{"title":"Nicholas M Katz","description":null,"products":[{"product_id":"random-matrices-frobenius-eigenvalues-and-monodromy-book-nicholas-m-katz-9780821810170","title":"Random Matrices, Frobenius Eigenvalues, and Monodromy","description":"The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and $L$-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinity and related techniques from orthogonal polynomials and Fredholm determinants.","brand":"WoB","offers":[{"title":"US \/ GOOD \/ SBYB","offer_id":50369541112081,"sku":"CIN0821810170G","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0821810170.jpg?v=1763474427"},{"product_id":"exponential-sums-hypergeometric-sheaves-and-monodromy-groups-book-nicholas-m-katz-9780691272252","title":"Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups","description":"\u003cp\u003e\u003cb\u003eAn examination of some of the remarkable connections between group theory and arithmetic algebraic geometry over finite fields\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003eExponential sums have been of great interest ever since Gauss, and their importance in analytic number theory goes back a century to Kloosterman. Grothendieck’s creation of the machinery of l-adic cohomology led to the understanding that families of exponential sums give rise to local systems, while Deligne, who gave his general equidistribution theorem after proving the Riemann hypothesis part of the Weil conjectures, established the importance of the monodromy groups of these local systems. Deligne’s theorem shows that the monodromy group of the local system incarnating a given family of exponential sums determines key statistical properties of the family of exponential sums in question. Despite the apparent simplicity of this relation of monodromy groups to statistical properties, the actual determination of the monodromy group in any particular situation is highly nontrivial and leads to many interesting questions.\u003cbr\u003e\u003cbr\u003eThis book is devoted to the determination of the monodromy groups attached to various explicit families of exponential sums, especially those attached to hypergeometric sheaves, arguably the simplest local systems on G_m, and to some simple (in the sense of simple to write down) one-parameter families of one-variable sums. These last families turn out to have surprising connections to hypergeometric sheaves. One of the main technical advances of this book is to bring to bear a group-theoretic condition (S+), which, when it applies, implies very strong structural constraints on the monodromy group, and to show that (S+) does indeed apply to the monodromy groups of most hypergeometric sheaves.\u003c\/p\u003e","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":51592945402129,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":51592945467665,"sku":"NIN9780691272252","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ GARDNERS","offer_id":51597867712785,"sku":"NGR9780691272252","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0691272255.jpg?v=1750848055"},{"product_id":"exponential-sums-hypergeometric-sheaves-and-monodromy-groups-book-nicholas-m-katz-9780691272269","title":"Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups","description":"\u003cp\u003e\u003cb\u003eAn examination of some of the remarkable connections between group theory and arithmetic algebraic geometry over finite fields\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003eExponential sums have been of great interest ever since Gauss, and their importance in analytic number theory goes back a century to Kloosterman. Grothendieck’s creation of the machinery of l-adic cohomology led to the understanding that families of exponential sums give rise to local systems, while Deligne, who gave his general equidistribution theorem after proving the Riemann hypothesis part of the Weil conjectures, established the importance of the monodromy groups of these local systems. Deligne’s theorem shows that the monodromy group of the local system incarnating a given family of exponential sums determines key statistical properties of the family of exponential sums in question. Despite the apparent simplicity of this relation of monodromy groups to statistical properties, the actual determination of the monodromy group in any particular situation is highly nontrivial and leads to many interesting questions.\u003cbr\u003e\u003cbr\u003eThis book is devoted to the determination of the monodromy groups attached to various explicit families of exponential sums, especially those attached to hypergeometric sheaves, arguably the simplest local systems on G_m, and to some simple (in the sense of simple to write down) one-parameter families of one-variable sums. These last families turn out to have surprising connections to hypergeometric sheaves. One of the main technical advances of this book is to bring to bear a group-theoretic condition (S+), which, when it applies, implies very strong structural constraints on the monodromy group, and to show that (S+) does indeed apply to the monodromy groups of most hypergeometric sheaves.