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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 PGL2(Fq) mod the unipotent subgroup, the split and nonsplit tori, respectively. Using the Kirillov models of the representations of PGL2(Fq) 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 types(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 several equivalence relations induced from the partitions of the sets of words of finite length. We have results on words over finite fieldsties of its equivalence classes and explicit relationships between two words are determined. Moreover, we deal with words of finite length over the ring Z/N Z where N is a positive integer. We(More)