Conrad, Brian 1970
Overview
Works:  20 works in 100 publications in 2 languages and 1,802 library holdings 

Genres:  Conference papers and proceedings 
Roles:  Author, Thesis advisor, Editor, Creator, 958 
Classifications:  QA564, 512.7 
Publication Timeline
.
Most widely held works by
Brian Conrad
Grothendieck duality and base change by
Brian Conrad(
Book
)
22 editions published between 2000 and 2002 in English and held by 347 WorldCat member libraries worldwide
Grothendieck's duality theory for coherent cohomology is a fundamental tool in algebraic geometry and number theory, in areas ranging from the moduli of curves to the arithmetic theory of modular forms. Presented is a systematic overview of the entire theory, including many basic definitions and a detailed study of duality on curves, dualizing sheaves, and Grothendieck's residue symbol. Along the way proofs are given of some widely used foundational results which are not proven in existing treatments of the subject, such as the general base change compatibility of the trace map for proper CohenMacaulay morphisms (e.g., semistable curves). This should be of interest to mathematicians who have some familiarity with Grothendieck's work and wish to understand the details of this theory
22 editions published between 2000 and 2002 in English and held by 347 WorldCat member libraries worldwide
Grothendieck's duality theory for coherent cohomology is a fundamental tool in algebraic geometry and number theory, in areas ranging from the moduli of curves to the arithmetic theory of modular forms. Presented is a systematic overview of the entire theory, including many basic definitions and a detailed study of duality on curves, dualizing sheaves, and Grothendieck's residue symbol. Along the way proofs are given of some widely used foundational results which are not proven in existing treatments of the subject, such as the general base change compatibility of the trace map for proper CohenMacaulay morphisms (e.g., semistable curves). This should be of interest to mathematicians who have some familiarity with Grothendieck's work and wish to understand the details of this theory
Pseudoreductive groups by
Brian Conrad(
Book
)
24 editions published between 2010 and 2015 in English and Undetermined and held by 249 WorldCat member libraries worldwide
"Pseudoreductive groups arise naturally in the study of general smooth linear algebraic groups over nonperfect fields and have many important applications. This selfcontained monograph provides a comprehensive treatment of the theory of pseudoreductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudoparabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, of general smooth connected algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book as it develops tools likely to be used in tackling other problems"
24 editions published between 2010 and 2015 in English and Undetermined and held by 249 WorldCat member libraries worldwide
"Pseudoreductive groups arise naturally in the study of general smooth linear algebraic groups over nonperfect fields and have many important applications. This selfcontained monograph provides a comprehensive treatment of the theory of pseudoreductive groups and gives their classification in a usable form. The authors present numerous new results and also give a complete exposition of Tits' structure theory of unipotent groups. They prove the conjugacy results (conjugacy of maximal split tori, minimal pseudoparabolic subgroups, maximal split unipotent subgroups) announced by Armand Borel and Jacques Tits, and also give the Bruhat decomposition, of general smooth connected algebraic groups. Researchers and graduate students working in any related area, such as algebraic geometry, algebraic group theory, or number theory, will value this book as it develops tools likely to be used in tackling other problems"
Arithmetic algebraic geometry(
Book
)
11 editions published between 2001 and 2008 in English and held by 248 WorldCat member libraries worldwide
11 editions published between 2001 and 2008 in English and held by 248 WorldCat member libraries worldwide
Complex multiplication and lifting problems by
ChingLi Chai(
Book
)
8 editions published in 2014 in English and held by 216 WorldCat member libraries worldwide
Abelian varieties with complex multiplication lie at the origins of class field theory, and they play a central role in the contemporary theory of Shimura varieties. They are special in characteristic 0 and ubiquitous over finite fields. This book explores the relationship between such abelian varieties over finite fields and over arithmetically interesting fields of characteristic 0 via the study of several natural CM lifting problems which had previously been solved only in special cases. In addition to giving complete solutions to such questions, the authors provide numerous examples to illustrate the general theory and present a detailed treatment of many fundamental results and concepts in the arithmetic of abelian varieties, such as the Main Theorem of Complex Multiplication and its generalizations, the finer aspects of Tate's work on abelian varieties over finite fields, and deformation theory. This book provides an ideal illustration of how modern techniques in arithmetic geometry (such as descent theory, crystalline methods, and group schemes) can be fruitfully combined with class field theory to answer concrete questions about abelian varieties. It will be a useful reference for researchers and advanced graduate students at the interface of number theory and algebraic geometry.  Provided by publisher
