有理数域上椭圆曲线具有固定弗罗贝尼乌斯域的无穷多素数
Infinitely many primes with a fixed Frobenius field for an elliptic curve over $\mathbb{Q}$
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中文总结 AI 辅助
该研究针对有理数域上特定非复乘椭圆曲线与虚二次域，证明存在无穷多素数使得椭圆曲线在该素数处的弗罗贝尼乌斯域等于给定虚二次域，建立了相关计数函数的下界与上界，方法涉及复乘理论等。
中文摘要 AI 辅助
1987年，Elkies证明了一个引人注目的结果：有理数域$\boldsymbol{\text{Q}}$上的每一条椭圆曲线都有无穷多超奇异素数。受该定理及其与Lang-Trotter猜想关联的启发，我们研究椭圆曲线弗罗贝尼乌斯域的类似问题。对于有理数域上某些非复乘（non-CM）椭圆曲线$E$和虚二次域$K$，我们证明存在无穷多素数$p$，使得$E$在$p$处的弗罗贝尼乌斯域等于$K$。更准确地说，设$\boldsymbol{\text{π}}_E(x,K)$表示满足$p \boldsymbol{\text{≤}} x$的此类素数的数量，我们对任意$\boldsymbol{\text{ε}}>0$建立了无条件界$\boldsymbol{\text{π}}_E(x,K) \boldsymbol{\text{≫}}_{E,K,\boldsymbol{\text{ε}}} (\boldsymbol{\text{log}}\boldsymbol{\text{log}}x)^{1-\boldsymbol{\text{ε}}}$。我们还证明了与$\boldsymbol{\text{π}}_E(x,K)$相关的受限计数函数的无条件幂节省上界。该方法结合了Deuring的复乘理论、奇异模的性质以及模曲线上的算术交理论。
英文摘要
In 1987, Elkies proved the striking result that every elliptic curve over $\mathbb{Q}$ has infinitely many supersingular primes. Motivated by this theorem and its connection with the Lang-Trotter conjecture, we study the analogous problem for Frobenius fields of elliptic curves. For certain families of non-CM elliptic curves $E/\mathbb{Q}$ and imaginary quadratic fields $K$, we prove that there exist infinitely many primes $p$ for which the Frobenius field of $E$ at $p$ equals $K$. More precisely, letting $π_E(x,K)$ denote the number of such primes with $p\leq x$, we establish the unconditional bound $π_E(x, K)\gg_{E, K, ε} (\log\log x)^{1-ε}$ for every $ε>0$. We also prove unconditional power-saving upper bounds for a restricted counting function associated with $π_E(x,K)$. The approach combines Deuring's theory of complex multiplication, properties of singular moduli, and arithmetic intersection theory on modular curves.
