splitting and maximality conditions.There is a height ordering on R H arising from the height ordering H on Sym n (Z 2 ), where H(f ) is defined as the maximum absolute value of the coefficients of f . Note that although two rings in R H may be isomorphic, their heights need not be equal. For example, if γ ∈ SL 2 (Z), and we define the action γf (x, y) := f ((x, y)γ) on the space of integral binary n-ic forms, then it is always true thatSuch orbits [f ] may be ordered by their Julia invariant, which is an invariant defined in [27] for the action of SL 2 (Z) on Sym n (Z 2 ) (see §3.3 for details). Thus, we also define the family R J to be the multiset of ringsordered by Julia invariant J, where J(R [f ] ) := J([f ]). Asymptotics on the size of R J were obtained by Bhargava-Yang [13].In this paper, we compute averages taken over certain families contained in R H or R J . Let R r 1 ,r 2 H ⊂ R H and R r 1 ,r 2 J ⊂ R J be the respective subfamilies consisting of all Gorenstein 1 integral domains whose fraction field has signature (r 1 , r 2 ), i.e., has r 1 real embeddings and r 2 pairs of conjugate complex embeddings. Also, let R r 1 ,r 2 H,max ⊂ R r 1 ,r 2 H (resp. R r 1 ,r 2 J,max ⊂ R r 1 ,r 2 J ) be the subfamily containing all maximal orders. It is worthwhile to note that a given order O in a number field with signature (r 1 , r 2 ) may occur in R r 1 ,r 2 H or R r 1 ,r 2 H,max an infinite number of times (up to isomorphism) but only occurs with finite multiplicity in R r 1 ,r 2 J or R r 1...

Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields, when they are ordered by their absolute discriminants.For an order O in a cubic field, we study the three groups: Cl 2 (O), the group of ideal classes of O of order 2; Cl + 2 (O), the group of narrow ideal classes of O of order 2; and I 2 (O), the group of ideals of O of order 2. We prove that the mean value of the difference |Cl 2 (O)| − 1 4 |I 2 (O)| is always equal to 1, regardless of whether one averages over the maximal orders in real cubic fields, over all orders in real cubic fields, or indeed over any family of real cubic orders defined by local conditions. For the narrow class group, we prove that the average value of the difference |Cl + 2 (O)| − |I 2 (O)| is equal to 1 for any such family. Also, for any family of complex cubic orders defined by local conditions, we prove similarly that the mean value of the difference |Cl 2 (O)| − 1 2 |I 2 (O)| is always equal to 1, independent of the family. The determination of these mean numbers allows us to prove a number of further results as by-products. Most notably, we prove-in stark contrast to the case of quadratic fields-that: 1) a positive proportion of cubic fields have odd class number; 2) a positive proportion of real cubic fields have isomorphic 2-torsion in the class group and the narrow class group; and 3) a positive proportion of real cubic fields contain units of mixed real signature. We also show that a positive proportion of real cubic fields have narrow class group strictly larger than the class group, and thus a positive proportion of real cubic fields do not possess units of every possible real signature.

We determine the mean number of 3-torsion elements in the class groups of quadratic orders, where the quadratic orders are ordered by their absolute discriminants. Moreover, for a quadratic order O we distinguish between the two groups: Cl 3 (O), the group of ideal classes of order 3; and I 3 (O), the group of ideals of order 3. We determine the mean values of both |Cl 3 (O)| and |I 3 (O)|, as O ranges over any family of orders defined by finitely many (or in suitable cases, even infinitely many) local conditions.As a consequence, we prove the surprising fact that the mean value of the difference |Cl 3 (O)|− |I 3 (O)| is equal to 1, regardless of whether one averages over the maximal orders in complex quadratic fields or over all orders in such fields or, indeed, over any family of complex quadratic orders defined by local conditions. For any family of real quadratic orders defined by local conditions, we prove similarly that the mean value of the difference |Cl 3 (O)| − 1 3 |I 3 (O)| is equal to 1, independent of the family.

We consider families of quartic number fields whose normal closures over \mathbb{Q} have Galois group isomorphic to D_4 , the symmetries of a square. To any such field L , one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image D_4 . We determine the asymptotic number of such D_4 -quartic fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes, at an infinite number of primes), and as a consequence, we also compute the asymptotic number of order-4 elements in class groups and narrow class groups of quadratic fields ordered by discriminant. Traditionally, there have been two approaches to counting quartic fields, using arithmetic invariant theory in combination with geometry-of-number techniques, and applying Kummer theory together with L -function methods. Both of these strategies fail in the case of D_4 -quartic fields ordered by conductor since counting quartic fields containing a quadratic subfield with large discriminant is difficult. However, when ordering by conductor, we utilize additional algebraic structure arising from the outer automorphism of D_4 combined with both approaches mentioned above to obtain exact asymptotics.

We determine the average number of 3-torsion elements in the ray class groups of fixed (integral) conductor c of quadratic fields ordered by absolute discriminant, generalizing Davenport and Heilbronn's theorem on class groups. A consequence of this result is that a positive proportion of such ray class groups of quadratic fields have trivial 3-torsion subgroup whenever the conductor c is taken to be a squarefree integer having very few prime factors none of which are congruent to 1 mod 3. Additionally, we compute the second main term for the number of 3-torsion elements in ray class groups with fixed conductor of quadratic fields with bounded discriminant.