Dirichlet's Class Number Formula November 13, 2012 Due Wednesday, November 14, 2012. You may work together but you may not show one another your written work. You will need to know some results from Chapters 12 and 13 in the book, which we will be coming to next in class. A common theme in number theory is that special values of L-functions en- code arithmetic information. A famous example is the conjecture of Birch and Swinnerton-Dyer expressing information about an elliptic curve in terms of its L- function. Dirichlet's class number formula is another famous example, using an L-function to compute the class number of a number field. It works especially well for imaginary quadratic fields. p In this problem p will be an odd prime. Let F = Q ( −p) and let R be the ring of integers in F . Let ζF be the Dedekind zeta function −1 X 1 Y 1 ζ (s) = = 1 − : F Nas Nps a p The sum is over all ideals of F and the product is over all prime ideals. a 1. Assume that p ≡ 3 mod 4. Let χ (a) = p . (a) Let q be a prime. Prove that if χ (q) = 0 then R has one prime ideal of norm q; if χ (q) = 1 then R has two prime ideals of norm q; and if χ (q) = −1 then R has one prime ideal of norm q2. (b) Prove that ζF (s) = ζ (s) L (s; χ) where ζ (s) is the Riemann zeta function and 1 X χ (n) L (s; χ) = : ns n=1 (c) Prove that the formula ζF (s) = ζ (s) L (s; χ) is also true if p ≡ 1 mod 4 but (a−1)=2 a with χ(a) = (−1) p when a is odd, χ(a) = 0 when a is even. Show that χ is a Dirichlet character mod 4p. 1 2. (This is in the book but write it up.) Let D be the discriminant of F . Show that D is −p if p ≡ 3 mod 4 and −4p if p ≡ 1 mod 4. Let h be the class number of R and let w be the number of roots of unity in F . Thus w = 6 if p = 3 and w = 2 otherwise. We will use the formula ζF (s) = ζ (s) L (s; χ) to prove that 2πh L (1; χ) = : (1) wpjDj I will prove this in class. But the idea now is that we can compute the class number h if we can find another way of computing L (1; χ). Note that L (1; χ) is given by a conditionally convergent series. You do not have to worry too much about convergence issues. 3. Assume that p ≡ 3 mod 4. Let g (χ) = Pp−1 χ (a) e2πia=p. p a=1 p (a) We know that jg (χ)j = p. Use this to show g (χ) = ± pi. It is proved in p the book that g (χ) = pi. This is harder, but we'll need it to determine a sign in Problem 4. (b) Show that p−1 1 1 X X e2πian=p L (s; χ) = χ (a) : g (χ) ns a=1 n=1 (c) Deduce that p−1 1 Y −χ(a) L (1; χ) = log 1 − e2πia=p : g (χ) a=1 This gives us a formula in closed form but it can be simplified! 4. (a) Write 1 − eiθ−1 = Reiφ. We don't need to compute R but show that 1 φ = 2 (π − φ). (b) Assuming that p ≡ 3 mod 4 show that p−1 π X L (1; χ) = − aχ (a) : p3=2 a=1 Hint: Save yourself some work by using the fact that L (1; χ) is real. (But remind me why you know this.) (c) Assume that p ≡ 3 mod 4 but p > 3 (so w = 2). Show that p−1 1 X h = − aχ (a) p a=1 and use this to compute h if p = 7; 11; 19; 23. 2.