On ℓ-torsion in class groups of number fields

Abstract

For each integer ℓ ≥ 1, we prove an unconditional upper bound on the size of the ℓ-torsion subgroup of the class group, which holds for all but a zero-density set of field extensions of Q of degree d, for any fixed d ∈ {2, 3, 4, 5} (with the additional restriction in the case d = 4 that the field be non-D 4 ). For sufficiently large ℓ (specified explicitly), these results are as strong as a previously known bound that is conditional on GRH. As part of our argument, we develop a probabilistic "Chebyshev sieve," and give uniform, power-saving error terms for the asymptotics of quartic (non-D 4 ) and quintic fields with chosen splitting types at a finite set of primes.for every ε > 0. (Throughout, ε > 0 is allowed to be arbitrarily small (possibly taking a different value in different occurrences), and A ≪ B indicates that |A| ≤ cB for an implied constant c, which we allow in any instance to depend on ℓ, d, ε.)It is conjectured thatfor every ε > 0, but improving on the trivial bound (1.1) has proved difficult. (Impetus for this conjecture may be found in Duke [Duk98], Zhang [Zha05, page 10], and Brumer and Silverman [BS96, "Question CL(ℓ, d)"].) For K quadratic, Gauss's genus theory [Gau01] implies (1.2) in the case ℓ = 2. Recently, [BST + 17] obtained nontrivial upper bounds for 2-torsion in fields of degree d for all d ≥ 3, proving |Cl K [2]| ≪ D 0.2784...+ε K for d = 3, 4 and |Cl K [2]| ≪ D 1/2−1/2d+ε K for d ≥ 5. For ℓ = 3, after initial incremental improvement in [HV06], [Pie05], [Pie06] over the trivial bound (1.1) for quadratic fields, Ellenberg and Venkatesh proved [EV07, Prop. 3.4, Cor. 3.7] that (1.3) |Cl K [3]| ≪ D 1/3+ε K 2010 Mathematics Subject Classification. 11R29, 11N36 11R45 .

Levels of distribution for sieve problems in prehomogeneous vector spaces

“…proving (19). With Z := qX −1/d+η this simplifies to X In what follows we will choose Z = Z(q) := qX −1/d+η for a fixed small η > 0 so as to guarantee (20), so…”

“…For each squarefree integer q, we require estimates for the number of x ∈ A(X) bad at each prime dividing q. These may be obtained via the geometry of numbers (see, among other references, [17,1,3,9,13,19]) or using Shintani zeta functions [27], and we develop a third (simpler) method here.…”

Genus theory and ε-conjectures on p-class groups

“…Cl= [9,3,3,3,3,3] We publish, for any further examples, a program computing all the cyclic cubic fields such that there are exceptional 3-classes; so in most cases the 3-rank is N (instead of N − 1) and many examples have a non-trivial 9-rank. 9)==3,a=-a); P=x^3-f/3*x-f*a/27);N=omega(f);L=List;K=bnfinit(P,1);C8=component(K,8); h=component(component(C8,1),1);Cl=component(component(C8,1),2); dim=component(matsize(Cl),2);rho=0;for(i=1,dim,z=component(Cl,i); v=valuation(z,3);if(v>=1,rho=rho+1;listput(L,3^v)));if(rho>=N, print(" f=",f," N=",N," P=",P," Cl=",L)))))} f=657 N=2 P=x^3-219*x-1241 Cl= [3,3] f=657 N=2 P=x^3-219*x+730…”

“…Computation of C K,p for imaginary quadratic fields. We shall illustrate the cases p = 2, then p = 3, in various intervals of negative discriminants to observe the local decreasing of the variable C K,p (9). Note that for p > 3, the group W K is trivial contrary to the cases p = 2 and 3 where W K may be Z/pZ, which must probably be discarded in our considerations.…”

Linnik’s problems and maximal entropy methods

“…Theorem 7.4 can be proven by the same method as Theorem 2.3. Such a bound however is only known for powers of two and conjectured otherwise (see for instance [16]).…”