We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In this paper, we study the common index divisor $I(\mathbb K_m)$ and determine explicitly the prime ideal decomposition for any prime number in any simplest quartic number fields $\mathbb{K}_m$. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant $\leq x$ and given index.
The Thue-Siegel method is applied to derive an upper bound for the number of solutions to Thue's equation $F(x,y) = 1$ where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in this argument to handle forms whose discriminant is small in absolute value. We then apply our results to bound the number of integral points on a certain family of elliptic curves.
We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the literature (see Cohen--Diaz y Diaz--Olivier for the statement without proof and see Klüners for a proof), but a power-saving for the error term has not been explicitly determined except in the case that the base field is $\mathbb{Q}$. In this article, we describe the argument for obtaining both the explicit main term and a power-saving error term for the number of $…
The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.
We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields with imaginary quadratic fields. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.
We show that an order in a quartic field has fewer than $3000$ essentially different generators as a $\mathbb Z$-algebra (and fewer than $200$ if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of $2^{72}$. Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most $10$ classes of integral binary quartic forms (and at most $7$ if the discriminant is sufficiently large). This significantly improves the previously best known bound of $2^{80}$.
Tate first proposed a method to determine $K_2\mathcal{O}_F,$ the tame kernel of $F,$ and gave the concrete computations for some special quadratic fields with small discriminant. After that, many examples for quadratic fields with larger discriminants are given, and similar works also have been done for cubic fields and for some special quartic fields with discriminants not large. In the present paper, we investigate the case of more general imaginary cyclic quartic field $F=\mathbb{Q}\Big(\sqrt{-(D+B\sqrt{D})}\Big)$ with class number one and large discriminants. The key problem is how to dec…
Let $K$ be a field of char $K\neq 2$. For $a\in K$, we give an explicit answer to the field isomorphism problem of the simplest quartic polynomial $X^4-aX^3-6X^2+aX+1$ over $K$ as the special case of the field intersection problem via multi-resolvent polynomials. From this result, over an infinite field $K$, we see that the polynomial gives the same splitting field over $K$ for infinitely many values $a$ of $K$. We also see by Siegel's theorem for curves of genus zero that only finitely many algebraic integers $a\in\mathcal{O}_K$ in a number field $K$ may give the same splitting field. By appl…
It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $ξ$ of the polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\neq 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in …
We prove that the number of quartic $S_4$--extensions of the rationals of given discriminant $d$ is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$. For a prime number $p$ we derive that the dimension of the space of octahedral modular forms of weight 1 and conductor $p$ or $p^2$ is bounded above by $O(p^{1/2}\log(p)^2)$.
We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the joint cuspidal equidistribution of the shape of quartic fields paired with the shape of its cubic resolvent, when the fields are ordered by discriminant. Our estimate saves a small power in the corresponding Weyl sums.
Let $\ell>2$ be a positive integer, $ζ_\ell$ a primitive $\ell$-th root of unity, and $K$ a number field containing $ζ_\ell+ζ_\ell^{-1}$ but not $ζ_\ell$. In a recent paper, Chonoles et. al. study iterated towers of number fields over $K$ generated by the generalized Rikuna polynomial, $r_n(x,t;\ell) \in K(t)[x]$. They note that when $K = \mathbb{Q}$, $t \in \{0,1\}$, and $\ell=3$, the only ramified prime in the resulting tower is 3, and they ask under what conditions is the number of ramified primes small. In this paper, we apply a theorem of Guàrdia, Montes, and Nart to derive a formula for …
For $n=3$, 4, and 5, we prove that, when $S_n$-number fields of degree $n$ are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices.
