If we focus on the values Q(x, y) for x and y integers then it is the same to consider Q′(x, y): = Q(ax by,cx dy), for any integers a, b, c and d, so long as ad − bc = 1.
Such a modular transformation preserves discriminants, so we may consider the forms with given discriminant D, and count them modulo modular transformations. For example, h(−7) = 1 because Q(x, y): = x(1) should vanish if and only if there are infinitely many rational solutions.
For example, the rotational symmetry of a regular polygon with a prime number of edges—a cyclic group of prime order—is a finite simple group.
The next example is the rotational symmetry of a regular dodecahedron, but to see it as part of a family it should be regarded differently, via its action on five embedded tetrahedrons, for instance.
The effort to classify them precipitated the discovery of new examples, including the monster, and six pariah groups which do not belong to any of the natural families, and are not involved in the monster.
It also precipitated monstrous moonshine, which is an appearance of monster symmetry in number theory that catalysed developments in mathematics and physics.
Murray Perahia has just released his new album presenting Bach’s “The French Suites”.A stronger form of the BSD conjecture asserts that(E), which ties together the p-fold symmetries in III(E) with the infinite families of rational solutions to E, so Selmer groups and Tate–Shafarevich groups are of primary importance. The moonshine that we establish for the O’Nan group (see Theorem 1) enables us to prove new constraints on class numbers h(D) (see Theorem 2), and empowers us to relate certain Selmer groups Sel that are compatible with the character table is enough to prove that corresponding matrices exist, and enough to confirm moonshine for the O’Nan group. We first realise the F of Skinner–Urban, shows that (8) holds modulo certain primes p, when certain conditions on the underlying elliptic curve E are satisfied. The resulting classification is a crowning achievement of twentieth century mathematics.Thompson was awarded a Fields medal for his contributions.