8 editions published in 2014 in English and held by 216 WorldCat member libraries worldwide
Abelian varieties with complex multiplication lie at the origins of class field theory, and they play a central role in the contemporary theory of Shimura varieties. They are special in characteristic 0 and ubiquitous over finite fields. This book explores the relationship between such abelian varieties over finite fields and over arithmetically interesting fields of characteristic 0 via the study of several natural CM lifting problems which had previously been solved only in special cases. In addition to giving complete solutions to such questions, the authors provide numerous examples to illustrate the general theory and present a detailed treatment of many fundamental results and concepts in the arithmetic of abelian varieties, such as the Main Theorem of Complex Multiplication and its generalizations, the finer aspects of Tate's work on abelian varieties over finite fields, and deformation theory. This book provides an ideal illustration of how modern techniques in arithmetic geometry (such as descent theory, crystalline methods, and group schemes) can be fruitfully combined with class field theory to answer concrete questions about abelian varieties. It will be a useful reference for researchers and advanced graduate students at the interface of number theory and algebraic geometry.  Provided by publisher
Classification of pseudoreductive groups by
Brian Conrad(
Book
)
11 editions published between 2015 and 2016 in English and held by 166 WorldCat member libraries worldwide
In the earlier monograph Pseudoreductive Groups, Brian Conrad, Ofer Gabber, and Gopal Prasad explored the general structure of pseudoreductive groups. In this new book, Classification of Pseudoreductive Groups, Conrad and Prasad go further to study the classification over an arbitrary field. An isomorphism theorem proved here determines the automorphism schemes of these groups. The book also gives a TitsWitt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudosplit forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in Classification of Pseudoreductive Groups will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic
11 editions published between 2015 and 2016 in English and held by 166 WorldCat member libraries worldwide
In the earlier monograph Pseudoreductive Groups, Brian Conrad, Ofer Gabber, and Gopal Prasad explored the general structure of pseudoreductive groups. In this new book, Classification of Pseudoreductive Groups, Conrad and Prasad go further to study the classification over an arbitrary field. An isomorphism theorem proved here determines the automorphism schemes of these groups. The book also gives a TitsWitt type classification of isotropic groups and displays a cohomological obstruction to the existence of pseudosplit forms. Constructions based on regular degenerate quadratic forms and new techniques with central extensions provide insight into new phenomena in characteristic 2, which also leads to simplifications of the earlier work. A generalized standard construction is shown to account for all possibilities up to mild central extensions. The results and methods developed in Classification of Pseudoreductive Groups will interest mathematicians and graduate students who work with algebraic groups in number theory and algebraic geometry in positive characteristic
Autour des schémas en groupes : group schemes, a celebration of SGA3 by
Sylvain Brochard(
Book
)
9 editions published between 2014 and 2016 in French and English and held by 33 WorldCat member libraries worldwide
9 editions published between 2014 and 2016 in French and English and held by 33 WorldCat member libraries worldwide
Autour des schémas en groupes : group schemes, a celebration of SGA3 by École d'été "Schémas en groupes"(
Book
)
in French and held by 25 WorldCat member libraries worldwide
Volume 1 "contains the first part of the lecture notes of the Summer school 'Group Schemes, introduction to the SGA3 seminar of DemazureGrothendieck, ' which was held at the Centre International de Rencontres Mathématiques (CIRM) at Luminy in September 2011. This summer school was devoted to the theory of group schemes and especially of reductive group schemes. The contributions in this first part are expanded versions of the talks introducing Grothendieck topologies (S. Brochard), group schemes of multiplicative type (J. Oesterlé) and reductive group schemes (B. Conrad)."
