论文标题
分裂减少的质量分布
Distribution of primes of split reductions for abelian surfaces
论文作者
论文摘要
让$ a $是一个绝对简单的Abelian表面，该表面是在数字字段$ k $上定义的，并具有交换性（几何）内态环。令$π_{a，\ text {split}}（x）$表示$ k $中的Primes $ \ Mathfrak {p} $的数量，以使每个prime均由$ x $限制为$ a $的$ x $，以及$ a $ a $ a $ a $ a $ a $ a $ a $ \ thefrak {p} $ splits $ a $ a $ a $ a $ \ a。众所周知，这种素数的密度为零。在Dedekind Zeta函数的广义Riemann假设下，并可能延长了字段$ K $，我们证明$π_{a，\ text {split}}}（x）\ ll_ {a，k} x^^^^{\ frac {\ frac {41}} {42}} {42}} {42} {42} {42} \ log x $ igrial in $ nif unial in $ ring of y $ y $ yir of y $ yir of y $ rim of y $ yomorphial of y $ yomorphim emorphism $π_{a，\ text {split}}（x）\ ll_ {a，f，f，k} \ frac {x^{x^{\ frac {\ frac {11} {12}}}}} {\ log x） $π_{a，\ text {split}}（x）\ ll_ {a，f，f，f，k} x^{\ frac {2} {3}}}（\ log x）^{\ frac {\ frac {1} {1} {3}}}}} $ a $ a $具有复杂的乘法。这些结果提高了J. Achter在2012年的界限和2014年的D. Zywina。我们还提供了其他可靠猜想的更好的界限。
Let $A$ be an absolutely simple abelian surface defined over a number field $K$ with a commutative (geometric) endomorphism ring. Let $π_{A, \text{split}}(x)$ denote the number of primes $\mathfrak{p}$ in $K$ such that each prime has norm bounded by $x$, of good reduction for $A$, and the reduction of $A$ at $\mathfrak{p}$ splits. It is known that the density of such primes is zero. Under the Generalized Riemann Hypothesis for Dedekind zeta functions and possibly extending the field $K$, we prove that $π_{A, \text{split}}(x) \ll_{A, K} x^{\frac{41}{42}}\log x$ if the endomorphism ring of $A$ is trivial; $π_{A, \text{split}}(x) \ll_{A, F, K} \frac{x^{\frac{11}{12}}}{(\log x)^{\frac{2}{3}}}$ if $A$ has real multiplication by a real quadratic field $F$; $π_{A, \text{split}}(x) \ll_{A, F, K} x^{\frac{2}{3}}(\log x)^{\frac{1}{3}}$ if $A$ has complex multiplication by a CM field $F$. These results improve the bounds by J. Achter in 2012 and D. Zywina in 2014. We also provide better bounds under other credible conjectures.
