This book resulted from a research conference in arithmetic geometry held at Arizona State University in March 1993. The papers describe important recent advances in arithmetic geometry. Several articles deal with $p$-adic modular forms of half-integral weight and their roles in arithmetic geometry. The volume also contains material on the Iwasawa theory of cyclotomic fields, elliptic curves, and function fields, including $p$-adic $L$-functions and $p$-adic height pairings. Other articles focus on the inverse Galois problem, fields of definition of abelian varieties with real multiplication, and computation of torsion groups of elliptic curves. The volume also contains a previously unpublished letter of John Tate, written to J. P. Serre in 1973, concerning Serre's conjecture on Galois representations. With contributions by some of the leading experts in the field, this book provides a look at the state of the art in arithmetic geometry.
Real Hilbertianity and the field of totally real numbers by M. D. Fried, D. Haran, and H. Volklein Galois groups with prescribed ramification by D. Harbater Note on the zeros of $p$-adic $L$-functions by T. Metsankyla La fonction $L p$-adique de Kubota-Leopoldt by B. Perrin-Riou Supersingular $p$-adic height pairings on elliptic curves by A. Plater Fields of definition of Abelian varieties with real multiplication by K. A. Ribet $p$-adic interpolation of half-integral weight modular forms by A. Sofer $\Lambda$-adic modular forms of half-integral weight and a $\Lambda$-adic Shintani lifting by G. Stevens The non-existence of certain Galois extensions of $\mathbb Q$ unramified outside $2$ by J. Tate Iwasawa theory and cyclotomic function fields by D. S. Thakur Slopes of modular forms by D. L. Ulmer On the Taylor coefficients of theta functions of $CM$ elliptic curves by F. R. Villegas Torsion groups of elliptic curves over cubic and certain biquadratic number fields by H. G. Zimmer.
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