Cubic Fields with Geometry

Abstract

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On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

“…)-many points P m,n = (12m, 108n) in E D ′ (Q), each one corresponding to an unramified cubic extension of K D , verifying at the same time Theorem 4.1 in [13] which states that the number of non-conjugate cubic fields of discriminant D is…”

“…Denote by µ m,n the element µ m,n = 27n+ is a virtual 3-unit since C m,n (x, 1) has integer coefficients. Basically each of our F m,n is the standard form of the corresponding C m,n (x, 1) (following the definitions of [13,Chapter 4]). Also, C m,n (x, 1) is irreducible because F m,n is.…”

“…There is a well defined map Φ (see for example [13,Section 4.5] or [17]), from the set of triples of conjugate fields of discriminant D or −27D ′ , to the set of ideal classes of order 3 in CL(K D ′ ) together with the trivial class. The trivial class is generated by the fundamental unit of O K D ′ .…”

“…The trivial class is generated by the fundamental unit of O K D ′ . In the case D < −4, [13, Theorem 7], the principal class always has a unique pre-image under Φ whereas, for D > 1 there is no contribution from the trivial class [13,Theorem 6]. In the case that D < −4 and r 3 (D ′ ) denotes the 3-rank of the ideal class group of the real quadratic field K D ′ , the resolvent of the imaginary field K D , the classical Reflection Theorem of Scholz says that r 3 (D ′ ) ≤ r 3 (D) ≤ r 3 (D ′ ) + 1.…”

Normal integral bases and Gaussian periods in the simplest cubic fields

“…We write ∆ n = n 2 + 3n + 9 = bc 3 with b, c ∈ Z >0 where b is cube-free, and furthermore b = de 2 with d, e ∈ Z >0 where d and e are square-free and (d, e) = 1. We use the following lemma in the book [9] by Hambleton and Williams (see also [2], [6] and [16]). Lemma 4.…”