We give two constructions of families of elliptic curves over cubic or quartic fields with three, respectively four, `integral' elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements linearly independent. For their integrality, we prove a new criterion that does not ignore any torsion. We also verify Beilinson's conjecture numerically for just over 90 of the curves.
Shifted moments of cubic and quartic Dirichlet $L$-functions
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）
We study the $2k$-th moment of central values of the family of primitive cubic and quartic Dirichlet $L$-functions. We establish sharp lower bounds for all real $k \geq 1/2$ unconditionally for the cubic case and under the Lindelöf hypothesis for the quartic case. We also establish sharp lower bounds for all real $0 \leq k<1/2$ and sharp upper bounds for all real $k \geq 0$ for both the cubic and quartic cases under the generalized Riemann hypothesis (GRH). As an application of our results, we establish quantitative non-vanishing results for the corresponding $L$-values.
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 are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively.
Large values of cubic and quartic Dirichlet $L$-functions of prime moduli
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）
We show that there are $O_\varepsilon(H^{1.5+\varepsilon})$ monic, cubic polynomials with integer coefficients bounded by $H$ in absolute value whose Galois group is $A_3$. We also show that the order of magnitude for $D_4$ quartics is $H^2 (\log H)^2$, and that the respective counts for $A_4$, $V_4$, $C_4$ are $O(H^{2.91})$, $O(H^2 \log H)$, $O(H^2 \log H)$. Our work establishes that irreducible non-$S_3$ cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden.
We prove that every elliptic curve defined over a totally real number field of degree 4 not containing $\sqrt{5}$ is modular. To this end, we study the quartic points on four modular curves.
Dirichlet series associated to quartic fields with given resolvent
设$k$为三次域。本文给出了狄利克雷级数$\sum_K|\Disc(K)|^{-s}$的一个显式公式，这里的和是在所有三次预解式域同构于$k$的四次域的同构类上.我们的工作是Cohen、Diaz y Diaz和Olivier未发表预印本的续集，我们包含了他们结果的完整证明，以免依赖未发表的工作。这是一个同伴以前的文件，我们计算的狄利克雷级数相关的立方领域有一个给定的二次预解式。
We study the behavior under twisting of the Selmer rank parities of a self-dual prime-degree isogeny on a principally polarized abelian variety defined over a number field, subject to compatibility relations between the twists and the isogeny. In particular, we study isogenies on abelian varieties whose Selmer rank parities are related to the rank parities of elliptic curves with $j$-invariant 0 or 1728, assuming the Shafarevich-Tate conjecture. Using these results, we show how to classify the conjectural rank parities of all quartic or sextic twists of an elliptic curve defined over a number …
We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the way, we also prove that a positive proportion of quartic rings of integers do not arise as the invariant order of an integral binary quartic form despite having no local obstruction.
Let $C$ be a smooth genus one curve described by a quartic polynomial equation over the rational field $\mathbb Q$ with $P\in C(\mathbb Q)$. We give an explicit criterion for the divisibility-by-$2$ of a rational point on the elliptic curve $(C,P)$. This provides an analogue to the classical criterion of the divisibility-by-$2$ on elliptic curves described by Weierstrass equations. We employ this criterion to investigate the question of extending a rational $D(q)$-quadruple to a quintuple. We give concrete examples to which we can give an affirmative answer. One of these results implies that a…
We propose three kinds of explicit formulas for the elliptic lambda function by the elliptic modular function. Further, we derive incredible cubic identities as a corollary of our explicit formulas and evaluate some singular values of the elliptic lambda function explicitly.