in French and held by 25 WorldCat member libraries worldwide
Volume 1 "contains the first part of the lecture notes of the Summer school 'Group Schemes, introduction to the SGA3 seminar of DemazureGrothendieck, ' which was held at the Centre International de Rencontres Mathématiques (CIRM) at Luminy in September 2011. This summer school was devoted to the theory of group schemes and especially of reductive group schemes. The contributions in this first part are expanded versions of the talks introducing Grothendieck topologies (S. Brochard), group schemes of multiplicative type (J. Oesterlé) and reductive group schemes (B. Conrad)."
Modular forms and the Ramanujan Conjecture by
Brian Conrad(
Book
)
2 editions published between 2008 and 2010 in English and held by 4 WorldCat member libraries worldwide
2 editions published between 2008 and 2010 in English and held by 4 WorldCat member libraries worldwide
Zerodistribution and size of the Riemann zetafunction on the critical line by Maksym Radziwill(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We investigate the analytic properties of the Riemann zetafunction. We focus on the average size of the Riemann zetafunction on the critical line and on the vertical distribution of its zeros on the critical line
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We investigate the analytic properties of the Riemann zetafunction. We focus on the average size of the Riemann zetafunction on the critical line and on the vertical distribution of its zeros on the critical line
Relèvements de représentations galoisiennes à valeurs dans des groupes algébriques by Hong Quoc Auguste Hoang Duc(
)
1 edition published in 2015 in French and held by 1 WorldCat member library worldwide
Soient 1 > N > H > H' > 1 une suite exacte centrale de groupes algébriques sur Q_p^alg et F un corps de nombres. Etant donnée une représentation Galoisienne r' : Gal_F > H', on s'intéresse à ses relèvements à valeurs dans H à travers le morphisme H > H'. Un relèvement r : Gal_F > H sera dit minimal, s'il est nonramifié aux places où r' est nonramifiée et est de Rham/semistable/cristalline aux places divisant p si r' l'est. Dans cette thèse, nous montrons l'existence de relèvements minimaux dans certains cas
1 edition published in 2015 in French and held by 1 WorldCat member library worldwide
Soient 1 > N > H > H' > 1 une suite exacte centrale de groupes algébriques sur Q_p^alg et F un corps de nombres. Etant donnée une représentation Galoisienne r' : Gal_F > H', on s'intéresse à ses relèvements à valeurs dans H à travers le morphisme H > H'. Un relèvement r : Gal_F > H sera dit minimal, s'il est nonramifié aux places où r' est nonramifiée et est de Rham/semistable/cristalline aux places divisant p si r' l'est. Dans cette thèse, nous montrons l'existence de relèvements minimaux dans certains cas
Finite Honda systems and supersingular elliptic curves by
Brian Conrad(
)
1 edition published in 1996 in English and held by 1 WorldCat member library worldwide
1 edition published in 1996 in English and held by 1 WorldCat member library worldwide
Algebraic modular forms on definite orthogonal groups by Daniel Kim Murphy(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
The present thesis examines explicit computation with modular forms on definite orthogonal groups over the rational numbers. The space of functions on the genus of a definite quadratic form affords a natural representation of the Hecke algebra, which realizes the function space as a space of modular forms. The algorithm of the thesis computes the genus of a quadratic form as well as Hecke operators on the associated space of modular forms. A formula derived in the thesis yields Satake parameters from Hecke eigenvalues. Explicit examples of Satake parameters verify the correctness of the algorithm and the formula by their agreement with the predictions of the Arthur conjectures
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
The present thesis examines explicit computation with modular forms on definite orthogonal groups over the rational numbers. The space of functions on the genus of a definite quadratic form affords a natural representation of the Hecke algebra, which realizes the function space as a space of modular forms. The algorithm of the thesis computes the genus of a quadratic form as well as Hecke operators on the associated space of modular forms. A formula derived in the thesis yields Satake parameters from Hecke eigenvalues. Explicit examples of Satake parameters verify the correctness of the algorithm and the formula by their agreement with the predictions of the Arthur conjectures
