Rings and ideals parameterized by binary <i>n</i>
-ic forms

Wood, Melanie Matchett

doi:10.1112/jlms/jdq074

Cited by 26 publications

(80 citation statements)

References 18 publications

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“…Let n ≥ 3 be a fixed odd integer. In this section, we begin by recalling from [29,38] how rings of rank n naturally arise from integral binary n-ic forms. We then recall the parametrization given in [39] of 2-torsion ideal classes in such rings by orbits of pairs of n-ary quadratic forms.…”

Section: Parametrizations Of 2-torsion Ideal Classes and Composition mentioning

confidence: 99%

“…For k > 0, define ζ k = f 0 θ k + · · · + f k−1 θ, and let ζ 0 = 1. It is shown in both [29] and [38] that R f = ζ 0 , . .…”

Section: Rings Associated To Binary N-ic Formsmentioning

confidence: 99%

“…T to X f (see [38,Theorem 2.4]). We are interested in counting the 2-torsion ideal classes of the rings R f associated to irreducible forms f when n is odd.…”

Section: Rings Associated To Binary N-ic Formsmentioning

confidence: 99%

“…We now give a short description of the organization of the paper. In Section 2, we recall and expand on the details of the construction of rings R f of rank n from binary n-ic forms f given in [29,38]. We also describe the correspondence given in [39] between SL n -orbits of pairs of n-ary quadratic forms and order 2 ideal classes of such rings R f .…”

Section: Introductionmentioning

confidence: 99%

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Odd degree number fields with odd class number

Ho¹,

Shankar²,

Varma³

2018

Duke Math. J.

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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...

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Section: Parametrizations Of 2-torsion Ideal Classes and Composition mentioning

confidence: 99%

“…For k > 0, define ζ k = f 0 θ k + · · · + f k−1 θ, and let ζ 0 = 1. It is shown in both [29] and [38] that R f = ζ 0 , . .…”

Section: Rings Associated To Binary N-ic Formsmentioning

confidence: 99%

“…T to X f (see [38,Theorem 2.4]). We are interested in counting the 2-torsion ideal classes of the rings R f associated to irreducible forms f when n is odd.…”

Section: Rings Associated To Binary N-ic Formsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Odd degree number fields with odd class number

Ho¹,

Shankar²,

Varma³

2018

Duke Math. J.

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“…Finally to complete our survey of the literature, let us mention that in the articles [13,14,15,16,17,18] equivalence classes of twisted binary cubics are parametrised by Galois extensions in the case of fields [14] or by cubic rings in the case of Z [15,16,17,18]. This is relevant to the present paper since, up to equivalence, certain types of algebras are completely determined by their associated twisted cubic.…”

Section: Introductionmentioning

confidence: 99%

Polynomial invariants and moduli of generic two-dimensional commutative algebras

Traubenberg

Slupinski

2018

Linear and Multilinear Algebra

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Let V be a two-dimensional vector space over a field F of characteristic not 2 or 3. We show there is a canonical surjection ν from the set of suitably generic commutative algebra structures on V modulo the action of GL(V ) onto the plane F 2 . In these coordinates, which are quotients of invariant quartic polynomials, properties such as associativity and the existence of zero divisors are described by simple algebraic conditions. The map ν is a bijection over the complement of a degenerate elliptic curve Γ and over Γ we give an explicit parametrisation of the fibre in terms of Galois extensions of F. Algebras in ν −1 (Γ) are exactly those which admit non-trivial automorphisms. We show how ν can be lifted to a map from the SL(V )−moduli space to an algebraic hypersurface Γ ′ in a four-dimensional vector space whose equation is essentially the classical Eisenstein equation for the covariants of a binary cubic. This map is the restriction of a surjective map from the set of stable commutative algebras on V modulo the action of SL(V ) onto Γ ′ .

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Arithmetic invariant theory

Bhargava

Gross

2014

Progress in Mathematics

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Rings and ideals parameterized by binary n -ic forms

Cited by 26 publications

References 18 publications

Odd degree number fields with odd class number

Odd degree number fields with odd class number

Polynomial invariants and moduli of generic two-dimensional commutative algebras

Arithmetic invariant theory

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