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On the intersection of modular correspondences. (English) Zbl 0811.11026

Starting with an intersection-theoretic interpretation of well-known results on the classical modular polynomial \(\varphi_ m\) the authors show by intersection-theoretic considerations on the three-dimensional scheme \(\text{spec } \mathbb{Z}[ j,j']\), too, that the ring \(A:= \mathbb{Z}[ j,j']/ (\varphi_{m_ 1}, \varphi_{m_ 2}, \varphi_{m_ 3})\) has finite cardinality if and only if there is no positive definite binary quadratic form over \(\mathbb{Z}\) which represents the three integers \(m_ 1\), \(m_ 2\), \(m_ 3\) and that in this case any prime \(p\) which divides \(\# A\) must satisfy \(p<4 m_ 1 m_ 2 m_ 3\). This is an analogy of Hurwitz’s result on the finite-dimensionality of the ring \(\mathbb{C}[ j,j' ]/( \varphi_{m_ 1}, \varphi_{m_ 2})\) provided the product \(m_ 1 m_ 2\) is not a square.
Clearly, here \(j\) is the elliptic modular function on the upper half plane and the modular polynomial \(\varphi_ m\) is defined by \[ \varphi_ m (j(\tau), j(\tau'))= \prod (j(\tau)- j(A \tau')) \] where the product is taken over all integral matrices \(A\bmod \text{SL}_ 2(\mathbb{Z})\) such that \(\text{det } A\) equals \(m\in \mathbb{N}_ +\).
To achieve this, the authors remark that \(A\) is finite if and only if the Cartier divisors \(T_{m_ i}\) given by \(\varphi_{m_ i}=0\), \(i=1,2,3\) intersect properly and then define the arithmetic intersection number \(T_{m_ 1} T_{m_ 2} T_{m_ 3}\) to be \(\log \# A\) which equals on the other hand a sum over rational primes \(\sum n(p)\log p\). The coefficient \(n(p)\) is nothing but the length of the finite \(\mathbb{Z}_ p\)-module \(A\otimes \mathbb{Z}_ p\) and is expressible as a sum of products of certain genus invariants of ternary quadratic forms, the diagonal of which is the triple \((m_ 1, m_ 2, m_ 3)\). Finally it turns out that these invariants are either Siegel local densities or local intersection multiplicities. As to the proof, they use the Serre-Tate lifting theorem and the deformation space of a formal group of height 2.
As a by-product astonishing relations between Fourier coefficients of certain Eisenstein series and various intersection numbers (on the surface space \(\mathbb{C}[ j,j' ]\)) are established.

MSC:

11E08 Quadratic forms over local rings and fields
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
11F32 Modular correspondences, etc.
14J30 \(3\)-folds
11F30 Fourier coefficients of automorphic forms
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References:

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