There is an exercise (3.21) in Silverman, Arithmetic of Elliptic Curves as-

Let $C$ be a curve of genus one. For any point $O \in C$, we can associate to the elliptic curve $(C,O)$ it's $j$-invariant $j(C,O)$. We prove in this exercise that the value of $j(C,O)$ is independent of the choice of the base point $O$. Thus we can assign a $j$- invariant to any curve $C$ of genus one.

My question- There is nothing mentioned about $C$ being non singular or an elliptic curve and Silverman has only defined $j$-invariant for elliptic curves. So, I'm confused by the language of this question.

$j$-invariant is defined for only elliptic curves in the book (Chapter 3), as they have a Weierstrass form, we define their $j$- invariant in terms of coefficients of Weierstrass equation.

Can someone help clarify this question for me? Thank you in advance.

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    $\begingroup$ See this question. $\endgroup$ – Dietrich Burde Sep 6 '19 at 14:38
  • $\begingroup$ I don't quite understand that answer. Also, the author has discussed a particular case of this situation there, when $C$ is a cubic, consisting of three lines. $\endgroup$ – Shreya Sep 7 '19 at 5:46
  • $\begingroup$ Silverman only defines genus for nonsingular curves (in Theorem II.5.4) $\endgroup$ – Wojowu Sep 7 '19 at 10:35
  • $\begingroup$ @Wojowu Thank you! This makes sense now. $\endgroup$ – Dietrich Burde Sep 7 '19 at 10:51

"My question- There is nothing mentioned about $C$ being non singular or an elliptic curve and Silverman has only defined $j$-invariant for elliptic curves."

Yes, indeed we need to assume that the curve is non-singular. But Silverman assumes this already implicitly in the definition in a curve of genus $1$, because the genus is only defined for non-singular curves - see the comment above.

The $j$-invariant of an elliptic curve $E$ is defined by $$ j(E)=\frac{c_4(E)^3}{\Delta} $$ from the Weierstrass equation. There we need that $\Delta\neq 0$, i.e., that the curve is non-singular. So we cannot use this definition if $\Delta=0$.

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  • $\begingroup$ $j$- invariant is not necessarily always defined like that, but only when char($K$) $ \neq 2,3$. Infact, my another question was going to be this, how is $j$- invariant defined in general? The above expression you've given, is for when $char(K) \neq 2$, but then Silverman in the appendix mentions a result saying that here's the j-invariant in the other case by simple computation, but where did that come from? $\endgroup$ – Shreya Sep 7 '19 at 11:29
  • $\begingroup$ Also, in the paragraph I've quoted from the book, in the last line it says, "Thus we can assign...", it seems to imple that j invariant can be assigned to any curve of genus $1$ and not just to smooth curves of genus $1$. And even if it what I interpreted is incorrect, and what you say is right, then wouldn't what we have to show ($j(C,O)$ being assigned to every such curve) be redundant in that case? $\endgroup$ – Shreya Sep 7 '19 at 11:34
  • $\begingroup$ The generalized Weierstrass equation also holds in characteristic $2$ and $3$, hence also the $j$-invariant. For details see here, section $7$: $j$-invariant in characteristic $2$ and $3$. $\endgroup$ – Dietrich Burde Sep 7 '19 at 11:56

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