Celebratio Mathematica — Gross

Dick Gross’s marvelous mentoring

Works connected to Don Bernard Zagier

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B. H. Gross and D. B. Zagier: “On singular moduli,” J. Reine Angew. Math. 355 (1985), pp. 191–220. Dedicated to J.-P. Serre. MR 772491 Zbl 0545.10015 article

The values of the modular function \( j(\tau) \) at imaginary quadratic in the upper half plane are known as singular noduii. They are all algebraic integers. In this paper we will study the prime factorizations of the differences of two singular moduli. These differences turn out to be highly divisible numbers. For instance, we will determine the set of primes dividing the absolute norm of \[ j(\tau) - 1728 = j(\tau) - j(i) \] (and the multiplicities with which they occur); they turn out to be contained among the prime divisors of the positive integers of the form \( d - x^2 \) , where \( d \) is the discriminant of \( \tau \), and hence smaller than or equal to \( d \), e.g.: \[ j\Bigl(\frac{1+i\sqrt{163}}{2}\Bigr) - 1728 = - 2^6 3^6 7^2 11^2 19^2 127^2 163. \]

B. H. Gross and D. B. Zagier: “Heegner points and derivatives of \( L \)-series,” Invent. Math. 84 (1986), pp. 225–320. To John Tate. This work expands on a short note published in C. R. Acad. Sci., Paris 297 (1983). Part II was published in Math. Ann. 278 (1987). MR 833192 Zbl 0608.14019 article

The main theorem of this paper gives a relation between the heights of Heegner divisor classes on the Jacobian of the modular curve \( X_0(N) \) and the first derivatives at \( s = 1 \) of the Rankin \( L \)-series of certain modular forms. In the first six sections of this chapter, we will develop enough background material on modular curves, Heegner points, heights, and \( L \)-functions to be able to state one version of this identity precisely. In §7 we will discuss some applications to the conjecture of Birch and Swinnerton-Dyer for elliptic curves. For example, we will show that any modular elliptic curve over \( \mathbb{Q} \) whose \( L \)-function has a simple zero at \( s = 1 \) contains rational points of infinite order. Combining our work with that of Goldfeld [1976], one obtains an effective lower bound for the class numbers of imaginary quadratic fields as a function of their discriminants (§8). In §9 we will describe the plan of proof and the contents of the remaining chapters.

Many of the results of this paper were announced in our Comptes Rendus note [1983]. A more leisurely introduction to Heegner points and Rankin \( L \)-series may be found in our earlier paper [1984].