Abstract
Évariste Galois' last letter, addressed to Auguste Chevalier, on the eve of the (so-called) duel on May 30, 1832 (which, perhaps, simpler and more accurately described by Alfred, who did not allow a priest to deprive him from the final moments on the following day with his elder brother Évariste, as murder), was written on seven pages and was divided into three memoirs. The first memoir consumes a little less than two pages. It gave rise to what has come to be known as Galois theory (as, in particular, told by Melvin Kiernan). Yet Galois went on with stunningly amazing constructions in the second memoir, which consumed a bit more than two pages. The third (and longest!) memoir begins on the fifth page and remains mysteriously unresolved, yet it undoubtedly inspired Alexander Grothendieck to formulate his period conjecture. The letter is concluded with a paragraph on the latest ``principal contemplations'', concerning ``the applications of the theory of ambiguity to transcendental analysis'', where Galois delivers his last puzzle to us, saying that ``one recognizes immediately lots of expressions to look for''. Unfortunately, the severity of the time pressure upon him permitted only succinct last instructions with no more last examples. Still and disgracefully, many ``historians'' keep on incessantly and mundanely telling us (and each other) that we ought not ``overestimate'' the significance of the letter, which was (contrary to their advice) eloquently and veraciously described by Hermann Weyl as ``the most substantial piece of writing in the whole literature of mankind''!
Highlights
AbstractEvariste Galois’ last letter, addressed to Auguste Chevalier, on the eve of the duel on May 30, 1832, was written on seven pages and was divided into three memoirs
We exploit the identification of the points on the torus C/Λβ, which might be viewed as the domain of Rβ, with the points on the elliptic curve Eβ, which might be viewed as the image of the functional pair (Rβ, Rβ)
Oblivion has entirely replaced marvelling at Galois key step, towards solving the quintic, in depressing the degree of the modular equation, of level 5, from 6 to 5,31 and Galois is merely mentioned, along with Abel, for determining that the quintic is not generally solvable via radicals
Summary
An isomorphism between elliptic curves as their elliptic modulus β undergoes permissible transformations (generated by S and T ) might explicitly be given as a linear map between first coordinates. The Jacobi elliptic sine function, corresponding to elliptic modulus β and denoted by snβ = snβ(·), satisfies the differential equation snβ2 = 1 − sn2β 1 − β2sn2β , and coincides, up to homothety and translation (of its argument), with a square root of the function R (analytically continued). As the elliptic modulus β = sin θ undergoes the transformations, which we earlier discussed, corresponding elliptic functions R(·, − sin θ), R(·, i tan θ) and R(·, − sec θ) coincide, up to homothety, translation and multiplicative constants, with the squares of the Jacobi elliptic functions snβ, cnβ and dnβ.
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