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 $\frac{K}{<q>}$ 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 $<q>$ 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

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

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 $\frac{K}{<q>}$ 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 e1ements 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 $<q>$ 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

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 $\frac{K}{<q>}$ 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 $<q>$ 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