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Arithmetic and geometry / edited by Luis Dieulefait, Universitat de Barcelona
[and five others].
pages cm. – (London Mathematical Society lecture note series ; 420)
Papers presented at the trimester on “Arithmetic and Geometry” at the Hausdorff
Research Institute for Mathematics (University of Bonn), January–April 2013.
ISBN 978-1-107-46254-0
1. Number theory – Congresses. 2. Algebraic number theory – Congresses.
3. Geometry of numbers – Congresses. I. Dieulefait, Luis, editor.
QA241.A695 2015
510–dc23
2015001829
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- 7. LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at http://www.cambridge.org/mathematics 299 Kleinian groups and hyperbolic 3-manifolds, Y. KOMORI, V. MARKOVIC & C. SERIES (eds) 300 Introduction to Möbius differential geometry, U. HERTRICH-JEROMIN 301 Stable modules and the D(2)-problem, F.E.A. JOHNSON 302 Discrete and continuous nonlinear Schrödinger systems, M.J. ABLOWITZ, B. PRINARI & A.D. TRUBATCH 303 Number theory and algebraic geometry, M. REID & A. SKOROBOGATOV (eds) 304 Groups St Andrews 2001 in Oxford I, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 305 Groups St Andrews 2001 in Oxford II, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 306 Geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds) 307 Surveys in combinatorics 2003, C.D. WENSLEY (ed.) 308 Topology, geometry and quantum field theory, U.L. TILLMANN (ed) 309 Corings and comodules, T. BRZEZINSKI & R. WISBAUER 310 Topics in dynamics and ergodic theory, S. BEZUGLYI & S. KOLYADA (eds) 311 Groups: topological, combinatorial and arithmetic aspects, T.W. MÜLLER (ed) 312 Foundations of computational mathematics, Minneapolis 2002, F. CUCKER et al (eds) 313 Transcendental aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) 314 Spectral generalizations of line graphs, D. CVETKOVIC, P. ROWLINSON & S. SIMIC 315 Structured ring spectra, A. BAKER & B. RICHTER (eds) 316 Linear logic in computer science, T. EHRHARD, P. RUET, J.-Y. GIRARD & P. SCOTT (eds) 317 Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI & N.P. SMART (eds) 318 Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY 319 Double affine Hecke algebras, I. CHEREDNIK 320 L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁR (eds) 321 Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) 323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) 324 Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) 325 Lectures on the Ricci flow, P. TOPPING 326 Modular representations of finite groups of Lie type, J.E. HUMPHREYS 327 Surveys in combinatorics 2005, B.S. WEBB (ed) 328 Fundamentals of hyperbolic manifolds, R. CANARY, D. EPSTEIN & A. MARDEN (eds) 329 Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) 330 Noncommutative localization in algebra and topology, A. RANICKI (ed) 331 Foundations of computational mathematics, Santander 2005, L.M PARDO, A. PINKUS, E. SÜLI & M.J. TODD (eds) 332 Handbook of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) 333 Synthetic differential geometry (2nd Edition), A. KOCK 334 The Navier–Stokes equations, N. RILEY & P. DRAZIN 335 Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER 336 Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE 337 Methods in Banach space theory , J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics 2007, A. HILTON & J. TALBOT (eds) 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) 348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) 349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH 352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds) 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) 359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) 360 Zariski geometries, B. ZILBER
- 8. 361 Words: Notes on verbal width in groups, D. SEGAL 362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA 363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) 364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) 365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) 367 Random matrices: High dimensional phenomena, G. BLOWER 368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) 369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ 370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH 371 Conformal fractals, F. PRZYTYCKI & M. URBANSKI 372 Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) 373 Smoothness, regularity and complete intersection, J. MAJADAS & A. G. RODICIO 374 Geometric analysis of hyperbolic differential equations: An introduction, S. ALINHAC 375 Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds) 376 Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) 377 An introduction to Galois cohomology and its applications, G. BERHUY 378 Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds) 379 Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) 380 Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T.C WALL (eds) 381 Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds) 382 Forcing with random variables and proof complexity, J. KRAJÍCEK 383 Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 384 Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) 386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER 387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds) 388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds) 389 Random fields on the sphere, D. MARINUCCI & G. PECCATI 390 Localization in periodic potentials, D.E. PELINOVSKY 391 Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER 392 Surveys in combinatorics 2011, R. CHAPMAN (ed) 393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds) 394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds) 395 How groups grow, A. MANN 396 Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA 397 Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds) 398 Mathematical models in contact mechanics, M. SOFONEA & A. MATEI 399 Circuit double cover of graphs, C.