Women in Numbers 2 : Research Directions in Number Theory.

Intro -- Preface -- Workshop project titles and lectures -- Contributions to this volume -- Workshop website -- Acknowledgements -- The Local Equivariant Tamagawa Number Conjecture for Almost Abelian Extensions -- 1. Introduction -- 2. The main theorem -- 3. The image of the -value -- 4. Descent from the main conjecture of Iwasawa theory -- References -- Images of Metabelian Galois Representations Associated to Elliptic Curves -- 1. Introduction -- 2. Algebraic geometry -- 3. Group theory -- 4. Number theory and cohomology of groups -- 5. Examples -- 6. Main theorems -- 7. Consequences of the conjecture -- References -- Newton Polygons for a Variant of the Kloosterman Family -- 1. Introduction -- 2. Background material -- 3. Newton polygons of non-diagonal Laurent polynomials -- 4. Computation of Hodge polygons -- References -- Comparing Arithmetic Intersection Formulas for Denominators of Igusa Class Polynomials -- 1. Introduction -- 2. The Bruinier-Yang formula -- 3. The Lauter-Viray formula -- 4. Relative integral bases -- 5. Equality of indices -- 6. Equality of summands -- Acknowledgements -- References -- An Algorithmic Approach to the Dwork Family -- 1. Introduction -- 2. Background -- 3. Dwork's construction -- 4. Computing the connection matrix -- 5. The differential equation associated to the connection -- 6. An illustrative example -- References -- Ranks "Cheat Sheet" -- 1. Mordell-Weil group, rank, and Tate-Shafarevich group -- 2. -function, analytic rank, and BSD (Birch and Swinnerton-Dyer) Conjecture -- 3. (Un)boundedness -- 4. Distribution -- 5. Averages -- 6. Parity -- 7. Quadratic Twists -- 8. Selmer Groups and Selmer Ranks -- 9. Open Questions -- Background Material -- Surveys -- References -- Fully Homomorphic Encryption for Mathematicians -- 1. Introduction -- 2. Some history and background.

3. Somewhat Homomorphic Encryption over the integers -- 4. The Gentry, Smart-Vercauteren, and Gentry-Halevi SHE schemes -- 5. LWE and Ring-LWE -- References -- Mahler Measure of Multivariable Polynomials -- 1. Definition of Mahler Measure and Lehmer's question -- 2. Mahler Measure in several variables -- 3. Examples -- 4. Other occurences of Mahler measure -- 5. An algebraic integration for Mahler measure: the role of the regulator -- 6. The measures of a family of genus-one curves -- 7. Mahler measures of families of 3-surfaces -- Acknowledgement -- References -- Mahler Measure of Some Singular 3-surfaces -- 1. Introduction -- 2. Background on 3-surfaces -- 3. Main results and the general strategy for the proof -- 4. The Transcendental Lattice and the Rank -- 5. Relating the Mahler Measure to a newform -- 6. Relating ( ( ), ) to a newform -- 7. Infinite section for ₁₈ -- Acknowledgements -- References -- Distribution of Squarefree Values of Sequences Associated with Elliptic Curves -- 1. Introduction -- 2. Statement of results -- 3. Preliminaries -- 4. Key Lemma -- 5. Proof of Theorem 2.1 -- 6. Averaging the constants over families of elliptic curves -- Acknowledgements -- References -- Recent Advances for Ramanujan Type Supercongruences -- 1. Introduction -- 2. Ramanujan type formulas for 1/ -- 3. Ramanujan supercongrences of van Hamme -- 4. Proofs of van Hamme's supercongruences -- 5. Hypergeometric methods for van Hamme's conjectures -- 6. Ramanujan supercongruences arising from K3 surfaces -- Acknowledgements -- References.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.