The 2‐Selmer group of a number field and heuristics for narrow class groups and signature ranks of units

Abstract

We investigate in detail a homomorphism which we call the 2-Selmer signature map from the 2-Selmer group of a number field K to a nondegenerate symmetric space, in particular proving the image is a maximal totally isotropic subspace. Applications include precise predictions on the density of fields K with given narrow class group 2-rank and with given unit group signature rank. In addition to theoretical evidence, extensive computations for totally real cubic and quintic fields are presented that match the predictions extremely well. In an Appendix with Richard Foote, we classify the maximal totally isotropic subspaces of orthogonal direct sums of two nondegenerate symmetric spaces over perfect fields of characteristic 2 and derive some consequences, including a mass formula for such subspaces. Contents

Signature ranks of units in cyclotomic extensions of abelian number fields

“…Computations suggest that the signature rank of the units in the real subfield of the cyclotomic field of m-th roots of unity is in fact always close to the maximal possible rank of ϕ(m)/2 (equivalently, the unit signature rank deficiency for these fields should be close to 0), i.e., nearly all possible signature types arise for units. This is in keeping with the heuristics in [4] suggesting that 'most' totally real fields have nearly maximal unit signature rank (although these abelian extensions are hardly 'typical').…”

Bounds of the rank of the Mordell–Weil group of Jacobians of Hyperelliptic Curves

“…We conclude this section quoting a conjectural density of cubic fields and quintic fields with maximal ρ ∞ , which is part of a broader conjecture of Dummit and Voight (see [5]).…”

“…We refer the reader to [5] for heuristics and conjectures about the dimension of the totally positive units (in particular, the conjecture on page 4). In the following theorem, we use the notation of [5].…”

“…For part (2), see Section 2 of [5], and in particular Equation (2.9). Part (3) is shown in [1], where it is shown that ρ + − ρ ≤ r 1 /2 , where r 1 is the number of real embeddings of L. Here r 1 = p is an odd prime, so the proof is concluded.…”

A Note on the Equivalence of the Parity of Class Numbers and the Signature Ranks of Units in Cyclotomic Fields

“…There are units of K + of every possible signature if and only if [E : E + ] = 2 ϕ(p n )/2 , which is equivalent to [E + : E 2 ] = 1 since 2 ϕ(p n )/2 = [E : E 2 ] = [E : E + ][E + : E 2 ], so conditions (b2) and (b3) are equivalent. Then |Cl st (K + )| = |Cl(K + )|[E + : E 2 ] (for additional details, see [6,Section 2]) shows both that (b1) is equivalent to (b2) and that (4) and (5) (in the version (a1) and (b2)) are equivalent.…”