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A central concern of number theory is the study of localtoglobal principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a localtoglobal principle: Weils conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of Gbundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weils conjecture, based on the geometry of the moduli stack of Gbundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting ladic sheaves. Using this theory, Dennis Gaitsgory and Jacob Lurie articulate a different localtoglobal principle: a product formula that expresses the cohomology of the moduli stack of Gbundles (a global object) as a tensor product of local factors.Using a version of the GrothendieckLefschetz trace formula, Gaitsgory and Lurie show that this product formula implies Weils conjecture. The proof of the product formula will appear in a sequel volume.
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This product may contain chemicals known to the State of California to cause cancer,
birth defects, or other reproductive harm.
I was hoping for easier explanations. This volume is very technical, and I found myself lost pretty quickly. I guess I should've done my research before purchasing. Maybe it’s just not the right fit for my level of math knowledge.
S
Sofia Reyes
Essential for advanced study
If you're delving into the world of function fields, this book is essential. The detail and rigor put into each theorem are impressive. It’s nicely bound too, which is a plus for my collection!
T
Tariq El-Masri
Thorough but very complex
Weil's Conjecture Volume I is not for the faint-hearted! I've struggled with some of the topics, but that's expected given the advanced nature of the content. If you're really into mathematics, this is a crucial read, but be prepared for a challenge.
E
Elijah Porter
Niche interest, but worthwhile
I bought this for my research on algebraic geometry. While it's quite dense and requires patience, the insights it offers are invaluable. Just know this isn't a light read—it's definitely for those with a strong background in mathematics.
A
Anjali Patel
Great resource, but hard to navigate
I appreciate the depth of this book, but the organization could have been clearer. Some sections felt a bit jumbled, making it hard to follow at times. I’m still glad I have it on my shelf as a reference.
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