-Q. ZHANG 400 Dense sphere packings: a blueprint for formal proofs, T. HALES 401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU 402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J.L. RODRIGO & W. SADOWSKI (eds) 403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS & A. SZANTO (eds) 404 Operator methods for boundary value problems, S. HASSI, H.S.V. DE SNOO & F.H. SZAFRANIEC (eds) 405 Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) 406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) 407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL 408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds) 409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds) 410 Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI 411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCÍA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds) 412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS 413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds) 414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT 417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTATA & M. POPA (eds) 418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) 419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM 420 Arithmetic and geometry, L. DIEULEFAIT et al (eds) 421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) 422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds) 423 Inequalities for graph eigenvalues, Z. STANIĆ 424 Surveys in combinatorics 2015, A. CZUMAJ et al (eds)
- 9. London Mathematical Society Lecture Note Series: 420 Arithmetic and Geometry Edited by LUIS DIEULEFAIT Universitat de Barcelona GERD FALTINGS Max-Planck-Institut für Mathematik (Bonn) D. R. HEATH-BROWN University of Oxford YU. V. MANIN Max-Planck-Institut für Mathematik (Bonn) B. Z. MOROZ Max-Planck-Institut für Mathematik (Bonn) JEAN-PIERRE WINTENBERGER Université de Strasbourg
- 10. University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107462540 c Cambridge University Press 2015 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2015 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data Arithmetic and geometry / edited by Luis Dieulefait, Universitat de Barcelona [and five others]. pages cm. – (London Mathematical Society lecture note series ; 420) Papers presented at the trimester on “Arithmetic and Geometry” at the Hausdorff Research Institute for Mathematics (University of Bonn), January–April 2013. ISBN 978-1-107-46254-0 1. Number theory – Congresses. 2. Algebraic number theory – Congresses. 3. Geometry of numbers – Congresses. I. Dieulefait, Luis, editor. QA241.A695 2015 510–dc23 2015001829 ISBN 978-1-107-46254-0 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. Contents Preface page vii Introduction ix List of participants x Trimester Seminar xiii Workshop on Serre’s conjecture xxiii The research conference xxviii 1 Galois groups of local fields, Lie algebras and ramification 1 Victor Abrashkin 2 A characterisation of ordinary modular eigenforms with CM 24 Rajender Adibhatla and Panagiotis Tsaknias 3 Selmer complexes and p-adic Hodge theory 36 Denis Benois 4 A survey of applications of the circle method to rational points 89 T.D. Browning 5 Arithmetic differential equations of Painlevé VI type 114 Alexandru Buium and Yuri I. Manin 6 Differential calculus with integers 139 Alexandru Buium 7 Un calcul de groupe de Brauer et une application arithmétique 188 Jean-Louis Colliot-Thélène 8 Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity 193 Luis Dieulefait and Ariel Pacetti v
- 12. vi Contents 9 Big image of Galois representations and congruence ideals 217 Haruzo Hida and Jacques Tilouine 10 The skew-symmetric pairing on the Lubin–Tate formal module 255 M. A. Ivanov and S. V. Vostokov 11 Equations in matrix groups and algebras over number fields and rings: prolegomena to a lowbrow noncommutative Diophantine geometry 264 Boris Kunyavskiı̆ 12 On the -adic regulator as an ingredient of Iwasawa theory 283 L. V. Kuz’min 13 On a counting problem for G-shtukas 318 Ngo Dac Tuan 14 Modular forms and Calabi-Yau varieties 351 Kapil Paranjape and Dinakar Ramakrishnan 15 Derivative of symmetric square p-adic L-functions via pull-back formula 373 Giovanni Rosso 16 Uniform bounds for rational points on cubic hypersurfaces 401 Per Salberger 17 Descent on toric fibrations 422 Alexei N. Skorobogatov 18 On filtrations of vector bundles over P1 Z 436 A. Smirnov 19 On the dihedral Euler characteristics of Selmer groups of Abelian varieties 458 Jeanine Van Order 20 CM values of higher Green’s functions and regularized Petersson products 493 Maryna Viazovska
- 13. Preface The trimester on “Arithmetic and Geometry” at the Hausdorff Research Institute for Mathematics (University of Bonn) took place in January – April 2013. In the next few pages the reader will find a list of the participants of the trimester, the descriptions of the session on Serre’s conjecture, conducted by L.V. Dieulefait and J.-P. Wintenberger, and of the session on counting rational points on algebraic varieties, conducted by D.R. Heath-Brown, the programmes of the workshop on Serre’s conjecture and of the final research conference, and a list of the talks given at the HIM during the trimester. The participants were invited to submit their papers for publication in this volume. The papers appearing in the volume have been carefully refereed. Acknowledgements. We wish to record our gratitude to the Hausdorff Research Institute, and in particular to its director, Professor Dr W. Lück, for the hospitality and financial support we received. L.V. Dieulefait, G. Faltings, D.R. Heath-Brown, Yu.I. Manin, B.Z. Moroz, and J.-P. Wintenberger (editors). vii