Equivariant torsion and base change by Michael Lipnowski(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
The CheegerMuller theorem provides a spectral means of obtaining information about torsion in the cohomology of compact manifolds. Using this remarkable theorem together with trace formula comparisons, we prove comparisons between torsion in the cohomology of two manifolds of arithmetic origin related by base change
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
The CheegerMuller theorem provides a spectral means of obtaining information about torsion in the cohomology of compact manifolds. Using this remarkable theorem together with trace formula comparisons, we prove comparisons between torsion in the cohomology of two manifolds of arithmetic origin related by base change
Moduli spaces of PTstable objects by ChiehCheng Lo(
)
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We develop techniques for performing semistable reduction on a flat family of objects in the heart of a tstructure on the bounded derived category of coherent sheaves of a smooth projective threefold. Then we show that, with respect to Bayer's PTstability function, the semistable objects in the heart of perverse sheaves form a proper Artin stack of finite type, provided the rank is nonzero, and the rank and degree are coprime
1 edition published in 2010 in English and held by 1 WorldCat member library worldwide
We develop techniques for performing semistable reduction on a flat family of objects in the heart of a tstructure on the bounded derived category of coherent sheaves of a smooth projective threefold. Then we show that, with respect to Bayer's PTstability function, the semistable objects in the heart of perverse sheaves form a proper Artin stack of finite type, provided the rank is nonzero, and the rank and degree are coprime
Padic Hodge theory in rigid analytic families by Rebecca Michal Bellovin(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
In this thesis, we study padic Hodge theory in rigid analytic families. Roughly speaking, padic Hodge theory is the study of padic representations of padic Galois groups. One introduces certain padic period rings B, such as B_{HT}, B_{dR}, B_{st}, and B_{cris}, and uses them to define functors D_B(.) from the category of padic Galois representations to various categories of linear algebra data. In the first half of this thesis, we study generalizations of these functors to families of padic Galois representations with rigid analytic coefficients. We prove that the functors D_{HT}(.) and D_{dR}(.) are coherent sheaves, and we prove that the Badmissible locus is a closed subspace of the base. In the second half of this thesis, we study the linear algebra data which arises from families of potentially semistable Galois representations valued in a connected reductive group G. We prove that for any G, the moduli space of linear algebra data is reduced and locally a complete intersection, and we deduce that potentially semistable deformation rings are generically smooth and equidimensional
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
In this thesis, we study padic Hodge theory in rigid analytic families. Roughly speaking, padic Hodge theory is the study of padic representations of padic Galois groups. One introduces certain padic period rings B, such as B_{HT}, B_{dR}, B_{st}, and B_{cris}, and uses them to define functors D_B(.) from the category of padic Galois representations to various categories of linear algebra data. In the first half of this thesis, we study generalizations of these functors to families of padic Galois representations with rigid analytic coefficients. We prove that the functors D_{HT}(.) and D_{dR}(.) are coherent sheaves, and we prove that the Badmissible locus is a closed subspace of the base. In the second half of this thesis, we study the linear algebra data which arises from families of potentially semistable Galois representations valued in a connected reductive group G. We prove that for any G, the moduli space of linear algebra data is reduced and locally a complete intersection, and we deduce that potentially semistable deformation rings are generically smooth and equidimensional
Gvalued flat deformations and local models by Brandon William Allen Levin(
)
