there exists a non-zero $l$-torsion $D$ for a prime number $l\neq p$. Note that $D$ is not a $p^e$-torsion. Take the prime decomposition $$D=\sum m_iQ_i-\sum n_jR_j.$$ Since $\deg D=\sum m_i-\sum n_j=0$, we obtain the following contradiction $$\begin{aligned}
p^eD&=&\sum m_ip^eQ_i-\sum n_jp^eR_j\\
&\sim&\sum m_ip^eA-\sum n_jp^eA\\
&=&(\sum m_i-\sum n_j)p^eA\\
&=&0.\end{aligned}$$

In this step, we assume $\deg A=1$ and we prove the assertion.

We fix a point $Q\in X$ as in Step \[good-1-form\]. If $A\sim A'$, then the corresponding trace maps are the same by Lemma \[D-linearequ\]. Therefore, we may assume that $Q\not\in {{\operatorname{Supp}}}A$. By Step \[good-1-form\], there exists an element $$\eta\in H^0(X, \omega_X(p^eA-(p^e-1)Q))\setminus H^0(X, \omega_X(p^eA-p^eQ)).$$ Take the local ring $(R, \mathfrak m)$ corresponding to the point $Q$. Note that $F^e_*R$ is a free $R$-module. Let $\{x\}$ be the $p$-basis. Then, we obtain $$\omega_R=\bigoplus_{0\leq i<p^e}R^{p^e}x^idx.$$ Thus, we can write $$\eta|_{{{\operatorname{Spec}}}R}=\sum_{0\leq i<p^e}f_i^{p^e}x^idx.$$ $\eta\not\in H^0(X, \omega_X(p^eA-p^eQ))$ means $f_i\not\in \mathfrak m$ for some $0\leq i<p^e$. $\eta\in H^0(X, \omega_X(p^eA-(p^e-1)Q))$ means $f_i\in \mathfrak m$ for every $0\leq i<p^e-1$. Therefore, we obtain $f_{p^e-1}\not\in \mathfrak m.$ Then, we can find $c\in k^{\times}$ and $\mu\in \mathfrak m$ such that $$f_{p^e-1}=c+\mu.$$ By Remark \[trace-local\], we see ${{\operatorname{Tr}}}_{X}^e(A)(\eta)|_{{{\operatorname{Spec}}}R}\neq 0.$

\[curve-trace-cor\] Let $X$ be a smooth projective curve. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor. If 