arXiv subjects
Publications and source records attributed to Jinxiang Zeng.
We construct plane models of the modular curve $X_H(\ell)$, and use their explicit equations to compute Galois representations associated to modular forms for values of $\ell$ that are significantly higher than in prior works.
In this paper, we present a probabilistic algorithm to compute the number of $\mathbb{F}_p$-points of modular curve $X_1(n)$. Under the Generalized Riemann Hypothesis(GRH), the algorithm takes $\textrm{O}(n^{56+δ+ε}\log^{9+ε} p)$ bit operations, where $δ$ is an absolute constant and $ε$ is any positive real number. As an application, we can compute $#X_1(17)(\mathbb{F}_p)\textrm{mod} 17$ for huge primes $p$. For example, we have $#X_1(17)(\mathbb{F}_{10^{1000}+1357})\textrm{mod} 17=3$.
In this paper we present a probabilistic algorithm to compute the coefficients of modular forms of level one. Focus on the Ramanujan's tau function, we give out the explicit complexity of the algorithm. From a practical viewpoint, the algorithm is particularly well suited for implementations.
For the integer $ D=pq$ of the product of two distinct odd primes, we construct an elliptic curve $E_{2rD}:y^2=x^3-2rDx$ over $\mathbb Q$, where $r$ is a parameter dependent on the classes of $p$ and $q$ modulo 8, and show, under the parity conjecture, that the elliptic curve has rank one and $v_p(x([k]Q))\not=v_q(x([k]Q))$ for odd $k$ and a generator $Q$ of the free part of $E_{2rD}(\mathbb Q)$. Thus we can recover $p$ and $q$ from the data $D$ and $ x([k]Q))$. Furthermore, under the Generalized Riemann hypothesis, we prove that one can take $r<c\log^4D$ such that the elliptic curve $E_{2rD}$ has these properties, where $c$ is an absolute constant.