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Arithmetic Geometry, Number Theory, and Computation

  • Conference proceedings
  • © 2021

Overview

Editors:
  1. Jennifer S. Balakrishnan

    1. Department of Mathematics and Statistics, Boston University, Boston, USA

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  2. Noam Elkies

    1. Department of Mathematics, Harvard University, Cambridge, USA

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  3. Brendan Hassett

    1. Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, USA

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  4. Bjorn Poonen

    1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

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  5. Andrew V. Sutherland

    1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

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  6. John Voight

    1. Department of Mathematics, Dartmouth College, Hanover, USA

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  • Presents number theory as a computational discipline
  • Focuses on key examples central to future research
  • Supports foundational work at the intersection of arithmetic geometry and data science

Part of the book series: Simons Symposia (SISY)

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Table of contents (21 papers)

  1. Front Matter

    Pages i-x
    Download chapter PDF

  2. A Robust Implementation for Solving the S-Unit Equation and Several Applications

    • Alejandra Alvarado, Angelos Koutsianas, Beth Malmskog, Christopher Rasmussen, Christelle Vincent, Mckenzie West
    Pages 1-41

  3. Computing Classical Modular Forms for Arbitrary Congruence Subgroups

    • Eran Assaf
    Pages 43-104

  4. Square Root Time Coleman Integration on Superelliptic Curves

    • Alex J. Best
    Pages 105-129

  5. Computing Classical Modular Forms

    • Alex J. Best, Jonathan Bober, Andrew R. Booker, Edgar Costa, John E. Cremona, Maarten Derickx et al.
    Pages 131-213

  6. Elliptic Curves with Good Reduction Outside of the First Six Primes

    • Alex J. Best, Benjamin Matschke
    Pages 215-235

  7. Efficient Computation of BSD Invariants in Genus 2

    • Raymond van Bommel
    Pages 237-258

  8. Restrictions on Weil Polynomials of Jacobians of Hyperelliptic Curves

    • Edgar Costa, Ravi Donepudi, Ravi Fernando, Valentijn Karemaker, Caleb Springer, Mckenzie West
    Pages 259-276

  9. Zen and the Art of Database Maintenance

    • Edgar Costa, David Roe
    Pages 277-292

  10. Effective Obstruction to Lifting Tate Classes from Positive Characteristic

    • Edgar Costa, Emre Can Sertöz
    Pages 293-333

  11. Conjecture: 100% of Elliptic Surfaces Over \(\mathbb {Q}\) have Rank Zero

    • Alex Cowan
    Pages 335-342

  12. On Rational Bianchi Newforms and Abelian Surfaces with Quaternionic Multiplication

    • J. E. Cremona, Lassina Dembélé, Ariel Pacetti, Ciaran Schembri, John Voight
    Pages 343-363

  13. A Database of Hilbert Modular Forms

    • Steve Donnelly, John Voight
    Pages 365-373

  14. Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB

    • Taylor Dupuy, Kiran Kedlaya, David Roe, Christelle Vincent
    Pages 375-448

  15. Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves

    • María Inés de Frutos-Fernández, Sachi Hashimoto
    Pages 449-460

  16. Curves with Sharp Chabauty-Coleman Bound

    • Stevan Gajović
    Pages 461-484

  17. Chabauty–Coleman Computations on Rank 1 Picard Curves

    • Sachi Hashimoto, Travis Morrison
    Pages 485-506

  18. Linear Dependence Among Hecke Eigenvalues

    • Dohyeong Kim
    Pages 507-519

  19. Congruent Number Triangles with the Same Hypotenuse

    • David Lowry-Duda appendix by Brendan Hassett, with an appendix by Brendan Hassett
    Pages 521-536

  20. Visualizing Modular Forms

    • David Lowry-Duda
    Pages 537-557
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Keywords

About this book

This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research.


Specific topics include
● algebraic varieties over finite fields
● the Chabauty-Coleman method
● modular forms
● rational points on curves of small genus
● S-unit equations and integral points.

Editors and Affiliations

  • Department of Mathematics and Statistics, Boston University, Boston, USA

    Jennifer S. Balakrishnan

  • Department of Mathematics, Harvard University, Cambridge, USA

    Noam Elkies

  • Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, USA

    Brendan Hassett

  • Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

    Bjorn Poonen, Andrew V. Sutherland

  • Department of Mathematics, Dartmouth College, Hanover, USA

    John Voight

About the editors

Jennifer Balakrishnan is Clare Boothe Luce Associate Professor of Mathematics and Statistics at Boston University. She holds a Ph.D. in Mathematics from the Massachusetts Institute of Technology.


Noam Elkies is Professor of Mathematics at Harvard University. He holds a Ph.D. in Mathematics from Harvard University.


Brendan Hassett is Professor of Mathematics at Brown University and Director of the Institute for Computational and Experimental Research in Mathematics. He holds a Ph.D. in Mathematics from Harvard University.


Bjorn Poonen is Distinguished Professor in Science at the Massachusetts Institute of Technology. He holds a Ph.D. in Mathematics from the University of California at Berkeley.


Andrew Sutherland is Principal Research Scientist at the Massachusetts Institute of Technology. He holds a Ph.D. in Mathematics from the Massachusetts Institute of Technology.


John Voight is Professor of Mathematics at Dartmouth College. He holds a Ph.D. in Mathematics from the University of California at Berkeley.

Bibliographic Information

  • Book Title: Arithmetic Geometry, Number Theory, and Computation

  • Editors: Jennifer S. Balakrishnan, Noam Elkies, Brendan Hassett, Bjorn Poonen, Andrew V. Sutherland, John Voight

  • Series Title: Simons Symposia

  • DOI: https://doi.org/10.1007/978-3-030-80914-0

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

  • Hardcover ISBN: 978-3-030-80913-3Published: 16 March 2022

  • Softcover ISBN: 978-3-030-80916-4Published: 16 March 2023

  • eBook ISBN: 978-3-030-80914-0Published: 15 March 2022

  • Series ISSN: 2365-9564

  • Series E-ISSN: 2365-9572

  • Edition Number: 1

  • Number of Pages: X, 587

  • Number of Illustrations: 12 b/w illustrations, 36 illustrations in colour

  • Topics: Algebraic Geometry, Number Theory, Theory of Computation

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