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We construct resolutions of Gvalued local Galois deformation rings by moduli spaces of Kisin modules with Gstructure when l = p. This generalizes Mark Kisin's work on potentially semistable deformation rings. In the case of flat deformations, we prove a structural result about these resolutions which relates the connected components of Gvalued flat deformation rings to the connected components of projective varieties in characteristic p, which are moduli spaces of linear algebra data. As a key step in the study of these resolutions, we prove a full faithfulness result in integral padic Hodge theory. We also generalize results of Pappas and Zhu on local models of Shimura varieties to groups arising from Weil restrictions
1 edition published in 2013 in English and held by 1 WorldCat member library worldwide
We construct resolutions of Gvalued local Galois deformation rings by moduli spaces of Kisin modules with Gstructure when l = p. This generalizes Mark Kisin's work on potentially semistable deformation rings. In the case of flat deformations, we prove a structural result about these resolutions which relates the connected components of Gvalued flat deformation rings to the connected components of projective varieties in characteristic p, which are moduli spaces of linear algebra data. As a key step in the study of these resolutions, we prove a full faithfulness result in integral padic Hodge theory. We also generalize results of Pappas and Zhu on local models of Shimura varieties to groups arising from Weil restrictions
General existence theorems in moduli theory by Jack Kingsbury Hall(
)
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
In this thesis, we prove that there is an algebraic stack parameterizing all curves. The curves that appear in this algebraic stack are allowed to be arbitrarily singular, nonreduced, disconnected, and reducible. We also prove the boundedness of the open substack parameterizing reduced and connected curves with fixed arithmetic genus g and at most e irreducible components. We also show that for essentially any algebraic stack, there is an algebraic stack, the Hilbert stack, parameterizing quasifinite maps to the stack. The technical heart of this result is a generalization of formal GAGA to a nonseparated morphism of algebraic stacks, something that was previously unknown for a morphism of schemes. We also employ derived algebraic geometry, in an essential way, to prove the algebraicity of the Hilbert stack. The Hilbert stack, for algebraic spaces, was claimed to exist by M. Artin (1974), but was left unproved due to a lack of foundational results for nonseparated algebraic spaces. Finally, we generalize the fundamental GAGA results of J.P. Serre (1956) in three waysto the nonseparated setting, to stacks, and to families. As an application of these results, we show that analytic compactifications of the moduli stack of smooth curves possessing modular interpretations are algebraizable
1 edition published in 2011 in English and held by 1 WorldCat member library worldwide
In this thesis, we prove that there is an algebraic stack parameterizing all curves. The curves that appear in this algebraic stack are allowed to be arbitrarily singular, nonreduced, disconnected, and reducible. We also prove the boundedness of the open substack parameterizing reduced and connected curves with fixed arithmetic genus g and at most e irreducible components. We also show that for essentially any algebraic stack, there is an algebraic stack, the Hilbert stack, parameterizing quasifinite maps to the stack. The technical heart of this result is a generalization of formal GAGA to a nonseparated morphism of algebraic stacks, something that was previously unknown for a morphism of schemes. We also employ derived algebraic geometry, in an essential way, to prove the algebraicity of the Hilbert stack. The Hilbert stack, for algebraic spaces, was claimed to exist by M. Artin (1974), but was left unproved due to a lack of foundational results for nonseparated algebraic spaces. Finally, we generalize the fundamental GAGA results of J.P. Serre (1956) in three waysto the nonseparated setting, to stacks, and to families. As an application of these results, we show that analytic compactifications of the moduli stack of smooth curves possessing modular interpretations are algebraizable
Controlling ramification in number fields by Simon RubinsteinSalzedo(
)
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
