It is believed that one can attach a smooth mod-p representation of a general linear group to a mod-p local Galois representation in a natural way that is called mod-p Langlands program. This conjecture is quite far from being understood beyond GL₂(ℚₚ). However, for a given mod-p local Galois representation one can construct a candidate on the automorphic side corresponding to the Galois representation for mod-p Langlands correspondence via global Langlands. In this talk, we introduce automorphic invariants on the candidate that determine the given Galois representation for a certain family of mod-p Galois representations.

Let J={a,b} be an unordered pair of F_q, and E_J the associated elliptic curve of the form y^3=(x-a)(x-b) over \F_q. We show that there are "only three possible values" for the trace of Frobenius of E_J. Furthermore, these three values can be computed via a certain Jacobi sum. As applications, we first compute the average analytic rank of a certain family of elliptic curves. Next, we generate elliptic curves with designated extremal primes. After computing a variant of the n-th moment of Traces of Frobenius, we give explicit values and average values on class numbers of every constant field extension of K_J=F_q(\sqrt[3]{(T-a)(T-b)}). Finally, we compute the exact values and the average values on Euler-Kronecker constants of K_J. This is a joint work with Jinjoo Yoo.