Tate’s work and the Serre–Tate correspondence

Abstract

Abstract. The Serre-Tate correspondence contains a lot of Tate's work in a casual form. We present some excerpts that show how some of Tate's best known contributions came into being. Excuse all these letters. I find that writing you is an excellent method of organizing my thoughts. Some of this work was only published much later, some was never published. We give below some excerpts that show how some of Tate's best known contributions came into being. There are many other topics on which Tate worked that appear in the correspondence and that we will not consider, such as Galois cohomology, class field theory, p-adic Hodge theory, Honda-Tate theory, Serre-Tate theory, elliptic curves with everywhere good reduction, modular forms, Stark's conjectures. . . . Serre and The Tate curveElliptic curves over C are usually thought of as C modulo a lattice Λ. Now, two homothetic lattices give isomorphic elliptic curves, and so one can take the lattice Λ, corresponding to an elliptic curve E, of the form 2πi(Z + Zτ ), with Im τ > 0. Setting q = e 2iπτ and using the exponential map, one obtains an isomorphism E(C) ∼ = C * /q Z of complex Lie groups. If the equation of the elliptic curve is y 2 = x 3 − g 2 x − g 3 , classical formulas express g 2 , g 3 as power series in q and x(w), y(w), if w ∈ C * , as series in q, w, and w −1 . These series have rational coefficients, and Tate had the amazing idea that they could be used to give a description of (special) elliptic curves over a p-adic field analogous to the above description over C. This first shows up, in the correspondence, in a letter by Serre of July 31, 1959.Il paraît que vous faites des choses rupinantes avec les courbes elliptiques sur les p-adiques (j non entier), m'a raconté Lang; vous savez faire marcher ce que nos pères appelaient les "fonctions loxodromiques" sur les p-adiques. C'est bien sympathique, et j'aimerais beaucoup avoir des détails, siça ne vous ennuie pas d'écrire.Si j'ai bien compris ce que me racontait Lang, votre théorie montre de façon amusante qu'une courbe elliptiqueà multiplication

Editorial Comment

“…The ring has a unit element given by the equivalence class of a point [Spec(k)]. This ring was introduced by Grothendieck in his correspondence with Serre [3]. Results of Bittner [1] and Looijenga [16] give an alternate description of this ring, showing that the ring is generated by isomorphism classes of smooth projective varieties subject to the relation:…”

Noncommutative geometry and motives (à quoi servent les endomotifs?)

“…Grothendieck's new intuition was that the whole philosophy of motives is regulated by the theory of (algebraic) correspondences: "...J'appelle motif sur k quelque chose comme un groupe de cohomologie ℓadique d'un schema algébrique sur k, mais considérée comme indépendant de ℓ, et avec sa structure entière, ou disons pour l'instant sur Q, déduite de la théorie des cycles algébriques..." (cf. [15], Lettre 16.8.1964).…”