# Help needed in clarification of a Question in Silverman's exercise.

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.

• See this question. – Dietrich Burde Sep 6 '19 at 14:38
• 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. – Shreya Sep 7 '19 at 5:46
• Silverman only defines genus for nonsingular curves (in Theorem II.5.4) – Wojowu Sep 7 '19 at 10:35
• @Wojowu Thank you! This makes sense now. – 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 $$\Delta=0$$.
• $j$- 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? – Shreya Sep 7 '19 at 11:29
• 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? – Shreya Sep 7 '19 at 11:34
• 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 $3$. – Dietrich Burde Sep 7 '19 at 11:56