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Extremal primes for elliptic curves without complex multiplication
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Fix an elliptic curve
E
E
over
Q
\mathbb {Q}
. An
extremal prime
for
E
E
is a prime
p
p
of good reduction such that the number of rational points on
E
E
modulo
p
p
is maximal or minimal in relation to the Hasse bound, i.e.,
a
p
(
E
)
=
±
[
2
p
]
a_p(E) = \pm \left [ 2 \sqrt {p} \right ]
. Assuming that all the symmetric power
L
L
-functions associated to
E
E
have analytic continuation for all
s
∈
C
s \in \mathbb {C}
and satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal primes when
E
E
is a curve without complex multiplication. In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes are less probable than primes where
a
p
(
E
)
a_p(E)
is fixed because of the Sato-Tate distribution.
American Mathematical Society (AMS)
Title: Extremal primes for elliptic curves without complex multiplication
Description:
Fix an elliptic curve
E
E
over
Q
\mathbb {Q}
.
An
extremal prime
for
E
E
is a prime
p
p
of good reduction such that the number of rational points on
E
E
modulo
p
p
is maximal or minimal in relation to the Hasse bound, i.
e.
,
a
p
(
E
)
=
±
[
2
p
]
a_p(E) = \pm \left [ 2 \sqrt {p} \right ]
.
Assuming that all the symmetric power
L
L
-functions associated to
E
E
have analytic continuation for all
s
∈
C
s \in \mathbb {C}
and satisfy the expected functional equation and the Generalized Riemann Hypothesis, we provide upper bounds for the number of extremal primes when
E
E
is a curve without complex multiplication.
In order to obtain this bound, we use explicit equidistribution for the Sato-Tate measure as in the work of Rouse and Thorner, and refine certain intermediate estimates taking advantage of the fact that extremal primes are less probable than primes where
a
p
(
E
)
a_p(E)
is fixed because of the Sato-Tate distribution.
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