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Abstract
Let
E ∕ ℚ be
an elliptic curve that has complex multiplication (CM) by an imaginary quadratic field
K . For a prime
p , there
exists
𝜃 p
∈
[ 0 , π ]
such that
p
+ 1
−
# E ( 𝔽 p )
= 2 p cos 𝜃 p .
Let
x
> 0 be large, and
let
I
⊆
[ 0 , π ] be a subinterval.
We prove that if
δ
> 0
and
𝜃
> 0 are fixed
numbers such that
δ
+
𝜃
< 5
2 4 ,
x 1 − δ
≤
h
≤
x , and
| I | ≥ x − 𝜃 , then
1
h ∑
x < p ≤ x + h
𝜃 p ∈ I
log p
∼ 1
2 1 π ∕ 2 ∈ I + | I |
2 π ,
where
1 π ∕ 2 ∈ I
equals 1 if
π
2
∈
I
and
0
otherwise. We also discuss an extension of this result to the distribution of the
Fourier coefficients of holomorphic cuspidal CM newforms.
Keywords
CM elliptic curves, equidistribution, Grossencharacter,
$L$-function, zero-density estimate
Mathematical Subject Classification
Primary: 11M41
Milestones
Received: 9 May 2021
Revised: 24 April 2022
Accepted: 5 May 2022
Published: 14 April 2023
Communicated by Amanda Folsom
© 2023 MSP (Mathematical Sciences
Publishers).