- Title Pages
- Let τ:ℕ→ℤ be defined by
- Contents
- Preface
- Acknowledgments
- Author information
- Dependencies between the chapters
- Chapter One Introduction, main results, context
- Chapter Two Modular curves, modular forms, lattices, Galois representations
- Chapter Three First description of the algorithms
- Chapter Four Short introduction to heights and Arakelov theory
- Chapter Five Computing complex zeros of polynomials and power series
- Chapter Six Computations with modular forms and Galois representations
- Chapter Seven Polynomials for projective representations of level one forms
- Chapter Eight Description of <i>X</i><sub>1</sub>(5<i>l</i>)
- Chapter Nine Applying Arakelov theory
- Chapter Ten An upper bound for Green functions on Riemann surfaces
- Chapter Eleven Bounds for Arakelov invariants of modular curves
- Chapter Twelve Approximating <i>V<sub>f</sub></i> over the complex numbers
- Chapter Thirteen Computing <i>V<sub>f</sub></i> modulo <i>p</i>
- Chapter Fourteen Computing the residual Galois representations
- Chapter Fifteen Computing coefficients of modular forms
- Epilogue
- Bibliography
- Index
Epilogue
Epilogue
- Chapter:
- (p.399) Epilogue
- Source:
- Computational Aspects of Modular Forms and Galois Representations
- Author(s):
- Bas Edixhoven, Jean-Marc Couveignes
- Publisher:
- Princeton University Press
This epilogue describes some work on generalizations and applications, as well as a direction of further research outside the context of modular forms. Theorems 14.1.1 and 15.2.1 will certainly be generalized to spaces of cusp forms of arbitrarily varying level and weight. This has already been done for the probabilistic variant of Theorem 14.1.1, in the case of square-free levels (and of level two times a square-free number). Some details and some applications of Bruin's work, as well as a perspective on point counting outside the context of modular forms are described. Deterministic generalizations of the two theorems mentioned above will lead to deterministic applications.
Keywords: modular forms, cusp forms, square-free levels, Peter Bruin
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- Title Pages
- Let τ:ℕ→ℤ be defined by
- Contents
- Preface
- Acknowledgments
- Author information
- Dependencies between the chapters
- Chapter One Introduction, main results, context
- Chapter Two Modular curves, modular forms, lattices, Galois representations
- Chapter Three First description of the algorithms
- Chapter Four Short introduction to heights and Arakelov theory
- Chapter Five Computing complex zeros of polynomials and power series
- Chapter Six Computations with modular forms and Galois representations
- Chapter Seven Polynomials for projective representations of level one forms
- Chapter Eight Description of <i>X</i><sub>1</sub>(5<i>l</i>)
- Chapter Nine Applying Arakelov theory
- Chapter Ten An upper bound for Green functions on Riemann surfaces
- Chapter Eleven Bounds for Arakelov invariants of modular curves
- Chapter Twelve Approximating <i>V<sub>f</sub></i> over the complex numbers
- Chapter Thirteen Computing <i>V<sub>f</sub></i> modulo <i>p</i>
- Chapter Fourteen Computing the residual Galois representations
- Chapter Fifteen Computing coefficients of modular forms
- Epilogue
- Bibliography
- Index
