Indivisibility of class numbers of imaginary quadratic fields

Beckwith, Olivia

doi:10.1186/s40687-017-0109-x

Cited by 9 publications

(8 citation statements)

References 38 publications

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“…Like the divisibility problem, it is quite useful to know the distribution of the real quadratic fields Q( √ D) for which h(D) is not divisible by a given integer g. In view of this, the question of indivisibility of class numbers of quadratic fields draws attention of mathematicians as this is closely related to the Gauss class number 1 conjecture (cf. [2], [3], [7], [11], [16], [17]). As in the problem of the simultaneous divisibility of class numbers, Byeon [3] addressed the question of simultaneous 3-indivisibility of class numbers of quadratic fields and proved the following.…”

Section: Introductionmentioning

confidence: 99%

On the simultaneous 3-divisibility of class numbers of triples of imaginary quadratic fields

Chattopadhyay

Muthukrishnan

2021

Acta Arith.

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For a square-free integer t, Byeon [3] proved the existence of infinitely many pairs of quadratic fields Q(√ D) and Q(√ tD) with D > 0 such that the class numbers of all of them are indivisible by 3. In the same spirit, we prove that for a given integer t ≥ 1 with t ≡ 0 (mod 4), a positive proportion of fundamental discriminants D > 0 exist for which the class numbers of both the real quadratic fields Q(√ D) and Q(√ D + t) are indivisible by 3. This also addresses the complement of a weak form of a conjecture of Iizuka in [8]. As an application of our main result, we obtain that for any integer t ≥ 1 with t ≡ 0 (mod 12), there are infinitely many pairs of real quadratic fields Q(√ D) and Q(√ D + t) such that the Iwasawa λ-invariants associated with the basic Z3-extensions of both Q(√ D) and Q(√ D + t) are 0. For p = 3, this supports Greenberg's conjecture which asserts that λp(K) = 0 for any prime number p and any totally real number field K.

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Section: Introductionmentioning

confidence: 99%

On the simultaneous 3-divisibility of class numbers of triples of imaginary quadratic fields

Chattopadhyay

Muthukrishnan

2021

Acta Arith.

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“…The only proven cases of the above conjecture are when n = 3, due to Bhargava [2]. In this paper, we provide evidence toward all cases of Conjecture 1 by computing the average size of the 2-torsion subgroups of ideal class groups of certain infinite families of number fields of fixed odd degree n; even though we do not average over the family of all number fields of a given signature ordered by discriminant, the mean values coincide with (1), conditional on a certain tail estimate. Unconditionally, we prove that an infinite number of odd degree n S n -fields with signature (r 1 , r 2 ) have odd class number.…”

Section: Introductionmentioning

confidence: 93%

“…Jochnowitz [26] also used such methods to generalize the results of [23,24,25] to the real quadratic case. The most general result was obtained by Wiles [37] and Beckwith [1] using trace formula methods in conjunction with the geometry of Shimura curves and the theory of mock modular forms of half-integer weight, respectively. Applications of such results include unconditional versions of modularity lifting theorems in the residually reducible case [33] as well as the nonvanishing of certain L-values associated to elliptic curves with rational torsion points [35].…”

Section: Introductionmentioning

confidence: 99%

“…Fix an odd integer n ≥ 3 and a pair of nonnegative integers (r 1 , r 2 ) such that r 1 + 2r 2 = n. Consider the set of isomorphism classes of degree n S nnumber fields with signature (r 1 , r 2 ). The average number of 2-torsion elements in the ideal class groups of such fields is 1 + 2 1−r 1 −r 2 (1) when these fields are ordered by discriminant.…”

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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“…Note that this theorem gives an estimate for odd order torsion as well, since a group with an element of order 2k must also have an element of order k. This theorem gives the strongest estimate for class number divisibility for most g. Gauss' genus theory shows that 2-torsion subgroup of Cl(D) has size 2 w(D) , where w(D) is the number of distinct prime factors of D. Furthermore, [18] gives a lower of estimate of X/ log X for the number of imaginary quadratic fields with an element of order 4 in their class group. For 3-torsion, Heath-Brown [12] proved the lower bound of X In her PhD thesis [2], the author studied the indivisibility of class groups for imaginary quadratic fields satisfying arbitrary local conditions, refining and quantifying a recent theorem of Wiles [21]. The theorem of that paper was analogous to the work of Horie and Nakagawa [15], who extended a famous theorem of Davenport and Heilbronn [6] to count imaginary quadratic fields satisfying arbitrary local conditions at primes and with trivial 3-torsion in their class group.…”

Section: Introductionmentioning

confidence: 99%

Class number divisibility for imaginary quadratic fields

Beckwith

2020

Res. number theory

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In this note we revisit classic work of Soundararajan on class groups of imaginary quadratic fields. Let A, B, g ≥ 3 be positive integers such that gcd(A, B) is square-free. We refine Soundararajan's result to show that if 4 ∤ g or if A and B satisfy certain conditions, then the number of negative square-free D ≡ A (mod B) down to −X such that the ideal class group of Q( √ D) contains an element of order g is bounded below by X 1 2 +ǫ(g)−ǫ , where the exponent is the same as in Soundararajan's theorem. Combining this with a theorem of Frey, we give a lower bound for the number of quadratic twists of certain elliptic curves with p-Selmer group of rank at least 2, where p ∈ {3, 5, 7}. 9 10 −ǫ .

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Indivisibility of class numbers of imaginary quadratic fields

Cited by 9 publications

References 38 publications

On the simultaneous 3-divisibility of class numbers of triples of imaginary quadratic fields

On the simultaneous 3-divisibility of class numbers of triples of imaginary quadratic fields

Odd degree number fields with odd class number

Class number divisibility for imaginary quadratic fields

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