- 15. Introduction The main theme of the trimester was the interplay of different methods used in modern number theory. We wish to emphasize the new results and conjec- tures in arithmetic geometry, having direct bearing on the classical number theoretic problems. Two sessions, on the recently proved Serre’s conjecture from 15 January to 14 February (organizers: L. Dieulefait and J.-P. Winten- berger) and on counting rational points on algebraic varieties from 15 March to 14 April (organizer: D.R. Heath-Brown), as well as a couple of shorter workshops, several seminars, and mini-courses were organized. The trimester culminated in a research conference from 15 to 19 April. The aim of the session “Serre’s conjecture” was to report on recent works linked to that conjecture, in particular about Galois representations and automorphic representations. During the weeks starting on 14 January and 21 January, Henri Carayol lectured on his work on the algebraic properties of Griffiths-Schmid varieties. The Griffiths-Schmid varieties are analytic vari- eties classifying Hodge structures. Studying their algebraic properties might be a step towards constructing Galois representations associated to automorphic representations appearing in the cohomology of these varieties. Our second theme related to the recent work of Michael Harris, Kai-Wen Lan, Richard Taylor and Jack Thorne, who have constructed Galois representations asso- ciated to not necessarily self-dual automorphic representations. The proof heavily relies on p-adic properties of automorphic representations. The aim of the session “counting rational points on algebraic varieties” was to report on recent works on the existence, frequency and distribution of ratio- nal points on algebraic varieties. Thus the main themes were local to global principles, Manin’s conjecture, developments of the Hardy-Littlewood method and the determinant method. ix
- 16. List of participants Victor Abrashkin (University of Durham) Rajender Adibhatla (Universität Regensburg) Shabnam Akhtari (University of Oregon) Paloma Bengoechea (College de France) Denis Benois (Université Bordeaux 1) Tobias Berger (University of Sheffield) Yuri Bilu (IMB Université Bordeaux I) Marco Boggi (University of Los Andes) Pierre Le Boudec (Institute of Advanced Study) Régis de la Bretéche (Institut de Mathématiques de Jussieu – Paris Rive Gauche, UMR 7586 Université Paris-Diderot) Christophe Breuil (Université Paris-Sud) Tim Browning (University of Bristol) Jörg Brüdern (Universität Göttingen) Roman Budylin (Steklov Mathematical Institute) Alexandru Buium (University of New Mexico) Alberto Cámara (University of Nottingham) Henri Carayol (L’Université de Strasbourg et du CNRS) Magnus Carlson (University of Stockholm) Tommaso Centeleghe (Universität Heidelberg) Huan Chen (Ecole normale superieure ENS) Narasimha Kumar Cheraku (Ruprecht Karls Universität Heidelberg) Przemyslaw Chojecki (Institut Mathématique de Jussieu) Laurent Clozel (Université de Paris Sud 11) Jean-Louis Colliot-Thélène (Université Paris-Sud) David Mendes da Costa (University of Bristol) Tuan Ngo Dac (Université de Paris 13) x
- 17. List of participants xi Ulrich Derenthal (Universität München) Fred Diamond (King’s College London) Rainer Dietmann (Royal Holloway, University of London) Luis Dieulefait (Universitat de Barcelona) Gerd Faltings (Max-Planck-Institut für Mathematik) Ivan Fesenko (University of Nottingham) Nuno Freitas (Universitat de Barcelona) Wojciech Jerzy Gajda (The Adam Mickiewicz University) David Geraghty (Institute for Advanced Study) Sergey Gorchinskiy (Steklov Mathematical Institute) Frank Gounelas (Humboldt University) Xavier Guitart (Universität Duisburg-Essen) Shuvra Gupta (University of Iowa) Shai Haran (Technion – Israel Institute of Technology) Michael Harris (Institut de Mathématiques de Jussieu) Roger Heath-Brown (University of Oxford) Florian Herzig (University of Toronto) Alexander Ivanov (Universität Heidelberg) Mikhail Ivanov (Saint-Petersburg State University) Andrew Kresch (Universität Zürich) Lars Kühne (SNS Pisa) Boris Kunyavskii (Bar-Ilan University) Leonid Kuzmin (National Research Center Kurchatov Institute) Kai-Wen Lan (University of Minnesota) Dmitry Logachev (Universidad Simon Bolivar) Oliver Lorscheid (IMPA) Daniel Loughran (Leibniz Universität Hannover) Yuri Manin (Max-Planck-Institut für Mathematik) Oscar Marmon (Georg-August-Universität Göttingen) Lilian Matthiesen (University of Bristol) David McKinnon (University of Waterloo) Boris Moroz (Universität Bonn) Jeanine Van Order (The Hebrew University of Jerusalem) Denis Osipov (Steklov Mathematical Institute) Ambrus Pál (Imperial College London) Aftab Pande (Cidade Universitária Ilha do Fundao) Alexej Parshin (Steklov Mathematical Institute) Florian Pop (University of Pennsylvania) Dinakar Ramakrishnan (Caltech) Giovanni Rosso (Université Paris 13)
- 18. xii List of participants Mohamed Saidi (Exeter University) Per Salberger (Chalmers University of Technology) Damaris Schindler (University of Bristol) Mehmet Haluk Sengun (University of Warwick) Evgeny Shinder (Max-Planck-Institut für Mathematik) Ceclia Salgado Guimaraes da Silva (UFRJ) Alexei Skorobogatov (Imperial College London) Arne Smeets (Université Paris-Sud 11) Aleksander Smirnov (Steklov Institute of Mathematics in St. Petersburg) Efthymios Sofos (University of Bristol) Cesar Alejandro Soto Posada (Universität Tübingen) Mike Swarbrick Jones (University of Bristol) Jack Thorne (Harvard University) Jacques Tilouine (Université Paris 13) Yuri Tschinkel (New York University) Pankaj Hemant Vishe (University of York) Sergei Vostokov (University of St. Petersburg) Yosuhiro Wakabayashi (Kyoto University) Gabor Wiese (Université du Luxembourg) Nicholas Williams (University of Exeter) Jean-Pierre Wintenberger (Université Strasbourg) Trevor Wooley (University of Bristol) Yanhong Yang (Universität Mainz) Adrin Zenteno (Universidad Nacional Autónoma de México)
- 19. Trimester Seminar January 8, Jeanine Van Order, Iwasawa main conjectures for GL(2) via Howard’s criterion (abstract). In this talk, I will present the Iwasawa main conjectures for Hilbert modular eigenforms of parallel weight two in dihe- dral or anticyclotomic extensions of CM fields. The first part will include an overview of known results, as well as some discussion of open problems and applications (e.g. to bounding Mordell-Weil ranks), and should be accessible to the non-specialist. The second part will describe the p-adic L-functions in more detail, as well as the non-vanishing criterion of Howard (and its implications for the main conjectures). January 15, Oliver Lorscheid, A blueprinted view on F1-geometry (abstract). A blueprint is an algebraic structure that “interpolates” between multiplicative monoids and semirings. The associated scheme theory applies to several problems in F1-geometry: Tits’s idea of Chevalley groups and build- ings over F1, Euler characteristics as the number of F1-rational points, total positivity, K-theory, Arakelov compactifications of