I am reading Silverman's article "A Survey of the Arithmetic of Elliptic Curves", which is published in Modular Forms and Fermat's Last Theorem. The article is (partly) accessible through this Google Books link.
The article uses a star notation I'm not familiar with. There are two particular cases I have in mind. First, speaking about an elliptic curve $E$ on page 18, Silverman writes:
Further, for any point $Q \in E$, there is the translation-by-$Q$ map $$\tau_Q : E \to E, \tau_Q(P) = P+Q.$$
Riemann-Roch tells us that an elliptic curve has a unique holomorphic differential (up to scalar). On the Weierstrass equations (1) and (2) it is given by
$$\omega_E = \frac{dx}{2y+a_1x+a_3} \ \ \text{ and } \ \ \omega_E = \frac{dx}{2y} \ \ \text{ respectively.}$$
The uniqueness of $\omega_E$ implies that it is translation invariant, $$\tau_Q^{*}(\omega_E) = \omega_E \ \ \text{for all } Q \in E.$$
I do not know what $\tau_Q^{*}$ means here.
Later, in defining the Weil pairing on the bottom of page 21 (unfortunately not accessible through Google Books), Silverman writes:
Let $S,T \in E[m]$. Choose a function $g$ on $E$ whose divisor satisfies $$\text{div}(g) = [m]^{*}(T) - [m]^{*}(O).$$
I know that $[m]P$ is the $m$-fold sum of $P$ with itself (if $m > 0$), but I don't know how $[m]^{*}$ is defined.
If $f: X \to Y$ is a morphism, where $X$ and $Y$ are objects of some category, and $F$ is a contravariant functor from that category to another one, we write $f^*$ for the induced morphism $F(Y) \to F(X)$. (If $F$ is contravariant, we write $f_*:F(X) \to F(Y)$.)