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Four quadratic forms whose cubes sum to zero
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Abstract
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Let u be an even integer not divisible by 3. For each z = (z1, z2, z3) E Z(3) satisfying z(2) = 0 mod 2 and Qu(z) = z(1)(2)-z(2)(2)-((u(6)-1)/3)z23 = 0, we find integral binary quadratic forms sz, tz such that sz(x, y)(3) + tz(x, y)(3) = sz(y, x)(3) + tz(y, x)(3). The coefficients of sz, tz are integer linear combinations of (z1, z2, z3). This yields a three parameter family of 4-tuples of binary quadratic forms spanning infinitely many GL2(Z) orbits, whose cubes add up to zero. The first couple of examples go back to Ramanujan.
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0.3
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