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Papers by Nicholas Katz

Research paper thumbnail of A note on Galois representations with big image

L’Enseignement Mathématique

Research paper thumbnail of Rigid local systems and alternating groups

Tunisian Journal of Mathematics

Contents 1. Introduction 1 2. The local systems in general 2 3. The candidate local systems for A... more Contents 1. Introduction 1 2. The local systems in general 2 3. The candidate local systems for Alt(2q) 4 4. Basic facts about H n 5 5. basic facts about H 2q−1 6 6. Basic facts about the group G geom for F (k, 2q − 1, ψ) 7 7. The third moment of F (k, 2q − 1, ψ) and of G(k, 2q − 1, ψ) 8 8. Exact determination of G arith 11 9. Identifying the group 12 References 25

Research paper thumbnail of On a Question of Zannier

Research paper thumbnail of The divisor function over arithmetic progressions

Research paper thumbnail of From Clausen to Carlitz: Low-Dimensional Spin Groups and Identities among Character Sums

Moscow Mathematical Journal

Research paper thumbnail of On the intersection matrix of a hypersurface

Annales scientifiques de l'École normale supérieure

Research paper thumbnail of Witt Vectors and a Question of Rudnick and Waxman

International Mathematics Research Notices

Research paper thumbnail of Internal reconstruction of unit-root <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span>-crystals via expansion-coefficients. With an appendix by Luc Illusie

Annales scientifiques de l'École normale supérieure

Internal reconstruction of unit-root F-crystals via expansioncoefficients. With an appendix by Lu... more Internal reconstruction of unit-root F-crystals via expansioncoefficients. With an appendix by Luc Illusie Annales scientifiques de l'É.N.S. 4 e série, tome 18, n o 2 (1985), p. 245-285. <http://www.numdam.org/item?id=ASENS_1985_4_18_2_245_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1985, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens), implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Research paper thumbnail of On the differentiation of De Rham cohomology classes with respect to parameters

Journal of Mathematics of Kyoto University

Research paper thumbnail of Corrections to : “A conjecture in the arithmetic theory of differential equations”

Bulletin de la Société mathématique de France, 1982

Research paper thumbnail of A conjecture in the arithmetic theory of differential equations

Bulletin de la Société mathématique de France, 1982

http://www.numdam.org/item?id=BSMF\_1982\_\_110\_\_203\_0 © Bulletin de la S. M. F., 1982, tous droit... more <http://www.numdam.org/item?id=BSMF_1982__110__203_0> © Bulletin de la S. M. F., 1982, tous droits réservés. L'accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html), implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Research paper thumbnail of Random Matrices, Frobenius Eigenvalues, and Monodromy

Colloquium Publications, 1998

The main topic of this book is the deep relation between the spacings between zeros of zeta and L... more 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 Well 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. It will be useful and interesting to researchers and graduate students working in any of these areas.

Research paper thumbnail of Groupes de Monodromie en Géométrie Algébrique

Lecture Notes in Mathematics, 1972

Research paper thumbnail of Slope filtration of F-crystals

Asterisque Societe Mathematique De France, 1979

Research paper thumbnail of Monodromy and the Tate Conjecture: Picard numbers and Mordell-Weil ranks in families

Israel Journal of Mathematics, 2000

Introduction We use results of Deligne on …-adic monodromy and equidistribution, combined with el... more Introduction We use results of Deligne on …-adic monodromy and equidistribution, combined with elementary facts about the eigenvalues of elements in the orthogonal group, to give upper bounds for the average &quot;middle Picard number&quot; in various equicharacteristic families of ...

