Hartshorne II.8.3(c)

Let $Y$ be a nonsingular plane cubic curve and $X = Y \times Y$

Show that $p_g(X)=1$ but $p_a(X)=-1$

Now for the $p_a$ part we have using Hilbert polynomials and previous exercise (Chapter 1 7.2e) that $p_a = -1$

But how can I find $p_g$?

Look at example II.8.20.3: $\omega_X\cong \mathcal{O}_X$, so you can combine this with part (b) of the exercise which shows that $\omega_{Y\times Y}\cong p_1^*\omega_Y\otimes p_2^*\omega_Y$ to see that $\omega_{Y\times Y}\cong\mathcal{O}_{Y\times Y}$. Therefore by an application of exercise II.4.5(d), the global sections are just $k$, showing $p_g=1$.