Abstract:
We shall proceed with the discussion (started on September 1, 2022) of the second memoir of the last letter of Evariste Galois and further demonstrate an underestimated, delayed, yet acceleratingly rising influence of the Galois constructive theory. In particular, we shall learn that the Galois elliptic function turns out to be the natural (“canonical”) object of investigation, which combined two traditional (historically formed and to this day artificially separated) approaches to the study of elliptic functions and integrals. We shall clearly see that neither the Jacobi-Abel elliptic function nor the Weierstrass elliptic function can replace the (simplest) Galois elliptic function as the fundamental object of a unified and systematic (algebraic) study of elliptic curves and modular forms.
A short list of acquainting literature follows:
1. Adlaj S. Elliptic integrals, functions, curves and polynomials. Computer assisted mathematics, 2019 (1): 3–8 (http://cte.eltech.ru/ojs/index.php/cam/article/view/1626).
2. Adlaj S. Multiplication and division on elliptic curves, torsion points and roots of modular equations. Zapiski Nauchnykh Seminarov POMI, 2019 (485): 24–57 (https://semjonadlaj.com/SP/ECMD.pdf).
3. Adlaj S. Galois elliptic function and its symmetries. Presented on 2019.04.18 at the annual Polynomial Computer Algebra Conference in the Euler International Mathematical Institute, St. Petersburg, Russia (https://pca-pdmi.ru/2019/files/37/PCA2019SA_slides.pdf).
4. Seliverstov A.V. On circular sections of a quadric surface. Computer Tools in Education, 2020 (4), 59-68 (http://cte.eltech.ru/ojs/index.php/kio/article/view/1665). In the latter article, the author acquaints the reader with the "Galois axis" in the context of ancient history of conic sections.
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