arithmetic curves; and it has multiple connections to other branches of algebraic geometry: Lambda- schemes (after Borger), log schemes (after Kato), relative schemes (after Toen and Vaquie), congruence schemes (after Berkovich and Deitmar), idempotent analysis, analytic spaces and tropical geometry. After a brief overview and an introduction to the basic definitions of this theory, we focus on the combinato- rial aspects of blue schemes. In particular, we explain how to realize Jacques Tits’s idea of Weyl groups as Chevalley groups over F1 and Coxeter complexes as buildings over F1. The central concepts are the rank space of a blue scheme and the Tits category, which make the idea of “F1-rational points” rigorous. January 16, Jean-Pierre Wintenberger, Introduction to Serre’s modu- larity conjecture (abstract). This lecture is intended for non-specialists. We state Serre’s modularity conjecture and give some consequences and hints on its proof. xiii
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- 24. The Project Gutenberg eBook of Beyond Rope and Fence
- 25. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Beyond Rope and Fence Author: David Grew Illustrator: Harold Sichel Release date: November 13, 2020 [eBook #63745] Most recently updated: October 18, 2024 Language: English Credits: Produced by The Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive) *** START OF THE PROJECT GUTENBERG EBOOK BEYOND ROPE AND FENCE ***
- 32. Beyond Rope and Fence D A V I D G R E W B o n i a n d L i v e r i g h t P u b l i s h e r s ~ N e w Yo r k
- 33. Copyright, 1922, by Boni and Liveright, Inc. ─────── PRINTED IN THE UNITED STATES OF AMERICA
- 34. To you, dear old Dora, who inspired this book, I dedicate it. I regret most poignantly that life has ordained that you may never know, despite my caresses and my quart measures filled to overflowing with oats, how deeply I have sympathised with you in those moments when you stood motionless before me and I could see by the strange, sad light in your eyes that you were dreaming of long departed, happy years of freedom on the plains. D. G.
- 36. CONTENTS CHAPTER PAGE Foreword i I. For the Love of Her Foal 1 II. To the North! 25 III. Death in the Howl of Coyotes 35 IV. A Seeking That Found 48 V. Man, the Usurper 59 VI. How Man Breaks the Spirit and the Body 75 VII. The Conspiracy of Man and Coyote 87 VIII. Retribution 116 IX. Slowly Man Crept Northward 123 X. The Doors of the Trap Shut 133 XI. Rope, Iron and Fire 163 XII. The Strength of the Weak 178 XIII. Labour Without Love or Wage 195 XIV. Only Justice Had Been Done 201 XV. The Trail of the Moose 225
- 38. FOREWORD In the fall of the year, the farmers and the ranchers of the northwest prairies of Canada release their horses for the winter. Strange as it may seem to those of us who shudder at the very thought of raging blizzards on the open plains, the horses that are left free to roam over unsheltered space and are obliged to dig down through feet of snow for their grass, not only survive the severest winters but are generally found fat and strong the next spring. If while you are out riding you happen upon a group of these free horses, they will stare at you curiously until they begin to fear that you have come to gather them up and to take them back to the farm yard, then with angry, defiant tossing of heads they will turn and gallop out of reach, going so fast that you will not see them for snow dust. The horse you are riding, if he has ever enjoyed a winter of that freedom, will struggle to get away from you so that he may join them. Because you will not let him go, he will show his displeasure like a petulant child and long after you have forced him to abandon the attempt to get loose, long after the happier group has disappeared, he will keep turning his head back and calling yearningly to them. The farmer who releases his horses in the fall rarely loses any of them. Every farmer knows every horse within a radius of twenty-five miles or more, knows them by name and colour, knows their histories and peculiarities. When the farmer is in doubt as to who some distant rider may be, you can hear him think aloud thus: “That’s Skinner’s sorrel, Billy. Skinner’s goin’ for his mail.” Or: “That’s Spicer’s white nag, Madge. I’ll bet Spicer’s comin’ to see
- 39. about them oats.” So in the spring of the year, when the farmers are all out searching for their horses, they know those they come upon, and if some farmer sees Skinner’s sorrel, Billy, he will drive him in the direction of Skinner’s homestead, talking to Billy as he does so, in some such fashion as this: “Well, Billy, you little devil, you ain’t any the worse for the worst winter in twenty years. You’re fat as a pig. Go on now, get home! I know you don’t like the idea of gettin’ back to work, but it’s soon seedin’ time, you know!” The farmer who works beside his horses daily, who gets to understand every expression of these beautiful, intelligent creatures, always talks seriously to them. This sounds strange to us until we have come in contact with these animals for a short time, when, hardly being conscious of it, we soon start talking to them ourselves. They certainly understand many words and I have seen evidences of horses recognising at once what sort of temper or mood men happen to be in as soon as they approach them. Just as they learn to understand us, we learn to understand them. Every neigh or whinny takes on the meaning of a word, and their scowling or angry shaking of heads, and their protests against certain discomforts we impose upon them appear as clearly as the similar expressions of people. The most amazing fact, however, that slowly dawns upon us, is the fact that these lovely animals live in a conscious world of their own, not half so different from ours as we had allowed ourselves to think. The rancher is not as intimate with the horses he breeds and rears in virtual wildness on the vast ranges which he leases from the government and about which he builds his barbed wire fences. Naturally so. He has from several hundred to several thousand horses and they are virtually in a wild state until he sells them, when they are broken-in and most of the untamed spirit is crushed out of them by heavy labour.