Research paper thumbnail of Bernard Dwork (1923–1998)

Notices of the American Mathematical Society, 1999

[Research paper thumbnail of Corrigendum to "G2 and hypergeometric sheaves" [Finite Fields Appl](https://a.academia-assets.com/images/blank-paper.jpg)

Research paper thumbnail of On a certain class of exponential sums

Research paper thumbnail of Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116)

Research paper thumbnail of The Grothendieck Festschrift : a collection of articles written in honor of the 60th birthday of Alexander Grothendieck

Research paper thumbnail of A note on Galois representations with big image

L’Enseignement Mathématique

Research paper thumbnail of Rigid local systems and alternating groups

Tunisian Journal of Mathematics

Contents 1. Introduction 1 2. The local systems in general 2 3. The candidate local systems for A... more Contents 1. Introduction 1 2. The local systems in general 2 3. The candidate local systems for Alt(2q) 4 4. Basic facts about H n 5 5. basic facts about H 2q−1 6 6. Basic facts about the group G geom for F (k, 2q − 1, ψ) 7 7. The third moment of F (k, 2q − 1, ψ) and of G(k, 2q − 1, ψ) 8 8. Exact determination of G arith 11 9. Identifying the group 12 References 25

Research paper thumbnail of On a Question of Zannier

Research paper thumbnail of The divisor function over arithmetic progressions

Research paper thumbnail of From Clausen to Carlitz: Low-Dimensional Spin Groups and Identities among Character Sums

Moscow Mathematical Journal

Research paper thumbnail of On the intersection matrix of a hypersurface

Annales scientifiques de l'École normale supérieure

Research paper thumbnail of Witt Vectors and a Question of Rudnick and Waxman

International Mathematics Research Notices

Research paper thumbnail of Internal reconstruction of unit-root <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span></span></span></span>-crystals via expansion-coefficients. With an appendix by Luc Illusie

Annales scientifiques de l'École normale supérieure

Internal reconstruction of unit-root F-crystals via expansioncoefficients. With an appendix by Lu... more Internal reconstruction of unit-root F-crystals via expansioncoefficients. With an appendix by Luc Illusie Annales scientifiques de l'É.N.S. 4 e série, tome 18, n o 2 (1985), p. 245-285. <http://www.numdam.org/item?id=ASENS_1985_4_18_2_245_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1985, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens), implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Research paper thumbnail of On the differentiation of De Rham cohomology classes with respect to parameters

Journal of Mathematics of Kyoto University

Research paper thumbnail of Corrections to : “A conjecture in the arithmetic theory of differential equations”

Bulletin de la Société mathématique de France, 1982

Research paper thumbnail of A conjecture in the arithmetic theory of differential equations

Bulletin de la Société mathématique de France, 1982

http://www.numdam.org/item?id=BSMF\_1982\_\_110\_\_203\_0 © Bulletin de la S. M. F., 1982, tous droit... more <http://www.numdam.org/item?id=BSMF_1982__110__203_0> © Bulletin de la S. M. F., 1982, tous droits réservés. L'accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html), implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Research paper thumbnail of Random Matrices, Frobenius Eigenvalues, and Monodromy

Colloquium Publications, 1998

The main topic of this book is the deep relation between the spacings between zeros of zeta and L... more 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 Well 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. It will be useful and interesting to researchers and graduate students working in any of these areas.

Research paper thumbnail of Groupes de Monodromie en Géométrie Algébrique

Lecture Notes in Mathematics, 1972

Research paper thumbnail of Slope filtration of F-crystals

Asterisque Societe Mathematique De France, 1979

Research paper thumbnail of Monodromy and the Tate Conjecture: Picard numbers and Mordell-Weil ranks in families

Israel Journal of Mathematics, 2000

Introduction We use results of Deligne on …-adic monodromy and equidistribution, combined with el... more Introduction We use results of Deligne on …-adic monodromy and equidistribution, combined with elementary facts about the eigenvalues of elements in the orthogonal group, to give upper bounds for the average &quot;middle Picard number&quot; in various equicharacteristic families of ...

Research paper thumbnail of Bernard Dwork (1923–1998)

Notices of the American Mathematical Society, 1999

[Research paper thumbnail of Corrigendum to "G2 and hypergeometric sheaves" [Finite Fields Appl](https://a.academia-assets.com/images/blank-paper.jpg)

Research paper thumbnail of On a certain class of exponential sums

Research paper thumbnail of Gauss Sums, Kloosterman Sums, and Monodromy Groups. (AM-116)

Research paper thumbnail of The Grothendieck Festschrift : a collection of articles written in honor of the 60th birthday of Alexander Grothendieck

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