(The) relation between the quadratic representation of a prime and binomial coefficients

Abstract

In this thesis, we consider the problem due to Gauss. When a prime can be represented as quadratic form, the first term is congruent to a product of binomial coefficients. In chapter 1, we introduce the history of the results about this problem and define some notations which are needed in the following chapters. In chapter 2, we describe the properties of some special sums, e.g. Gauss sums, Jacobi sums and Eisenstein sums, and p-adic gamma function. They play an important role to prove our main theorem in chapter 3. To relate the Gauss sums with binomial coefficients, we use Gross-Koblitz formula. In chapter 3, We generalize the problem of Gauss to the primes of the form p=tn+r when p splits as $\mathfrak{p}_1 \mathfrak{p}_2$ in $\mathbb{Q}(\sqrt{-t})$. When a prime can be represented as quadratic form, the first term is congruent to a product of binomial coefficients. Let $q=p^f$ where f is the order of r modulo t, $χ = ω^{\frac{q-1}{t}}$ where ω is the Teichm$ü$ller character on $\mathbb{F}_q$ and g(χ) be the Gauss sum. For suitable $τ_i ∈ Gal(\mathbb{Q}(ζ_t, ζ_p)/\mathbb{Q})$ $(i=1, …, g)$, we show that ◁수식 삽입▷(원문을 참조하세요) such that $4p^h = a^2 + tb^2 $ for some integers a and b where h is the class number of $\mathbb{Q}(\sqrt{-t})$. And we explicitly compute a mod t (or t/4) and a mod p, in particular, a is congruent to a product of binomial coefficients modulo p. In chapter 4, we give some examples.