On the syntomic regulator of the Hesse cubic curves and $p$-adic hypergeometric functions

We introduce a new type of $p$-adic hypergeometric function, which satisfies congruence relations similar to Dwork’s $p$-adic hypergeometric function. Also, we prove that the syntomic regulators of the Hesse cubic curves are expressed in terms of the special values of our new $p$-adic hypergeometric functions. We also show that there is a transformation formula between our new $p$-adic hypergeometric function and a $p$-adic hypergeometric function of logarithmic type defined by Asakura.