Math 115 - Spring 1994 - Ribet - Midterm 2

Math 115 - Spring 1994 - Ribet - Midterm 2 - E in terms of...

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Math 115 Professor K. A. Ribet Take-home Exam, due April 7, 1994 Let p be a prime number greater than 3. Let N p be the number of solutions to y 2 = x 3 - x in F p . 1. Show that N p = p + a F p a 3 - a p . 2. Prove that N p = p if p 3 (mod 4). 3. Suppose from now on that p 1 mod 4. Recall from class that p may be written in the form r 2 + s 2 where r and s are integers, cf. Proposition 8.3.1 of the text. Since p is odd, r and s cannot have the same parity we will suppose that r is odd and that s is even . Show that r and s are then determined up to sign. (This is a restatement of problem 12 on page 106 of the book.) 4. While I’m at it, let me assign problem 13 on page 106. This came up in class. 5. Let E = p - N p = - a F p a 3 - a p = - a F p a - a 3 p ; we think of E as an error term. Here is a table giving the value of E for twenty-one small primes p : p 13 17 29 37 41 53 61 73 89 97 101 109 113 137 149 157 173 181 193 197 229 E 6 2 - 10 - 2 10 14 - 10 - 6 10 18 - 2 6 - 14 - 22 14 22 - 26 - 18 - 14 - 2 30 6. Calculate r and s for a fair number of the twenty-one primes p which appear in the table. Following in the 1814 footsteps of Gauss, conjecture a rule which determines
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Unformatted text preview: E in terms of r and s . For example, decide what E ought to be when p = 144169 = (315) 2 + (212) 2 . 7. Let χ be a character of order 4 on F * p , so that χ 2 is the quadratic symbol ( · p ) . Show that E =-2 Re J , where J = J ( χ, χ 2 ). Check this general formula by calculating E and J explicitly in the case where p = 5 and χ is the character mapping 2 to i . 8. Regard J as an element of Z [ i ]. Show that J + 1 is divisible by (2 + 2 i ). (See page 168 of the book if you get stuck.) 9. Suppose that J = α + iβ where α and β are integers. Explain why α is odd, β is even, α + β + 1 is divisible by 4 and α 2 + β 2 = p . Recapitulate what you have learned in the form of a rule for calculating N p when p is congruent to 1 mod 4....
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