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.

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Annexin A2 (AnxA2) has been shown to play multiple roles in growth, development, and metabolism, but the functions of AnxA2 and the signaling pathways associated with AnxA2 are still not fully understood. In this study, we aim to reveal whether and how AnxA2 could be involved in milk synthesis and proliferation of bovine mammary epithelial cells (BMECs). Using gene function study approaches, we found that AnxA2 positively regulates PIP3 level, phosphorylation of mTOR, and protein levels of SREBP-1c and Cyclin D1 leading to milk synthesis and cell proliferation. We further observed that both AnxA2-36 kD phosphorylated form and AnxA2-33 kD protein could be induced from AnxA2-36 kD protein in BMECs under methionine, leucine, estrogen or prolactin stimulation. These above results strongly demonstrate that AnxA2 functions as a critical regulator for amino acid or hormone-induced milk synthesis and cell proliferation via the PI3K-mTOR-SREBP-1c/Cyclin D1 signaling pathway.