This thesis focuses on two aspects of limited ramification and is split up into two independent sections. The first section (which comprises the second and third chapters) is on the distribution of class groups of cyclic cubic fields. We propose an explanation for the discrepancy between the observed number of cyclic cubics whose 2class group is C_2 x C_2 and the number predicted by the CohenLenstra heuristics, in terms of an invariant living in a quotient of the Schur multiplier group. We also show that, in some cases, the definition of the invariant can be simplified greatly, and we compute 10^5 examples. The second section (which comprises the fourth and fifth chapters) discusses branched covers of algebraic curves, especially covers of elliptic curves with one branch point. We produce some techniques that allow us to write down explicit equations for such maps, and then we give examples of number fields which arise from such covers. Finally, we present some possibilities for future works that the author hopes to pursue
1 edition published in 2012 in English and held by 1 WorldCat member library worldwide
This thesis focuses on two aspects of limited ramification and is split up into two independent sections. The first section (which comprises the second and third chapters) is on the distribution of class groups of cyclic cubic fields. We propose an explanation for the discrepancy between the observed number of cyclic cubics whose 2class group is C_2 x C_2 and the number predicted by the CohenLenstra heuristics, in terms of an invariant living in a quotient of the Schur multiplier group. We also show that, in some cases, the definition of the invariant can be simplified greatly, and we compute 10^5 examples. The second section (which comprises the fourth and fifth chapters) discusses branched covers of algebraic curves, especially covers of elliptic curves with one branch point. We produce some techniques that allow us to write down explicit equations for such maps, and then we give examples of number fields which arise from such covers. Finally, we present some possibilities for future works that the author hopes to pursue
Some problems in multiplicative number theory by Junsoo Ha(
)
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
Let q be a power of a prime p. In the first part of this thesis, we establish the upper bound of the least prime primitive root mod q by p^3.1. We say a polynomial in F_q [T] is msmooth if all of its irreducible factors are of degree less than or equal to m. Let N(n, m) be the number of solutions to the polynomial equation X+Y=2Z where all variables are msmooth polynomials of degree n. In the second part of this thesis, we establish a lower bound on N(n, m) when (8+d) log_q n <= m <= n^1/2 for small d, and prove the analog of the xyz conjecture of Lagarias and Soundararajan in the polynomial rings over finite fields
1 edition published in 2014 in English and held by 1 WorldCat member library worldwide
Let q be a power of a prime p. In the first part of this thesis, we establish the upper bound of the least prime primitive root mod q by p^3.1. We say a polynomial in F_q [T] is msmooth if all of its irreducible factors are of degree less than or equal to m. Let N(n, m) be the number of solutions to the polynomial equation X+Y=2Z where all variables are msmooth polynomials of degree n. In the second part of this thesis, we establish a lower bound on N(n, m) when (8+d) log_q n <= m <= n^1/2 for small d, and prove the analog of the xyz conjecture of Lagarias and Soundararajan in the polynomial rings over finite fields
Autour des schémas en groupes : école d'été "Schémas en groupes" by École d'Été Schémas en Groupes(
Book
)
in French and held by 0 WorldCat member libraries worldwide
in French and held by 0 WorldCat member libraries worldwide
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Related Identities
 Prasad, Gopal Editor
 Gabber, Ofer 1958
 Rubin, Karl Editor
 Chai, ChingLi Author
 Oort, Frans 1935
 Brochard, Sylvain 1979 Author Editor
 Oesterlé, Joseph Collector Editor
 Edixhoven, B. (Bas) 1962 Editor
 SpringerLink (Service en ligne)
 Chaudouard, PierreHenri (1973....).
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Associated Subjects
Abelian varieties Arithmetical algebraic geometry Duality theory (Mathematics) Forms, Modular Galois theory Geometry, Algebraic Group schemes (Mathematics) Group theory Linear algebraic groups Mathematics Multiplication, Complex Number theory Representations of groups Schemes (Algebraic geometry) Topology
Alternative Names
Brian Conrad Amerikaans wiskundige
Brian Conrad amerikansk matematikar
Brian Conrad amerikansk matematiker
Brian Conrad matematico americano
Brian Conrad mathématicien américain
Brian Conrad USamerikanischer Mathematiker
Conrad, B.
Conrad B. 1970....
Conrad, Brian.
Conrad, Brian David
Conrad, Brian David 1970
Брайан Конрад
برایان کنراد ریاضیدان آمریکایی
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