# Short introduction to heights and Arakelov theory

# Short introduction to heights and Arakelov theory

This chapter discusses bounding the heights of the coefficients of minimal polynomial *P*. As was hinted at in Chapter 3, such bounds are obtained using Arakelov theory, a tool that is discussed in this chapter. It is not at all excluded that a direct approach to bound the coefficients of *P* exists, thus avoiding the complicated theory that we use. On the other hand, it is clear that the use of Arakelov theory provides a way to split the work into smaller steps, and that the quantities occurring in each step are intrinsic in the sense that they do not depend on coordinate systems or other choices one could make. This method does not depend on cancellations of terms in the estimates we will do; all contributions encountered can be bounded appropriately.

*Keywords:*
modular forms, minimal polynomial, Arakelov theory, height functions, arithmetic surfaces

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