2022, Vol. 3, Issue 2, Part B
p-adic analysis and its role in arithmetic geometry
Author(s): Kumar Aditay
Abstract: p-adic analysis, derived from the completion of the rational numbers with respect to p-adic valuations, establishes a non-Archimedean metric framework that significantly shapes arithmetic geometry. This paper delineates the core principles of p-adic numbers, their associated analytic constructs, including power series and distributions, and their critical applications to Diophantine equations, elliptic curves, and the Langlands program. Particular focus is accorded to p-adic L-functions, interpolation techniques via the Kubota-Leopoldt method, and the integration of p-adic Hodge theory in linking Galois representations to crystalline cohomology. Through a synthesis of historical developments and recent progress, this examination underscores the essential contributions of p-adic analysis to conjectures such as Birch and Swinnerton-Dyer, as well as to advancements in explicit class field theory, thereby affirming its profound influence on the arithmetic properties of algebraic varieties.
DOI: https://doi.org/10.22271/math.2022.v3.i2b.261
Pages: 144-146 | Views: 179 | Downloads: 83
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How to cite this article:
Kumar Aditay. p-adic analysis and its role in arithmetic geometry. Journal of Mathematical Problems, Equations and Statistics. 2022; 3(2): 144-146. DOI: 10.22271/math.2022.v3.i2b.261