We give an asymptotic formula for the number of $D_4$ quartic extensions of a function field with discriminant equal to some bound, essentially reproducing the analogous result over number fields due Cohen, Diaz y Diaz, and Olivier, but with a stronger error term. We also study the relative density of $D_4$ and $S_4$ quartic extensions of a function field and show that with mild conditions, the number of $D_4$ quartic extensions can far exceed the number of $S_4$ quartic extensions
We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic ri…
A standard formula (1) leads to a proof of HT90, but requires proving the existence of $θ$ such that $α\ne 0$, so that $β=α/σ(α)$. We instead impose the condition (M), that taking $θ=1$ makes $α=0$. Taking $n=3$, we recover Shanks's simplest cubic fields. The "simplest" number fields of degrees 3 to 6, Washington's cyclic quartic fields, and a certain family of totally real cyclic extensions of $\Q(\cos(π/4m))$ all have defining polynomials whose zeroes satisfy \eqref{e:M}. Further investigation of (M) for $n=4$ leads to an elementary algebraic construction of a 2-parameter family of octic pol…
In this paper, we consider maximal and irreducible quartic orders which arise from integral binary quartic forms, via the construction of Birch and Merriman, and whose field of fractions is a quartic $D_4$-field. By a theorem of M. Wood, such quartic orders may be regarded as quartic $D_4$-fields whose ring of integers has a monogenic cubic resolvent. We shall give the asymptotic number of such objects when ordered by conductor, as well as estimate the asymptotic number of such objects when ordered by discriminant.
We consider families of number fields of degree 4 whose normal closures over $\mathbb{Q}$ have Galois group isomorphic to $D_4$, the symmetries of a square. To any such field $L$, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image $D_4$. We determine the asymptotic number of such quartic $D_4$-fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes…
Using elementary number theory we study Diophantine equations over the rational integers of the following form, $y^2=(x+a)(x+a+k)(x+b)(x+b+k)$, $y^2=c^2x^4+ax^2+b$ and $y^2=(x^2-1)(x^2-α^2)(x^2-(α+1)^2).$ We express their integer solutions by means of the divisors of the discriminant of $f(x),$ where $y^2=f(x)$.
When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S_4, while the remaining 17% have Galois group D_4. We study these proportions over a general number field F. We find that asymptotically 100% of quadratic number fields have more D_4 extensions than S_4 and that the ratio between the number of D_4 and S_4 quartic extensions is biased arbitrarily in favor of D_4 extensions. Under GRH, we give a lower bound that holds for general number fields.
We determine the shapes of all degree $4$ number fields that are Galois. These lie in four infinite families depending on the Galois group and the tame versus wild ramification of the field. In the $V_4$ case, each family is a two-dimensional space of orthorhombic lattices and we show that the shapes are equidistributed, in a regularized sense, in these spaces as the discriminant goes to infinity (with respect to natural measures). We also show that the shape is a complete invariant in some natural families of $V_4$-quartic fields. For $C_4$-quartic fields, each family is a one-dimensional spa…
It is shown that the quartic Fermat equation $x^4 +y^4=1$ has nontrivial integral solutions in the Hilbert class field $Σ$ of any quadratic field $K=\mathbb{Q}(\sqrt{-d})$ whose discriminant satisfies $-d \equiv 1$ (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in $K=\mathbb{Q}(\sqrt{-p})$, for $p$ $( > 7)$ a prime congruent to $7$ (mod 8), but does have a nontrivial solution in the odd degree extension $Σ$ of $K$. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in deta…
Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types of polynomials modulo primes, and cycle types of the Galois groups of polynomials. One remarkable example is the removal of all artificial constraints from the Kummer-Dedekind Theorem that relates splitting and factorization patterns. Finally, we present an elementary proof that the discriminant of the splitting field of a monic irreducible polynomial with i…
In 2012 the first named author conjectured that totally real quartic fields of fundamental discriminant are determined by the isometry class of the integral trace zero form; such conjecture was based on computational evidence and the analog statement for cubic fields which was proved using Bhargava's higher composition laws on cubes. Here, using Bhargava's parametrization of quartic fields we prove the conjecture by generalizing the ideas used in the cubic case. Since at the moment, for arbitrary degrees, there is nothing like Bhargava's parametrizations we cannot deal with degrees $n > 5$ in …
