Products and sums of powers of binomial coefficients mod $p$ and solutions of certain quaternary Diophantine systems

Abstract:

In this paper we prove that certain products and sums of powers of binomial coefficients modulo $p = qf + 1$, $q = {a^2} + {b^2}$, are determined by the parameters x occurring in distinct solutions of the quaternary quadratic partition \[ \begin {array}{*{20}{c}} {16{p^\alpha } = {x^2} + 2q{u^2} + 2q{v^2} + q{w^2},\quad (x,u,v,w,p) = 1,} \\ {xw = a{v^2} - 2buv - a{u^2},\quad x \equiv 4\pmod q,\alpha \geqslant 1.} \\ \end {array} \] The number of distinct solutions of this partition depends heavily on the class number of the imaginary cyclic quartic field \[ K = Q\left ( {i\sqrt {2q + 2a\sqrt q } } \right ),\] as well as on the number of roots of unity in K and on the way that p splits into prime ideals in the ring of integers of the field $Q({e^{2\pi ip/q}})$. Let the four cosets of the subgroup A of quartic residues be given by ${c_j} = {2^j}A,j = 0,1,2,3$, and let \[ {s_j} = \frac {1}{q}\sum \limits _{t \in {c_j}} {t,\quad j = 0,1,2,3.} \] Let ${s_m}$ and ${s_n}$ denote the smallest and next smallest of the ${s_j}$ respectively. We give new, and unexpectedly simple determinations of ${\Pi _{k \in {c_n}}}kf!$ and ${\Pi _{k \in {c_{n + 2}}}}kf!$, in terms of the parameters x in the above partition of $16{p^\alpha }$, in the complicated case that arises when the class number of K is $> 1$ and ${s_m} \ne {s_n}$.