Rankin–Selberg L-functions and the reduction of CM elliptic curves
Rankin–Selberg L-functions and the reduction of CM elliptic curves
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Rankin-Selberg L 函数和 CM 椭圆曲线的简化
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发表时间：
2015
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通讯作者：
M. Young
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作者：
Sheng;R. Masri;M. Young
设q为素数，$$K={mathbb Q}(sqrt{-D})$$ K= q （-D）为虚二次域，使得q在K中不存在。若$$mathfrak {q}$$ q是K的Hilbert类域中q以上的素数，则从$$overline{{mathbb Q}}$$ q¯复乘整数环的椭圆曲线集$${mathcal {O}}_K$$ OK到$${mathbb {F}}_{q^2}.$$ Fq2上的超奇异椭圆曲线集存在一个约简映射$$egin{aligned} r_{mathfrak q}:;{mathcal {Eell ell }}({mathcal {O}}_K) longrightarrow {mathcal {Eell ell }}^{ss}({mathbb F}_{q^2}) end{aligned}$$ rq:E l＿r (OK) E l＿r (ss) （Fq2）。我们证明了CM椭圆曲线能约简到一个给定的超奇异椭圆曲线的数目的一个一致渐近公式，并利用这一结果推导出对于$$D gg _{varepsilon } q^{18+varepsilon }.$$ D ^ εq18+ε的约简映射是满射的。这可以看作是关于等差数列中最小素数的林尼克定理的类比。我们也用相关的思想证明了Rankin-Selberg L-函数的中心值的平均$$egin{aligned} sum _{chi }L(f imes Theta _chi ,1/2) end{aligned}$$∑χL（f×Θχ,1/2）的统一渐近公式$${L(f imes {Theta _{chi}},s)}$$ L(f×Θχ,s)，其中f是一个固定的权值2，水平q是算术归一化的Hecke顶点形式，$$Theta _chi $$ Θχ在权值1上变化，水平D是与一个理想类群特征相关的级数$$chi $$ χ k。我们将这一结果应用于研究阿贝尔变、次凸、和$$L^4$$自同态形式的L4范数。
Let q be a prime and $$K={mathbb Q}(sqrt{-D})$$K=Q(-D) be an imaginary quadratic field such that q is inert in K. If $$mathfrak {q}$$q is a prime above q in the Hilbert class field of K, there is a reduction map $$egin{aligned} r_{mathfrak q}:;{mathcal {Eell ell }}({mathcal {O}}_K) longrightarrow {mathcal {Eell ell }}^{ss}({mathbb F}_{q^2}) end{aligned}$$rq:Eℓℓ(OK)⟶Eℓℓss(Fq2)from the set of elliptic curves over $$overline{{mathbb Q}}$$Q¯ with complex multiplication by the ring of integers $${mathcal {O}}_K$$OK to the set of supersingular elliptic curves over $${mathbb {F}}_{q^2}.$$Fq2. We prove a uniform asymptotic formula for the number of CM elliptic curves which reduce to a given supersingular elliptic curve and use this result to deduce that the reduction map is surjective for $$D gg _{varepsilon } q^{18+varepsilon }.$$D≫εq18+ε. This can be viewed as an analog of Linnik’s theorem on the least prime in an arithmetic progression. We also use related ideas to prove a uniform asymptotic formula for the average $$egin{aligned} sum _{chi }L(f imes Theta _chi ,1/2) end{aligned}$$∑χL(f×Θχ,1/2)of central values of the Rankin–Selberg L-functions $${L(f imes {Theta _{chi}},s)}$$L(f×Θχ,s) where f is a fixed weight 2, level q arithmetically normalized Hecke cusp form and $$Theta _chi $$Θχ varies over the weight 1, level D theta series associated to an ideal class group character $$chi $$χ of K. We apply this result to study the arithmetic of Abelian varieties, subconvexity, and $$L^4$$L4 norms of autormorphic forms.