Exercise 8.3

Let χ be a non trivial multiplicative character of 𝔽 p and ρ be the character of order 2. Show that ∑ ⁡ t χ ( 1 − t 2 ) = J ( χ , ρ ) .[Hint: Evaluate J ( χ , ρ ) using the relation N ( x 2 = a ) = 1 + ρ ( a ) .]

Answers

Proof.

J ( χ , ρ ) = ∑ a + b = 1 χ ( a ) ρ ( b ) = ∑ a + b = 1 χ ( a ) ( N ( x 2 = b ) − 1 ) = ∑ a + b = 1 χ ( a ) N ( x 2 = b ) − ∑ a + b = 1 χ ( a ) .

As χ ≠ 𝜀 ,

∑ a + b = 1 χ ( a ) = ∑ a ∈ 𝔽 p χ ( a ) = 0 .

Let C = { x 2 | x ∈ 𝔽 ∗ } the set of squares in 𝔽 p ∗ , C ¯ its complementary in 𝔽 p ∗ :

𝔽 p = { 0 } ∪ C ∪ C ¯ .

Then

J ( χ , ρ ) = ∑ a + b = 1 χ ( a ) N ( x 2 = b ) = ∑ a + b = 1 , b = 0 χ ( a ) N ( x 2 = b ) + ∑ a + b = 1 , b ∈ C χ ( a ) N ( x 2 = b ) + ∑ a + b = 1 , b ∈ C ¯ χ ( a ) N ( x 2 = b ) = χ ( 1 ) + 2 ∑ b ∈ C χ ( 1 − b ) ,

because N ( x 2 = b ) = 0 if x ∈ C ¯ , and N ( x 2 = b ) = 2 if x ∈ C . As each b ∈ C has two roots, and as the set of roots of two distinct b are disjointed,

J ( χ , ρ ) = χ ( 1 ) + ∑ t ∈ 𝔽 p ∗ χ ( 1 − t 2 ) = ∑ t ∈ 𝔽 p χ ( 1 − t 2 ) .

Conclusion : if χ is a non trivial multiplicative character of 𝔽 p and ρ the character of order 2,

J ( χ , ρ ) = ∑ t ∈ 𝔽 p χ ( 1 − t 2 ) .

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2022-07-19 00:00
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