First published by Oxford University Press, this text is a systematic introduction to algebraic geometry with a clear arithmetic purpose . Its central theme is the study of arithmetic curves—geometric objects defined over the integers or over complete discrete valuation rings. In essence, it develops the modern tools of algebraic geometry (schemes, sheaves, cohomology) specifically to understand the deep properties of curves over non-algebraically closed fields, and crucially, over Dedekind rings.
For graduate students and researchers straddling the line between pure algebraic geometry and number theory, finding the right textbook can feel like a quest for the Holy Grail. You need a text that doesn't just teach schemes and cohomology, but one that builds a bridge to Diophantine geometry, class field theory, and the arithmetic of curves. Enter Qing Liu’s —a masterpiece that delivers exactly that. liu algebraic geometry and arithmetic curves pdf
), Liu’s work bridges the gap between abstract scheme theory and the concrete study of arithmetic surfaces and curves over Dedekind domains. First published by Oxford University Press, this text
The prevalence of the PDF version of this text has had a distinct impact on how the subject is learned. Unlike Hartshorne, which often requires significant "filling in the gaps" by the reader (a pedagogical style that can be discouraging without a mentor), Liu is relatively self-contained. For graduate students and researchers straddling the line
The first few chapters provide a comprehensive introduction to the category of schemes, morphisms, and sheaves. Unlike other texts, Liu introduces the necessary commutative algebra (like flatness and completion) exactly when it is needed to understand the geometry. 2. Cohomology and Finiteness
Liu, Qing. Algebraic Geometry and Arithmetic Curves . Oxford Graduate Texts in Mathematics, 2002. (Translated by Reinie Erné)
Qing Liu’s work remains a cornerstone of modern number theory. Whether you are studying the Néron models of abelian varieties or the intersections of divisors on an arithmetic surface, this book will likely be your primary guide.