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Additional resources for Advanced Analytic Number Theory, Part I: Ramification Theoretic Methods
We deﬁne the trace (resp. the norm) of Fq over Fp as m−1 F TrFqp x := x + xp + · · · + xp and Fq 1+p+···+pm−1 NFp x := x . 5) One can easily check that these maps send Fq to Fp and that the trace is Fp -linear (resp. the norm, multiplicative). We ﬁrst use the trace to construct an additive character: if q = pm and a ∈ Fq , we deﬁne it by and denote it as ⎞ ⎛ F 2πi TrFqp a ⎠. 6) ψ(a) := exp ⎝ p Now we can generalize the calculation over Fp . 10. Lemma. Let b ∈ Fq . Then we have the formula ψ(ab) = a∈Fq q 0 if b = 0, if b = 0.
6) ψ(a) := exp ⎝ p Now we can generalize the calculation over Fp . 10. Lemma. Let b ∈ Fq . Then we have the formula ψ(ab) = a∈Fq q 0 if b = 0, if b = 0. 7) Proof. The formula is obviously true for b = 0. If b = 0, the map F a → TrFqp (ab) from Fq to Fp is Fp -linear and surjective. Therefore, every element of Fp appears pm−1 times in the image of the trace, and hence m−1 a∈Fq ψ(ab) = p x∈Fp exp(2πix/p) = 0. Convention. The unitary character χ0 is deﬁned by χ0 (a) = 1 for every a ∈ Fq . , a homomorphism), other than the unitary character (over F∗q ), we extend it by χ(0) = 0.
6) Using this, ﬁnd a formula for Np in terms of Gauss sums of the form Np = p − 0 + τ (1) p 2 1 G(χ2 ) + 2 2 G(χ3 ) , where | i | = 1. 7) Conclude that Np 1 for every p = 2, 17. 23. Exercise. In this exercise, we ask you to prove that the equation 2y 2 = x4 − 17 has solutions modulo N for every N , but does not have any rational solutions over Q. 1) Assume that there exist x = a/b and y = c/d, which are a solution to the equation, with a, c ∈ Z, b, d ∈ N∗ and gcd(a, b) = gcd(c, d) = 1. Prove that b4 divides d2 and that d2 divides 2b4 and deduce from this that d = b2 and 2c2 = a4 − 17b4 .