The content discusses the rationality of Weil sums of binomials, which are sums of the form W K,s
u = Σ_x∈K ψ(xs - ux), where K is a finite field, ψ is the canonical additive character of K, u is an element of K×, and s is a positive integer relatively prime to |K×|.
The key insights are:
The Weil spectrum, which is the set of distinct values that the Weil sums W K,s
u can take as u runs through K×, always contains at least three distinct values if s is nondegenerate (i.e., not a power of the characteristic p modulo |K×|).
It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers.
The main result of the paper shows that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where |K| = 5 and s ≡ 3 (mod 4).
The paper uses tools from algebraic number theory, p-adic analysis, and the study of certain algebraic sets over finite fields to prove this result.
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arxiv.org
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by Daniel J. Ka... klokken arxiv.org 04-09-2024
https://arxiv.org/pdf/2306.14414.pdfDypere Spørsmål