\u003c\/p\u003e","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":51597867942161,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ GARDNERS","offer_id":51597868073233,"sku":"NGR9780691272269","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"US \/ NEW \/ INGRAM","offer_id":51612192801041,"sku":"NIN9780691272269","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/0691272263.jpg?v=1751136449"},{"product_id":"random-matrices-frobenius-eigenvalues-and-monodromy-book-nicholas-m-katz-9781470475079","title":"Random Matrices, Frobenius Eigenvalues, and Monodromy","description":"The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. The authors draw upon many disparate areas of mathematics from algebraic geometry, moduli spaces, mondromy, equidistribution, and the Weil conjectures to probability theory and the compact classical groups.","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":51629712212241,"sku":"","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ GARDNERS","offer_id":51629712703761,"sku":"NGR9781470475079","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/1470475073.jpg?v=1761990458"},{"product_id":"twisted-l-functions-and-monodromy-book-nicholas-m-katz-9780691091518","title":"Twisted L-Functions and Monodromy","description":"For hundreds of years, the study of elliptic curves has played a central role in mathematics. This book explores: What is the average rank of elliptic curves, and how does the rank vary in various kinds of families of elliptic curves? It is suitable for those interested in number theory and algebraic geometry.","brand":"WoB","offers":[{"title":"- \/ - \/ -","offer_id":51803201011985,"sku":null,"price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ GARDNERS","offer_id":51803201470737,"sku":"NGR9780691091518","price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ VERY_GOOD \/ INTERNAL","offer_id":53130938515729,"sku":"GOR003837762","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9780691091518.jpg?v=1752236676"},{"product_id":"convolution-and-equidistribution-book-nicholas-m-katz-9780691153315","title":"Convolution and Equidistribution","description":"Convolution and Equidistribution explores an important aspect of number theory--the theory of exponential sums over finite fields and their Mellin transforms--from a new, categorical point of view. The book presents fundamentally important results and a plethora of examples, opening up new directions in the subject. The finite-field Mellin transform (of a function on the multiplicative group of a finite field) is defined by summing that function against variable multiplicative characters. The basic question considered in the book is how the values of the Mellin transform are distributed (in a probabilistic sense), in cases where the input function is suitably algebro-geometric. This question is answered by the book's main theorem, using a mixture of geometric, categorical, and group-theoretic methods. By providing a new framework for studying Mellin transforms over finite fields, this book opens up a new way for researchers to further explore the subject.","brand":"WoB","offers":[{"title":"US \/ NEW \/ INGRAM","offer_id":53081715441937,"sku":"NIN9780691153315","price":0.0,"currency_code":"GBP","in_stock":false},{"title":"GB \/ NEW \/ GARDNERS","offer_id":53241053675793,"sku":"NGR9780691153315","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9780691153315.jpg?v=1769956373"},{"product_id":"arithmetic-moduli-of-elliptic-curves-book-nicholas-m-katz-9780691083520","title":"Arithmetic Moduli of Elliptic Curves","description":"This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's \"Fundamenta Nova\" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld.","brand":"WoB","offers":[{"title":"US \/ NEW \/ INGRAM","offer_id":53362523373841,"sku":"NIN9780691083520","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9780691083520.jpg?v=1774991043"},{"product_id":"exponential-sums-and-differential-equations-book-nicholas-m-katz-9780691085999","title":"Exponential Sums and Differential Equations","description":"Deals with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. This book discusses results about differential equations and their differential galois groups (G) and one-parameter families of exponential sums and their geometric monodromy groups (G).","brand":"WoB","offers":[{"title":"- \/ - \/ INTERNAL","offer_id":53381303959825,"sku":null,"price":0.0,"currency_code":"GBP","in_stock":true},{"title":"US \/ NEW \/ INGRAM","offer_id":53381304353041,"sku":"NIN9780691085999","price":0.0,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9780691085999.jpg?v=1775525199"}],"url":"https:\/\/www.worldofbooks.com\/en-gb\/collections\/author-books-by-nicholas-m-katz.oembed","provider":"World of Books ","version":"1.0","type":"link"}