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Talk on the boat: Don Zagier's work on singular moduli

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Benedict Gross
Mon, 2011-06-27 10:00 - 11:00

Singular moduli are the values of the modular function $j(\tau)$ at the points $z$ in the upper half plane that satisfy a quadratic equation with rational coefficients. In other words, they are the $j$-invariants of elliptic curves with complex multiplication.
These invariants were studied intensively by the leading number theorists of the nineteenth century. They are algebraic integers, which generate certain abelian extensions of the imaginary quadratic fields $\mathbb Q(z)$. The theory was believed to have been brought to a very satisfying completion in the early twentieth century. That was before Don got his hands on it.
In early $1983$ Don sent me an amazing letter from Japan containing a proof of a factorization formula for the integer which is the norm of the difference of two singular moduli of relatively prime discriminants $D$ and $D'$. This was a completely new aspect of the theory, which Don had discovered by extensive numerical experimentation. One particularly striking fact (which should have been noticed earlier) is that any prime p dividing this norm must divide an integer of the form $(DD' - x^2)/4$. This letter (in its original handwritten form, as well as a Latex version prepared by Carl Erickson) is reproduced below.

Don's proof involved the study of a Hilbert modular Eisenstein series for the real quadratic field $\mathbb Q (\sqrt{DD'})$. At the end of the letter, he challenged me to find an algebraic proof, which I sketched in a letter of reply (also reproduced below) and reproduced in the talk.
In $2002$, Don discovered another wonderful formula, relating the integers which are the traces of singular moduli to the Fourier coefficients of a meromorphic modular form of weight $3/2$. I will put this result into the context of computing the images of Heegner points in the Jacobians of modular curves. In this case, the Jacobian of the curve of level $1$ is trivial, but the generalized Jacobian relative to the divisor $2(\infty)$ is isomorphic to the additive group.

Datei Slides the talk (pdf)759.18 KB
Datei Dons Letter (pdf)497.08 KB
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