What is it about?
If you take the very familiar Pascal's triangle and turn it 45 degrees, you get an infinite square matrix called Pascal's matrix. This matrix and its finite truncations appear in papers going as far back as 1871. In this paper, we find a surprising new property of the matrix, connecting its eigenvectors with local zeta functions and the problem of counting points on an elliptic curve over a finite field.
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Why is it important?
One of the central topics of study in arithmetic geometry is counting the number of solutions of an algebraic equation (or system of equations) over a finite field. In this paper, we find an unexpected link between the number of solutions of the equation y^2 = x(x-1)(x-z) over a finite field with seemingly unrelated topics in signal processing. This seems to complements a growing body of evidence that time and band limiting has something to do with the Riemann hypothesis and its variants.
Perspectives
This article is part of a broader push to further explore interactions between arithmetic geometry and integrable systems.
William Casper
California State University Fullerton
Read the Original
This page is a summary of: Binomial prolate spheroidal functions, Pascal matrices, and arithmetic of elliptic curves, Proceedings of the National Academy of Sciences, June 2026, Proceedings of the National Academy of Sciences,
DOI: 10.1073/pnas.2529171123.
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