- 40. A rancher can rarely tell you how many horses he has. During the spring when colts are most often born, his stock may double for all he knows. He does not attempt to find out until the fall, when he rounds them up. The young colts are separated from their mothers and branded. The poor young things are tied and thrown and the red hot iron, with the shape of each rancher’s particular brand, is pressed upon the shoulder till the insignia is burned through hair and skin, where the mark remains as long as the creature lives. The ranch horses are wilder and more spirited than the farm horses, but when the latter are released for the winter, they often mix with the former, breaking up into groups of those who seem to feel themselves more congenial to each other. Every animal has a character and personality of his own, and while he will get along beautifully with one horse, he will fight all the time with another. From my observation, it seems to me that the wild free horse does much less quarrelling than the horse that has toiled on the farm, which would indicate quite clearly how much like ours his nature is. Very few of the great herds that rustle for themselves all winter long die while they are away. Those that die are horses that either have been kept in the barn too late in the season or else that were in a starved condition when they were released. A horse that has been kept in the barn till after the cold season has set in and has been inured to the warmth of the barn, when suddenly exposed to the unsheltered open plains, if the weather happens to be severe, will sometimes die because it finds it is unable to adjust itself to the change in temperature. But there is one peculiarity of horse nature which sometimes kills the best horse, not only in the wilds but in the pasture or barn yard, if no one is about to come to its assistance. Every horse loves to roll. He will lie down on a sandy spot or on the snow and roll over from side to side. It sometimes happens that he selects a spot that has a deep rut, or that is near a wall, a stone, or a straw-stack. He will roll over and strike the wall or the straw-stack or get caught in the rut in such a way that he cannot force himself back. He will remain
- 41. helpless on his back till some one comes to his rescue. If he gets no assistance he will die in a very short time, sometimes within less than an hour. But I am interested in the horse as a fellow being, subject as we are to limitations; and, to a degree less perhaps than we are, capable of joy and sorrow. In so far as these beautiful creatures are able to communicate to others an indication of the emotions out of which their lives are built, I have taken my story directly from them. My story, too, comes fresh from the prairies. I did most of its planning while riding on horseback over hundreds of miles of rolling Alberta plains, often coming upon hills from which I could see a perfectly circular horizon without a sign of human life, save perhaps some telltale arrangement of stones, laid on the hilltop by Indians whom fate had long since swept from the plains of their fatherland. At such times my pony, whose wild and exciting history forms the greater part of this story, seemed as much moved by the open vastness and the stillness as I; and, each in his own way, we held communion with the spirit of the wilderness. D. G. Langmark, Alberta, Canada.
- 42. BEYOND ROPE AND FENCE
- 43. CHAPTER I FOR THE LOVE OF HER FOAL OLLING hills and shallow valleys—an ocean of brown waves with fast drying sloughs, like patches of sunshine on the surface of the sea—such was the Canadian prairie that autumn day—such were the miles and miles of Alberta range, bounded by a barbed wire fence that was completely lost in the unobstructed play of sunshine. It was an open wilderness, so vast that it seemed to stretch on almost endlessly beyond the horizon, which lay desolate and unbroken like a rusty, iron ring, girding the earth. Its immensity, by an inexorable contrast, dwarfed everything that crept over the surface of the plains into a helpless puniness. The hundred horses on the range, scattered and grouped by their predilections for each other, looked, in the distance, like ants crawling over the surface of a rock. Within sight of each other, bound by the ties of race, they nevertheless had their loves and their preferences. Most of the mothers with their little colts grazed in a group by themselves; while a few mothers, as if they felt that their children were better than their neighbour’s children, kept themselves apart from the herd, though always within sight. Among the latter was a shapely, light-brown or buckskin mare who was grazing peacefully about her precious, buckskin coloured daughter. The little one was asleep on the grass. Her graceful little legs were stretched as far as she could stretch them. Her lovely little
- 44. head lay flat on the ground. Her fluffy tail was thrown back on the grass with a delicious carelessness. She was only six months old, but already the very image of her mother. From the white strip on her forehead and the heavy black mane down to the unequal white spots on her two hind fetlocks, she was like her. Only her wiry, delicately wrought little legs seemed somewhat too long for her. Suddenly the old mare’s head went up high in the air; her grinding teeth ceased grinding as a broken machine comes to a dead stop; and the round, dilated, knowing eyes pierced the slight haze in the atmosphere. The little head on the grass raised just a bit, looked inquiringly at her beloved mother—quite near; then with the innocent confidence of childhood, dropped back again, rubbing the soft fragrant grass in an ecstasy of contentment. But the old mare continued to gaze intently, standing motionless as a stone. She saw that all the other horses were gazing just as intently as she was. Small moving objects—two men on horseback— had broken over the line of shadow along the southern horizon. One of them was loping away to the right and the other to the left. The old buckskin mare had already lived more than twenty years. Not only had she herself suffered at the hands of man, but she had had so many of her babies taken from her and cruelly abused—often before her very eyes. Her mother’s heart began beating fast and apprehensively. The other mares, not far from her, also showed signs of extreme nervousness. The buckskin saw them run off for a short distance as if in panic, then stop and gaze anxiously at the approaching riders. It was time to act. She looked questioningly a moment toward the north; but she realised that that direction would soon be closed to her, for she could tell that the riders, loping straight north, meant to turn in time and come back upon them. She called nervously to her little one. The little thing sprang to its feet, sidled up to her and gazed at the dark specks that were coming
