## 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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