Hw1: Take lattice (x,y,z,t): z=ax+by (mod p) and t=bx-ay (mod p). Compute volume and use Minkowski to prove Lagrange thm for primes.
Finally note that: (a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=
(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 +
(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 +
(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 +
(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2.
Hw2: Prove weak version of Szemeredi-Trotter: Incidences < O(mn^{1/2}+n). (where m is the number of points and n the number of lines).