- 45. together in the north, with fear glowing moist in her large, round eyes. Until she had seen a group of horsemen dismount, one day, she had thought that man was a monstrous sort of horse with a frightful hump on its back. What little she had been able to learn about him since that time had served only to intensify her fear of him; and despite her abiding confidence in her mother, she trembled timorously as she heard the ominous hoof-beats in the distance. The animals instinctively gathered into a bunch and started away at full speed. While one of the horsemen remained some distance behind, ready to prevent the group from going off to either side, the other plunged into the midst of them and deftly separated the mothers and their colts from the rest of the bunch. Then they allowed the single horses to run off to the north at their will; while they came together behind the mothers and their colts and drove them southward toward the long line of shadow that lay like a black elongated reptile, below the horizon and parallel to it. That long line of shadow, which widened as they neared it, was a great canyon which the Red Deer River had cut out of the level plains. From the jaws of the mouth of the canyon, which were a mile or so apart, the floor of the prairies fell away sheer in places, to a depth of a thousand feet. In many spots there were several parallel cuts in the edge of that floor. Where, during the ages, the elements had been unable to remove the loose earth, it lay along the bank in steep hills which rose up from the bottom of the canyon like gigantic teeth, all crumbling more or less, all dotted with stones and covered here and there with blotches of sagebrush and cacti. In the centre of the flat-bottomed canyon, as if an ancient torrential flood had spent itself and narrowed down at last to a small, shining stream, a quarter of a mile in width, ran the Red Deer River. In the middle of the half-mile wide space between the river and the hills that made the wall of the canyon, stood the buildings of the ranch. The house, a small shingled structure, stood on the east end of the spacious, sandy yard; while opposite and facing it was
- 46. the long, red barn with its open door below and the gaping window space in the loft above. North of the barn and against its blind wall there was a big corral, divided into two parts by a partition. The corral walls as well as the partition were made of logs laid horizontally, a foot apart and rising to a height of some eight feet. Each of these two sections had huge swinging gates which opened inward. As helplessly as the waters of Niagara, the frantic mothers, stealing side glances at their little ones and feeling them at their sides, poured down the steep incline, between the giant teeth, into the mouth of the canyon, slipping, sliding, and leaping downward riskily, in haste and fear. On the level bottom of the canyon, the buckskin mare made an attempt to turn from the path which led to the rancher’s buildings in the hope of getting to the river beyond; but one of the horsemen divined her rebellious intention and shot by her like a flash of light, heading her off and forcing her back. She realised the futility of baffling their superior wills; but went back with an angry shake of her wise old head and a deliberate scowl of hatred for the tormenting man and the servile horse under him who was betraying his kind. However the old mare happened to feel, the little buckskin, since the forces of evil had as yet made no attempt to separate her from her mother, shook the fear from her heart and took all the delight there was to take in this unexpected excitement of the day. Healthy to the last cell in her body, the race had merely accelerated the circulation of her blood; and the ease with which she was able to keep up with her mother made her conscious of a great and thrilling power. Her eyes and nostrils dilated, her mane bristling and her tail unfurled, her springy legs carrying her with ease, there was an expression of boundless joy in the motion of her graceful body. The gates of the corral stood wide open. Being so driven that they could not swerve from the path, half the group poured into one section of the corral and the other half into the other. When they turned at the opposite walls realising that there, there was no way
- 47. out again, and came back toward the gates, they saw the men closing them. Only the soul that has been trapped knows the crushing torment of four relentless walls. Round and round they went, madly and stupidly, and clouds of beaten earth rose from under their feet and choked them. Finally becoming aware of the fact that the men were not pursuing them any longer, they packed into a corner of the corral and, looking over the corral walls and between the logs, sought to learn what they were doing. They saw one man building a fire in the open, but a few paces from the corral; while the other was calmly and portentously making preparations that were only too familiar to the old mares. The little buckskin, beside her mother, always beside her mother, clinging to that big beloved body as the soul clings to life, was wedged into the very corner and right against the logs of the wall, so that her frightened eye, in the middle of the open space between two logs, could see the rancher’s house some four rods away. Her sides were still throbbing violently when she saw the house door open. A little girl appeared. The little filly did not know what kind of animal that was except that she guessed that it was some sort of man. She perceived with renewed trepidation that the little girl was hopping and skipping directly toward her. In her fright she pressed tight against her mother, but her mother, much more concerned with the men and apparently indifferent to the little girl, would not move an inch. When suddenly the little buckskin felt the touch of the little girl’s hand on her back, she called out frantically to her mother. But the old mare bent down her long neck, touched the little head with her soft, warm lips, murmured reassuringly and then looked away again. By that time the filly realised, uncomfortable though she was, that the little hand was not going to hurt her. The little girl climbed up two of the logs, moved slowly toward the little buckskin’s head, talking softly and coaxingly as she moved. The filly listened with ears pricked high. In the stream of meaningless prattle, the foal became aware of the existence of the combination
