令E/Q 為定義在有理數上的具有複乘數(CM)橢圓曲線及 P 為曲線上秩為無窮的有理點。本計畫的目的在於探討所有質數 p 使得當E/Q 為超奇異簡化(supersingular reduction)且 P modulo p 為一質點所形成的集合的密度問題。倘若 E/Q 為平常簡化 (ordinary reduction)，在GRH 是正確的前提下， Gupta 跟 Murty 證明了這個密度的存在性，並且在某些額外的條件下它們也證明了這個密度是正數。此計畫的初步目標是希望去證明存在無窮多個質數使得 E/Q 為超奇異簡化 (supersingular reduction) 且 P modulo p 為一質點。此計畫的終極目標是希望去證明所有質數 p 使得當 E/Q 為超奇異簡化且 P modulo p 為一質點所形成的集合的密度的存在性。 ;Let E/Q be an elliptic curve defined over the rational numbers, and let P be a rational point of infinite order. The purpose of this project is to study the problem about the density of the set of rational primes p for which E/Q has supersingular reduction and P modulo p is a primitive point. If E/Q has complex multiplication by a maximal order of an imaginary quadratic field, under generalized Riemann hypothesis (GRH),Gupta and Murty proved the existence of the density of ordinary primes for which P modulo p is a primitive point, and showed that the density is positive by imposing an additional assumption.The first aim of this project is to show that there are infinitely many primes p for which E/Q has supersingular reduction and P modulo p is a primitive point. The ultima goal is to prove the existence of the density of the set of rational primes p for which E/Q is supersingular and P modulo p is a primitive point.
