Abstract
In the 1960s, Birch proved that the traces of Frobenius for elliptic curves taken at random over a large finite field is modeled by the semicircular distribution (i.e. the usual SatoâTate for non-CM elliptic curves). In analogy with Birchâs result, a recent paper by Ono, the author, and Saikia proved that the limiting distribution of the normalized Frobenius traces
A
λ
âą
(
p
)
{A_{\lambda}(p)}
of a certain family of
K
âą
3
{K3}
surfaces
X
λ
{X_{\lambda}}
with generic Picard rank 19 is the
O
âą
(
3
)
{O(3)}
distribution. This distribution, which we denote by
1
4
âą
Ï
âą
f
âą
(
t
)
{\frac{1}{4\pi}f(t)}
, is quite different from the semicircular distribution. It is supported on
[
-
3
,
3
]
{[-3,3]}
and has vertical asymptotes at
t
=
±
1
{t=\pm 1}
. Here we make this result explicit. We prove that if
p
â„
5
{p\geq 5}
is prime and
-
3
â€
a
<
b
â€
3
{-3\leq a<b\leq 3}
, then
|
#
âą
{
λ
â
đœ
p
:
A
λ
âą
(
p
)
â
[
a
,
b
]
}
p
-
1
4
âą
Ï
âą
â«
a
b
f
âą
(
t
)
âą
đ
t
|
â€
98.28
p
1
/
4
.
\biggl{\lvert}\frac{\#\{\lambda\in\mathbb{F}_{p}:A_{\lambda}(p)\in[a,b]\}}{p}-%
\frac{1}{4\pi}\int_{a}^{b}f(t)\,dt\biggr{\rvert}\leq\frac{98.28}{p^{1/4}}.
As a consequence, we are able to determine when a finite field
đœ
p
{\mathbb{F}_{p}}
is large enough for the discrete histograms to reach any given height near
t
=
±
1
{t=\pm 1}
. To obtain these results, we make use of the theory of RankinâCohen brackets in the theory of harmonic Maass forms.