- 48. of sounds, “Queen,” as one becomes aware of a constantly repeated melody in a piece of music. By the time the little girl had carefully pushed her head through the space between two logs, directly in front of the filly’s muzzle, the little buckskin, though frightened again, became exceedingly curious. There was something very disarming about that soft voice and the soothing repetition of the word, “Queen.” She cautiously stretched her muzzle, sniffing at the little mouth, moving it closer and closer and just when she touched the little girl’s face, with a cry of delight the little girl kissed her fervently on the nose. She drew her muzzle away quickly and looked with a frightened eye. It had interrupted her attempt to sniff, however, and once more assured that there was nothing harmful about the little girl, she made a second attempt. The little girl continued calling her, “Queen,” coaxingly, till the little muzzle touched her lips again and once more she kissed her, crying out again with delight. This sweet, unofficial christening might have resulted in a beautiful, enduring friendship, but a sudden, terrific patter of feet in the next corral came through the air accompanied by a nauseating cloud of smoke, and all was confusion again. Round and round their section of the corral they swept again till they realised that the men were not yet molesting them. When they stopped to investigate, little Queen saw a man in the other section of the corral rush toward a mare with a long hideous stick. She saw him strike the colt that tried to follow her and saw the colt run back into the corral while the mother had run out. She could not quite understand what he was doing; but she experienced an overwhelming fear of losing her mother, and clung to her beloved sides with more tenacity than ever. The other section of the corral was finally cleared of all the mares who, standing on the outside, would not go away; but in concert rent the air with their cries of protest. Queen was so curious that, despite her beating heart, she moved to where she could see what was going on. She saw ropes flash through the air and immediately after, a little colt fell to the dusty ground. The cry from the little
- 49. one’s mother was answered by a stifled cry from the ground and as Queen, unable to stand still for fear, listened to that cry, there suddenly began coming to her the odour of blood and burning flesh. Madness seized upon them once more and the dizzying whirl round the choking corral gave them some relief. They finally stopped to rest a while, only to have another colt thrown and his cries and the smell of burning flesh set them through the frenzied motion round the corral, all over again. Most of the afternoon it took before all the colts in the first section had been branded and mutilated. It was a noisy, dusty, cruel process; and the men, perspiring heavily, their faces wet and black with the dust that settled on them, looked like tormenting imps of hell; but they were no more to be blamed for the cruelty that was theirs to do than were their helpless victims. All that clamour of pain and struggle could not disturb the mist- like loneliness that brooded over the far-reaching distance. On the other side of the river, visible beyond less rugged banks, stretched a lifeless country of hills and plains, so desolate and so motionless that the very stones that dotted them seemed with their feeble reflections to be futilely protesting against their destitution. A pause came to the torturous struggle. The gate of the first corral was opened and the sickened little colts shambled out into the open where their frantic mothers caressed them, then led them away to the east. The men walked off and disappeared in the house. Taking advantage of the silence and the respite, the still captive colts, one after another, took to sucking. It was not very long, however, before they were interrupted by the reappearance of the men. The skin on every captive began to tremble and the eight mothers with their eight colts packed into one corner. One man, carrying a long stick, entered the section and advanced to the middle while the other stationed himself at the gate. First the man with the stick forced the group to move into the opposite corner, then, after a long struggle, he singled out the buckskin mare. He had driven her toward the gate but a few feet, when little Queen,
- 50. bending so low that she passed under the stick, rushed out of reach of it and gained her mother’s side. Had it not been for the vigilance of the man at the gate they would have both escaped. It was getting to be late in the afternoon and the man was tired and impatient. As with most impatient people, his common sense gave way to his impatience. He was not only determined to get the buckskin mare out first, but he was even more anxious to punish her. He singled her out again and reaching her, struck her with his stick. In pain and fright, the mare rushed for the gate. It was partially opened and she was half way out when a cry from little Queen, who saw her leaving her, brought her to her senses. Rebelliously, she reared and fell with full force upon the gate. It swung violently backward, striking the man who held it so severely that it knocked him off his feet and sent him rolling to the wall. The second man who was trying to prevent Queen from following her mother was away over at the other end of the corral. The gateman’s cry and the image of him on the dusty ground, so confused the other that for a few moments he stood still, unable to move a muscle. When he saw his partner pick himself up, he realised that he should have hurried to the gate and closed it; but by that time the whole group had escaped and were racing for the hills, the buckskin mare in the lead and her precious Queen eagerly behind her. With a majestic toss of her head, conscious of having scored a victory, and determined to keep it, the buckskin mare fled across the flats. It was now not only the overwhelming desire to get away. Vaguely she realised that she had crossed the man’s will and that that was a punishable offence. The mothers whose foals had been branded were off on a field at the foot of the hills. The field had yielded a crop of oats and the oats had been reaped and taken from the field; but there was still enough grain left to make it worth their while to remain there. If, when they followed the fugitives with their eyes, they had any desire to go along, they knew that their sickened colts would not go with them.
