Titlebook: Arithmetic Geometry, Number Theory, and Computation; Jennifer S. Balakrishnan,Noam Elkies,John Voight Conference proceedings 2021 The Edit

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https://doi.org/10.1007/978-3-030-80914-0elliptic curves; subgroups; zeta functions; Hecke operators; Birch-Swinnerton-Dyer conjecture; Jacobians;

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978-3-030-80916-4The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerl

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Implementing Remote Data Portal Hosts,n . + . = 1, . in the computer algebra package SageMath. This paper outlines the mathematical basis for the implementation. We discuss and reference the results of extensive computations, including exponent bounds for solutions in many fields of small degree for small sets .. As an application, we p

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Object-Oriented Application Design,ving the . and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to curves in the set. We also discuss the connect

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https://doi.org/10.1007/978-1-4302-0144-1ns and Modular Forms Database [.]. This report explains the improvements made to the implementation of the algorithm described in [.] that were needed to do the computation of the Tamagawa numbers and the real period in reasonable time. We also explain some of the more technical details of the algor

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Business Framework Implementation,es of a finite reduction. This gives an upper bound on the “middle Picard number” of the hypersurface. The improvement over existing methods is that our method relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more. The obstr

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Business Framework Implementation,lliptic surfaces have rank 0 when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain .-functions, or from understanding of the local behavio