For a binary quadratic form $Q$, we consider the action of $\mathrm{SO}_Q$ on a two-dimensional vector space. This representation yields perhaps the simplest nontrivial example of a prehomogeneous vector space that is not irreducible, and of a coregular space whose underlying group is not semisimple. We show that the nondegenerate integer orbits of this representation are in natural bijection with orders in cubic fields having a fixed "lattice shape". Moreover, this correspondence is discriminant-preserving: the value of the invariant polynomial of an element in this representation agrees with…
Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The …
Let $K$ be a number field. The \textit{integral trace form} is the integral quadratic form given by $\text{tr}_{K/\mathbb{Q}}(x^2)|_{O_{K}}.$ In this article we study the existence of non-conjugated number fields with equivalent integral trace forms. As a corollary of one of the main results of this paper, we show that any two non-totally real number fields with the same signature and same prime discriminant have equivalent integral trace forms. Additionally, based on previous results obtained by the author and the evidence presented here, we conjecture that any two totally real quartic fields…
We compute the higher ramification groups and the Artin conductors of radical extensions of the rationals. As an application, we give formulas for their discriminant (using the conductor-discriminant formula). The interest in such number fields is due to the fact that they are among the simplest non-abelian extensions of the rationals (and so not classified by Class Field Theory). We show that this extensions have non integer jumps in the superior ramification groups, contrarily to the case of abelian extensions (as prescribed by Hasse-Arf theorem).
We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually can not hold, which implies the non-monogenity of the fields. The fields that we consider are higher degree number fields, o…
In this paper we prove an explicit formula for the arithmetic intersection number (CM(K).G1)_{\ell} on the Siegel moduli space of abelian surfaces, generalizing the work of Bruinier-Yang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number (CM(K).G1)_{\ell} under str…
The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the denominators of the discriminants of the Igusa class polynomials all have the property 1) that they are bounded by d, the absolute value of the discriminant of K, and 2) that they divide d-x^2, for some integer x whose square is less than d. A slightly stronger condition is given in Section 3. Such primes are primes of bad reduction for the genus 2 curve and primes of …
Let $K$ be an imaginary quadratic field of discriminant less than or equal to -7 and $K_{(N)}$ be its ray class field modulo $N$ for an integer $N$ greater than 1. We prove that singular values of certain Siegel functions generate $K_{(N)}$ over $K$ by extending the idea of our previous work. These generators are not only the simplest ones conjectured by Schertz, but also quite useful in the matter of computation of class polynomials. We indeed give an algorithm to find all conjugates of such generators by virtue of Gee and Stevenhagen.
Let $N(5,D_5,X)$ be the number of quintic number fields whose Galois closure has Galois group $D_5$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect that $N(5,D_5,X) \sim C X^{1/2}$ for some constant $C$. The best known upper bound is $N(5,D_5,X)\ll X^{3/4 + ε}$, and we show this could be improved by counting points on a certain variety defined by a norm equation; computer calculations give strong evidence that this number is $\ll X^{2/3}$. Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for $A_4$ q…
We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields $K$ (the trivial bound being $O_ε(|{\rm Disc}(K)|^{1/2+ε})$ by Brauer--Siegel). This yields corresponding improvements to: 1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves; 2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves; 3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves; and 4) bounds of Baily and Wong on the number of $A_4$-quarti…
For a quartic primitive CM field $K$, we say that a rational prime $p$ is {\it evil} if at least one of the abelian varieties with CM by $K$ reduces modulo a prime ideal $\gerp| p$ to a product of supersingular elliptic curves with the product polarization. We call such primes {\it evil primes for $K$}. In \cite{GL}, we showed that for fixed $K$, such primes are bounded by a quantity related to the discriminant of the field $K$. In this paper, we show that evil primes are ubiquitous in the sense that, for any rational prime $p$, there are an infinite number of fields $K$ for which $p$ is evil …
