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In this work, we state and prove Lerch's theorems for Fermat and Euler quotients over function fields defined analogously to the number fields. The Fermat's little theorem states that if p is a prime and a is an integer not divisible by p, then a p−1 ≡ 1 mod p. This gives rise to the definition of the Fermat quotient of p with base a, q(a, p) = a p−1 − 1 p(More)
In this paper we study Cayley graphs on PGL 2 (F q) mod the unipotent subgroup , the split and nonsplit tori, respectively. Using the Kirillov models of the representations of PGL 2 (F q) of degree greater than one, we obtain explicit eigenvalues of these graphs and the corresponding eigenfunctions. Character sum estimates are then used to conclude that two(More)
In this paper, we study the Drinfeld cusp forms for Γ 1 (T) and Γ(T) using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for Γ 1 (T) of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that(More)
In this work, we study the elliptic curve E : y 2 = f (x), where f (x) is a cubic permutation polynomial over some finite commutative ring R. In case R is the finite field F q , it turns out that the group of rational points on E is cyclic of order q + 1. This group is a product of cyclic groups if R = Z n , the ring of integers modulo a square-free n. In(More)