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Complex Multiplication by Reinhard Schertz

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9

Berwick’s congruences

In 1927 the English mathematician W.E.H. Berwick published an article on numerical computation of singular values of j including a series of conjectures concerning congruences satisfied by these singular values, without giving any idea of proof. In the framework of complex multiplication these congruences constitute a rather untypical result. Therefore, it was not surprising that the first proof of some of these congruences by Gross and Zagier (1985) did not use the classical methods of complex multiplication but was relying on counting isogenies of elliptic curves. Moreover, Gross and Zagier only treated congruences modulo powers of the primes p = 2, 3, 5, 7 and 11 without giving any hint on how similar congruences modulo ...

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