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Some analogue of Hilbert's irreducibility theorem and the distribution of algebraic number fields
... for p = 2; also, let λ 1 (2) = 1 and λ n (2) = 1/2 for n ≥ 2. Then a result of Yamamura [26,Proposition 3] states that λ n ( p) is the density of monic polynomials of degree n over Z p having discriminant indivisible by p 2 . Let λ n := p λ n ( p), where the product is over all primes p. ...
... We now estimate (35). The Q-invariant can also be given a weight by viewing the torus T as sitting inside G 0 and using (26). Namely, ...
We determine the density of monic integer polynomials of given degree that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials f(x), such that f(x) is irreducible and is the ring of integers in its fraction field, is positive, and is in fact given by . It also follows from our methods that there are monogenic number fields of degree n having associated Galois group and absolute discriminant less than X, and we conjecture that the exponent in this lower bound is optimal.
We study an infinite family of j-invariant zero elliptic curves and their -isogenous curves , where D and are fundamental discriminants of a specific form, and is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field always has non-trivial 3-class group. We prove a series of results related to the set of rational points , and the -equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant D=48035713, and we show that every curve in these classes should violate the Hasse Principle.
For arbitrary with we denote by the discriminants which are squarefree, and we define the family of elliptic curves . These curves admit a rational 3-isogeny . In this paper we show that the rank of the -Selmer group of and of the -isogenous curves have specific values related to the 3-rank of the ideal class group of the quadratic field . Employing a known result on the parity of \Sha[E_{D}], we obtain that the rank of these curves is bounded below by 1 for and by 2 for . Finally, we give an explicit, infinite subfamily of curves with , where p are primes of a specific form.
In this paper, we determine all the imaginary abelian number fields with class number one. There exist exactly 172 imaginary abelian number fields with class number one. The maximal conductor of these fields is 10921 = 67 · 163, which is the conductor of the biquadratic number field Q(√-67, √-163).
The purpose of this article is to describe simple ways to construct quadratic number fields each having an unramified extension which properly contains the Hilbert class field of its genus field (in the wide sense). The motivation of this study is the author’s observation that under the Generalized Riemann Hypothesis (GRH), for most quadratic number fields of small conductors, their maximal unramified extensions coincide with the Hilbert class fields of their genus fields. More precisely, under GRH, among the 305 imaginary quadratic number fields with discriminants larger than —1000, at most 16 fields are exceptional [39], [40], and among the 1690 real quadratic number fields with discriminants less than or equal to 5565, only 4 fields are exceptional [41].
The maximal unramified extensions of the imaginary quadratic number fields with class number two are determined explicitly under the Generalized Riemann Hypothesis.
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