We consider the differential KZ equations over
C
\mathbb C
in the case, when the hypergeometric solutions are one-dimensional hyperelliptic integrals of genus
g
g
. In this case the space of solutions of the differential KZ equations is a
2
g
2g
-dimensional complex vector space.
We also consider the same differential equations modulo
p
s
p^s
, where
p
p
is an odd prime number and
s
s
is a positive integer, and over the field
Q
p
\mathbb Q_p
of
p
p
-adic numbers.
We describe a construction of polynomial solutions of the differential KZ equations modulo
p
s
p^s
. These polynomial solutions have integer coefficients and are
p
s
p^s
-analogs of the hyperelliptic integrals. We call them the
p
s
p^s
-hypergeometric solutions. We consider the space
M
p
s
\mathcal M_{p^s}
of all
p
s
p^s
-hypergeometric solutions, which is a module over the ring of polynomial quasi-constants modulo
p
s
p^s
. We study basic properties of
M
p
s
\mathcal M_{p^s}
, in particular its natural filtration, and the dependence of
M
p
s
\mathcal M_{p^s}
on
s
s
.
We show that the
p
p
-adic limit of
M
p
s
\mathcal M_{p^s}
as
s
→
∞
s\to \infty
gives us a
g
g
-dimensional vector space of solutions of the differential KZ equations over the field
Q
p
\mathbb Q_p
. The solutions over
Q
p
\mathbb Q_p
are power series at a certain asymptotic zone of the KZ equations.
In the appendix written jointly with Steven Sperber we consider all asymptotic zones of the KZ equations in the special case
g
=
1
g=1
of elliptic integrals. It turns out that in this case the
p
p
-adic limit of
M
p
s
\mathcal M_{p^s}
as
s
→
∞
s\to \infty
gives us a one-dimensional space of solutions over
Q
p
\mathbb Q_p
at every asymptotic zone. We apply Dwork’s theory of the classical hypergeometric function over
Q
p
\mathbb Q_p
and show that our germs of solutions over
Q
p
\mathbb Q_p
defined at different asymptotic zones analytically continue into a single global invariant line subbundle of the associated KZ connection. Notice that the corresponding KZ connection over
C
\mathbb C
does not have proper nontrivial invariant subbundles, and therefore our invariant line subbundle is a new feature of the KZ equations over
Q
p
\mathbb Q_p
.
Also in the appendix we follow Dwork and describe the Frobenius transformations of solutions of the KZ equations for
g
=
1
g=1
. Using these Frobenius transformations we recover the unit roots of the zeta functions of the elliptic curves defined by the affine equations
y
2
=
β
x
(
x
−
1
)
(
x
−
α
)
y^2= \beta \,x(x-1)(x-\alpha )
over the finite field
F
p
\mathbb F_p
. Here
α
,
β
∈
F
p
×
,
α
≠
1
\alpha ,\beta \in \mathbb F_p^\times , \alpha \ne 1
. Notice that the same elliptic curves considered over
C
\mathbb {C}
are used to construct the complex holomorphic solutions of the KZ equations for
g
=
1
g=1
.