b^p)\\
&=&\sum_J b^pd(\varphi(a_J))\wedge d(\varphi(x_J)).\end{aligned}$$

We state the following vanishing result of $F$-split varieties for a later use.

\[F-split-B-vanish\] Let $X$ be an $n$-dimensional $F$-split projective variety and let $A$ be an ample [$\mathbb{Z}$]{}-divisor. Then, $$H^1(X, B_X^n(A))=0.$$

Consider the exact sequence $$0\to B_X^n\to F_*\omega_X\overset{C_X^n}\to \omega_X\to 0.$$ Then, by Proposition \[F-split-surje\], the trace map $$C_X^n={{\operatorname{Tr}}}_{X}^1(A):H^0(X, \omega_X(pA))\to H^0(X, \omega_X(A))$$ is surjective. Therefore, we obtain the exact sequence $$0\to H^1(X, B_X^n(A))\to H^1(X, \omega_X(pA)).$$ Since $F$-split varieties satisfy the Kodaira vanishing theorem ([@MR Proposition 2]), we obtain the vanishing $H^1(X, B_X^n(A))=0$.

The trace map of Frobenius for curves
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In this section, we calculate the trace map $${{\operatorname{Tr}}}_{X}^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A))$$ for the case where $X$ is a curve. By Question \[Tango-cex\] and its answer, ${{\operatorname{Tr}}}_X^e(A)$ is not surjective in general. But, for almost all the case, ${{\operatorname{Tr}}}_X^e(A)$ is a non-zero map.

\[curve-trace-nonzero\] Let $X$ be a smooth projective curve whose genus $g(X)$ is not zero. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor. Then, for every $e\in\mathbb Z_{>0}$, the trace map $${{\operatorname{Tr}}}_{X}^e(A):H^0(X, \omega_X(p^eA))\to H^0(X, \omega_X(A))$$ is a non-zero map.

Fix $e\in\mathbb Z_{>0}$. Since $A$ is ample, we see $\deg A\geq 1.$ We consider the two cases: $\deg A>1$ and $\deg A=1$.

In this step, we assume 