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Register a new userOn singular moduli for arbitrary discriminants
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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.
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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 $f$ 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 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.
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.
The discriminant of a polynomial of the form $pm x^n pm x^m pm 1$ has the form $n^n pm m^m(n-m)^{n-m}$ when $n,m$ are relatively prime. We investigate when these discriminants have prime power divisors. We explain several symmetries that appear in the classification of these values of $n,m$. We prove that there are infinitely many pairs of integers $n,m$ for which this discriminant has no prime cube divisors. This result is extended to show that for infinitely many fixed $m$, there are infinitely many $n$ for which the discriminant has no prime cube divisor.
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