Frey states in 'Links between stable elliptic curves and certain Diophantine equations' the following

"The most important fact about elliptic curves with reduction of muItipIicative type is due to Tate: Let K be a finite extension field of the field $\mathbb{Q}_l$ of $l$-adic numbers with $\delta_E \in K^{\times2}$ ($\delta$ the Hasse invariant) and assume that E has reduction of multiplicative type mod $l$. Then the group of K-rational points of E, E(K), is analytically isomorphic to $\frac{K}{\langle q \rangle}$ where q, the $l$-adic period of E, is an element in $G_l$ with $j_E = \frac{1}{q} + \sum\limits_{i\geq 0} a_iq^i$ . The elements $a_i$ are the integers occurring in the usual Fourier expansion of the (classical) j-function over $\mathbb{C}$ (with $q= e^{2\pi\tau i}$)."

I have been able to find some related information on different areas, for example the quotient of a field K by $\langle q\rangle $ but nothing that relates the torsion points and the j-invariant in any way. The way that he introduces it makes it seem as though there is an important paper on the subject.

Source: G.Frey, Links between stable elliptic curves and certain Diophantine equations, Ann.Univ. Saraviensis, 1(1986), 1-40


2 Answers 2


This theory, due to Tate as Frey recalls, was mostly unpublished for a long time, but Tate's paper appeared in ``A review of non-Archimedean elliptic functions'', in Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press (1995), 162—184.

It is also used extensively in Serre's Abelian $\ell$-adic representations and elliptic curves, Research Notes in Mathematics, 7. A K Peters (1998).


If you want some other references, the basic theory of Tate's parametrization of $p$-adic elliptic curves with non-integral $j$-invariant is covered in:

  1. Chapter 5 of my book Advanced Topics in the Arithmetic of Elliptic Curves, Springer.
  2. Robert's Elliptic Curves, Springer Lecture Note 326.
  3. (in somewhat less detail) Lang's Elliptic Functions.

But of course, I had a copy of Tate's unpublished letter on the subject, as undoubtedly did Robert and Lang.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.