ﻻ يوجد ملخص باللغة العربية
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.
In a paper of P. Paillier and J. Villar a conjecture is made about the malleability of an RSA modulus. In this paper we present an explicit algorithm refuting the conjecture. Concretely we can factorize an RSA modulus n using very little information
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.
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 factoriz
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 o
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