- 51. The buckskin mare gave them hardly a glance. She struck up the steep incline with risky speed, bent upon getting out of the men’s reach, as soon as was possible. The men, on the other hand, were at a disadvantage. Before they could saddle their ponies, the mares, they knew, would be off somewhere at the other end of the range. They realised, too, that the mares were now so excited that they would have very great difficulty in rounding them up. They were angry at the rebellious mare, but these animals were their property and they did not want to hurt them. Another struggle at that time, they felt, might even endanger their own lives. The man who had been knocked over was not only as tired as the other fellow was, but he was aching from head to foot. Besides, the afternoon was rapidly giving way to early evening. They decided to finish the branding on the following day. But to the buckskin mare the spaces behind her seemed peopled with imaginary pursuers, and she struggled up the slippery incline as if her very life depended upon getting to the top and away. The rest of the mares that fled with her and their little ones seemed to find greater difficulty in getting to the top, but they followed as eagerly. Rocks and sand rolled thunderously down behind them and the dust rose from the mouth of the canyon like volcanic smoke. When they finally reached the level plains above, the old mare was white with foam. They had that afternoon been rounded up in a hollow toward the northeast of where they now were and fear of being rounded up again sent the buckskin mare to the west. Her usual fear of man, many times intensified by the feeling that now she would be severely punished for breaking loose, aroused in her old head the instinctive desire of the animal that is pursued, to get under cover. Though there was neither sight nor sound of any one behind her, she ran with might and main for the coulee that she knew was a mile and a half to the west, and until she had turned over the lip of the coulee and had reached the very end of its slope, she did not slacken her pace, several times almost breaking a leg in badger holes that she avoided by only a hair’s breadth. Down in the gulch there was a path, made by the water of the melted snow in
- 52. spring as it had wound its way to the river. Along this path, which led northward, they trotted without stopping till they came to where the range fence forced them to halt. Here at last they rested, though the buckskin mare kept anxious vigil for the first sign of any one pursuing them. The mothers began grazing slowly while their young, moving with them, strove to get the milk they felt belonged to them. As soon as the colts had had all the milk there was for them they went leisurely in search of tender grasses and soon all were grazing as if nothing had ever happened. But the buckskin mare was still worried. She walked to the two wires that barred her way and with her head above the upper wire she gazed to the north. A quarter of a mile away, the coulee ended. Its floor curved upward like the bottom of a ship. Where it ended and the prairie floor began there was a cluster of sagebrush. The evening was rapidly turning the sage into a silhouette against the bright background of the sky. Fear of pursuit came back with the coming of the night and the old mare roused herself. With a sudden impulse she backed away from the wires and dropped to her knees. Pushing her head under the lower wire she moved cautiously forward, an inch at a time. Slowly she felt the wire move backward over her body and each time the barb dug through her skin she stopped and tried to crouch lower. With a sharp scratch it rolled over her withers and stuck painfully into her back. She tried again to crouch down lower, but failing to rid herself of the barb, she rested a moment. The barb hurt her considerably and she made a strenuous effort to lower herself out of its reach, and in so doing pressed her outstretched muzzle right into a rosebush. While the pain of thorns still pricked her lips there was a sudden flash of white right before her eyes and a thump on the ground as if a rock had been thrown at her. With all the strength in her body, forgetting in her fright the wire on her back, she sprang backward to her feet, snapping the lower wire and stretching the upper one as if it had been a string.
- 53. Her frightened jump, the momentary struggle with the upper wire that had caught in her mane, and the cry that escaped her, set the group into a stampede, and she herself, when finally freed from the entangling wire, dashed off to the rear for a dozen rods. The slopes of the coulee were dotted with the mares and colts who had fled in every direction. Outside the range and on the rim of the coulee lay a silly rabbit, stretching himself and gazing down with foolish eyes. There was nothing dangerous visible and nothing in the air to worry her, so the old mare started slowly and cautiously back again toward the one wire now hanging limply, and, in one place, less than two feet from the ground. There she sniffed about carefully and suddenly raising her head, she caught sight of the rabbit, as he was bounding away. There were many things that the old buckskin was afraid of, but a rabbit was not one of them. Realising that she had allowed herself to become alarmed at nothing, she went at her task with greater determination. She was about to get down to her knees again when she realised that the remaining wire was now low enough for her to step over it. Carefully lifting each leg, her skin quivering with her excitement, the buckskin mare stepped over the wire into freedom; and little Queen, frightened to see her mother beyond the fence, made it with a single leap. The old buckskin was for running now as fast as she could for the north, but she wanted the rest of the mares to go with her. She turned to look at them. There they were grazing at various points with absolute indifference to the great achievement she had consummated. She called to them to follow, but beyond a busy reply they paid no heed to her. When, however, they heard the sound of her tearing the more abundant grass outside the range, they awoke to the fact that they were not getting all they might get. Whereas the ideal of liberty had been an abstraction to them, the fact of abundant grass was a reality, and it was not many minutes before, one by one, they had all made their way over the hanging wire.
- 54. The late autumn nights had steadily grown colder and, since hollows are colder than the higher portions of the prairie at night, they moved rapidly to the plains above. Round about them lay the silent night, dark and infinite, and the stars looked down upon its hidden desolation. Closely together they grazed, lips fairly touching lips, without protest or impatience. As they grazed, they moved on to the north, and the rhythmic tear-tear of grass interspersed with rhythmic footfalls was the accompanying cadence of their half- unconscious flight. Some four miles from the range, they slept for the night on a low round hill and when dawn came they found the earth covered white with frost. The sun rose, putting a slight tinge of red into the whiteness, and Queen was so curious about it she went looking for the spots where it was thickest and licked it off the sage or rosebushes. To warm up they raced for half an hour, following the old buckskin to the north, then spent the rest of the morning grazing and moving leisurely. It was well on toward the middle of the day when an open triangle of honking geese, high in the air, made them look up. The old mare watched the geese move across the sky till they were lost in the south and was just about to return to her grazing when she saw two small objects appear on the horizon. They were so far away that they were indiscernible, but she did not wait to make certain what they were. With a call that frightened the little herd she turned north and fled. For several hours they raced on toward the heart of the wilderness; then complaint on the part of the little ones, who did not like this endless running, stopped them. But they had rested only a few minutes when they discovered the rancher and his assistant rounding a hill about two miles behind them. The frantic mothers, remembering yesterday’s struggle, fled at top speed, never slackening for a moment till, nearly twelve miles farther north, the little ones deliberately hung back. When, however, half an hour later, their pursuers surprised them by coming up on top of a hill only half
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