The effective Shafarevich conjecture for abelian varieties of -type

In this article we establish the effective Shafarevich conjecture for abelian varieties over ${\mathbb Q}$ of ${\text {GL}_2}$ -type. The proof combines Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory. Our result opens the way for the effective study of integral points on certain higher dimensional moduli schemes such as, for example, Hilbert modular varieties.

An effective Arakelov-theoretic version of the hyperbolic isogeny theorem

For any integereand hyperbolic curveXover$\overline{\mathbb Q}$, Mochizuki showed that there are only finitely many isomorphism classes of hyperbolic curvesYof Euler characteristicewith the same universal cover asX. We use Arakelov theory to prove an effective version of this finiteness statement.

The Gross-Zagier Formula on Shimura Curves

This comprehensive account of the Gross–Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross–Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross–Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross–Zagier formula is reduced to local formulas. This book will be of great use to students wishing to enter this area and to those already working in it.

Decomposition of the Geometric Kernel

This chapter describes the decomposition of the geometric kernel. It considers the assumptions on the Schwartz function and decomposes the height series into local heights using arithmetic models. The intersections with the Hodge bundles are zero, and a decomposition to a sum of local heights by standard results in Arakelov theory is achieved. The chapter proceeds by reviewing the definition of the Néeron–Tate height and shows how to compute it by the arithmetic Hodge index theorem. When there is no horizontal self-intersection, the height pairing automatically decomposes to a summation of local pairings. The chapter proves that the contribution of the Hodge bundles in the height series is zero. It also compares two kernel functions and states the computational result. It concludes by deducing the kernel identity.

Applying Arakelov theory

This chapter starts applying Arakelov theory in order to derive a bound for the height of the coefficients of the polynomials Pno hexa conversion found as in (8.2.10). It proceeds in a few steps. The first step is to relate the height of the bₗ(Qsubscript x,i) as in the ptrevious chapter to intersection numbers on Xₗ. The second step is to get some control on the difference of the divisors D₀ and Dₓ as in (3.4). Certain intersection numbers concerning this difference are bounded in Theorem 9.2.5, in terms of a number of invariants in the Arakelov theory on modular curves Xₗ. These invariants will then be bounded in terms of l in Chapter 11. The chapter formulates the most important results, Theorem 9.1.3, Theorem 9.2.1, and Theorem 9.2.5, in the context of curves over number fields, that is, outside the context of modular curves.