Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTrinomials, singular moduli and Riffauts conjecture
88
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Riffaut (2019) conjectured that a singular modulus of degree $hge 3$ cannot be a root of a trinomial with rational coefficients. We show that this conjecture follows from the GRH, and obtain partial unconditional results.
rate research
Read More
We prove that $|x-y|ge 800X^{-4}$, where $x$ and $y$ are distinct singular moduli of discriminants not exceeding $X$. We apply this result to the primitive element problem for two singular moduli. In a previous article Faye and Riffaut show that the number field $mathbb Q(x,y)$, generated by two singular moduli $x$ and $y$, is generated by $x-y$ and, with some exceptions, by $x+y$ as well. In this article we fix a rational number $alpha e0,pm1$ and show that the field $mathbb Q(x,y)$ is generated by $x+alpha y$, with a few exceptions occurring when $x$ and $y$ generate the same quadratic field over $mathbb Q$. Together with the above-mentioned result of Faye and Riffaut, this gives a drastic generalization of a theorem due to Allombert et al. (2015) about solution of linear equations in singular moduli.
Let d1 and d2 be discriminants of distinct quadratic imaginary orders O_d1 and O_d2 and let J(d1,d2) denote the product of differences of CM j-invariants with discriminants d1 and d2. In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer J(d1,d2) in the case that d1 and d2 are relatively prime and discriminants of maximal orders. To compute this formula, they first reduce the problem to counting the number of simultaneous embeddings of O_d1 and O_d2 into endomorphism rings of supersingular curves, and then solve this counting problem. Interestingly, this counting problem also appears when computing class polynomials for invariants of genus 2 curves. However, in this application, one must consider orders O_d1 and O_d2 that are non-maximal. Motivated by the application to genus 2 curves, we generalize the methods of Gross and Zagier and give a computable formula for v_p(J(d1,d2)) for any distinct pair of discriminants d1,d2 and any prime p>2. In the case that d1 is squarefree and d2 is the discriminant of any quadratic imaginary order, our formula can be stated in a simple closed form. We also give a conjectural closed formula when the conductors of d1 and d2 are relatively prime.
Let $f$ and $g$ be weakly holomorphic modular functions on $Gamma_0(N)$ with the trivial character. For an integer $d$, let $Tr_d(f)$ denote the modular trace of $f$ of index $d$. Let $r$ be a rational number equivalent to $iinfty$ under the action of $Gamma_0(4N)$. In this paper, we prove that, when $z$ goes radially to $r$, the limit $Q_{hat{H}(f)}(r)$ of the sum $H(f)(z) = sum_{d>0}Tr_d(f)e^{2pi idz}$ is a special value of a regularized twisted $L$-function defined by $Tr_d(f)$ for $dleq0$. It is proved that the regularized $L$-function is meromorphic on $mathbb{C}$ and satisfies a certain functional equation. Finally, under the assumption that $N$ is square free, we prove that if $Q_{hat{H}(f)}(r)=Q_{hat{H}(g)}(r)$ for all $r$ equivalent to $i infty$ under the action of $Gamma_0(4N)$, then $Tr_d(f)=Tr_d(g)$ for all integers $d$.
We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross-Kohnen-Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight 3/2. Moreover, we show that any harmonic Maass forms of weight 0 defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross-Kohnen-Zagier theorem and Zagiers modularity of traces of singular moduli, together with new geometric interpretations of the traces with non-positive index.
We call a polynomial monogenic if a root $theta$ has the property that $mathbb{Z}[theta]$ is the full ring of integers in $mathbb{Q}(theta)$. Consider the two families of trinomials $x^n + ax + b$ and $x^n + cx^{n-1} + d$. For any $n>2$, we show that these families are monogenic infinitely often and give some positive densities in terms of the coefficients. When $n=5$ or 6 and when a certain factor of the discriminant is square-free, we use the Montes algorithm to establish necessary and sufficient conditions for monogeneity, illuminating more general criteria given by Jakhar, Khanduja, and Sangwan using other methods. Along the way we remark on the equivalence of certain aspects of the Montes algorithm and Dedekinds index criterion.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
