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Register a new userA Cubic analogue of the Friedlander-Iwaniec spin over primes
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Jori Merikoski
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In 1998 Friedlander and Iwaniec proved that there are infinitely many primes of the form $a^2+b^4$. To show this they defined the spin of Gaussian integers by the Jacobi symbol, and one of the key ingredients in the proof was to show that the spin becomes equidistributed along Gaussian primes. To generalize this we define the cubic spin of ideals of $mathbb{Z}[zeta_{12}]=mathbb{Z}[zeta_3,i]$ by using the cubic residue character on the Eisenstein integers $mathbb{Z}[zeta_3]$. Our main theorem says that the cubic spin is equidistributed along prime ideals of $mathbb{Z}[zeta_{12}]$. The proof of this follows closely along the lines of Friedlander and Iwaniec. The main new feature in our case is the infinite unit group, which means that we need to show that the definition of the cubic spin on the ring of integers lifts to a well-defined function on the ideals. We also explain how the cubic spin arises if we consider primes of the form $a^2+b^6$ on the Eisenstein integers.
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Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the action of the Hecke operators. The goal of that paper was to test the modularity of elliptic curves over F. In the present paper, we complement and extend this prior work in two ways. First, we tabulate more elliptic curves than were found in our prior work by using various heuristics (old and new cohomology classes, dimensions of Eisenstein subspaces) to predict the existence of elliptic curves of various conductors, and then by using more sophisticated search techniques (for instance, torsion subgroups, twisting, and the Cremona-Lingham algorithm) to find them. We then compute further invariants of these curves, such as their rank and representatives of all isogeny classes. Our enumeration includes conjecturally the first elliptic curves of ranks 1 and 2 over this field, which occur at levels of norm 719 and 9173 respectively.
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Let ${mathbb F}_q$ be the finite field with $q=p^k$ elements with $p$ being a prime and $k$ be a positive integer. For any $y, zinmathbb{F}_q$, let $N_s(z)$ and $T_s(y)$ denote the numbers of zeros of $x_1^{3}+cdots+x_s^3=z$ and $x_1^3+cdots+x_{s-1}^3+yx_s^3=0$, respectively. Gauss proved that if $q=p, pequiv1pmod3$ and $y$ is non-cubic, then $T_3(y)=p^2+frac{1}{2}(p-1)(-c+9d)$, where $c$ and $d$ are uniquely determined by $4p=c^2+27d^2,~cequiv 1 pmod 3$ except for the sign of $d$. In 1978, Chowla, Cowles and Cowles determined the sign of $d$ for the case of $2$ being a non-cubic element of ${mathbb F}_p$. But the sign problem is kept open for the remaining case of $2$ being cubic in ${mathbb F}_p$. In this paper, we solve this sign problem by determining the sign of $d$ when $2$ is cubic in ${mathbb F}_p$. Furthermore, we show that the generating functions $sum_{s=1}^{infty} N_{s}(z) x^{s}$ and $sum_{s=1}^{infty} T_{s}(y)x^{s}$ are rational functions for any $z, yinmathbb F_q^*:=mathbb F_qsetminus {0}$ with $y$ being non-cubic over ${mathbb F}_q$ and also give their explicit expressions. This extends the theorem of Myerson and that of Chowla, Cowles and Cowles.
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