I am confused by the reference to sheaves of regular functions, and suspect the OP is confused. If $E$ is an elliptic curve over $\mathbb{F}_p$, then the cohomology groups of the sheaf of regular functions on $E$ is an $\mathbb{F}_p$ vector space, and Weil cohomology theories are normally required to take values in a characteristic $0$ vector space. If you take a cohomology theory with values in a vector space of characteristic $p$, then you can only count fixed points modulo $p$. It is true that, for $X$ a projective variety over $\mathbb{F}_p$, we have
$$\# X(\mathbb{F}_p) \equiv \sum_j (-1)^j\ \mathrm{Tr}(\mathrm{Frob}^{\ast} : H^j(X, \mathcal{O}) \to H^j(X, \mathcal{O})) \bmod p.$$
A reference for this is W. Fulton, A fixed point formula for varieties over finite fields. But I suspect that isn't what the OP wants.

If you want to count points, not just count modulo $p$, then you need some definition of $H^j$ which takes values in modules over a characteristic zero ring. For an elliptic curve, you can use the Tate module: $T_{\ell}(E) := \lim_{\infty \leftarrow n} E[\ell^n]$ where $\ell$ is a prime not equal to $p$ and the maps in the inverse limit are multiplication by $\ell$.

You can make an ad hoc definition $H^1(E) = T_{\ell}(E)$ and $H^2(E) = \bigwedge^2 H^1(E)$. With this definition, as anon says, Lefschetz trace goes back to the 1930's. (See Silverman III.7.1 for $T_{\ell}(E) \cong \mathbb{Z}_{\ell}^2$, so $\bigwedge^2 T_{\ell}(E) \cong \mathbb{Z}_{\ell}$. Silverman III.8.3 implies that $\mathrm{Frob}$ acts on $\bigwedge^2 T_{\ell}(E)$ by $p$. The computations in Silverman V.1 and V.2 make it clear that $\# E(\mathbb{F}_p) = 1-\mathrm{Tr}\left( \mathrm{Frob}^{\ast} T_{\ell}(E) \to T_{\ell}(E)\right) + p$.)

If you want to show this matches some other definition of $H^j$, you have to say what definition you are using.