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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

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This theory, due to Tate as Frey recalls, was mostly unpublished for a long time, but Tate's paper appeared in ``A review of non-Archimedean elliptic functions'', in Elliptic curves, modular forms, & Fermat's last theorem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).

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But of course, I had a copy of Tate's unpublished letter on the subject, as undoubtedly did Robert and Lang.

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