Aligning vertical dots

I'd place the vertical dots under the equals sign, to be honest.

Anyway, here's how you can do it and also how to align the summation signs by making the upper limits zero width:

\documentclass{article}

\usepackage{amsmath,mathtools}

\begin{document}

\begin{align*}
  F_1 &= \sum_{i=1}^{\mathclap{r^2+1}} a_{i1}x_i^r \equiv 0 \pmod{p},\\
  F_2 &= \sum_{i=1}^{\mathclap{r^2+1}} a_{i2}x_i^r \equiv 0 \pmod{p},\\
      & \phantom{\;=}\vdotswithin{\displaystyle\sum}\\
  F_r &= \sum_{i=1}^r a_{ir}x_i^r \equiv 0 \pmod{p}.
\end{align*}

\end{document}

You can set it in a similarly-sized box with the aid of eqparbox:

\documentclass{article}

\usepackage{amsmath,eqparbox}

\begin{document}

\begin{align*}
  F_1 = &\eqmakebox[sum]{$\displaystyle\sum_{i=1}^{r^2+1}$} a_{i1}x_i^r \equiv 0 \pmod{p},\\
  F_2 = &\eqmakebox[sum]{$\displaystyle\sum_{i=1}^{r^2+1}$} a_{i2}x_i^r \equiv 0 \pmod{p},\\
        &\eqmakebox[sum]{$\vdots$} \\
  F_r = &\eqmakebox[sum]{$\displaystyle\sum_{i=1}^r$} a_{ir}x_i^r \equiv 0 \pmod{p}.
\end{align*}

\end{document}

All content with the same <tag> in \eqmakebox[<tag>] are set in a box of similar width.

Since it is a visual arrangement anyway, I suggest to add the preferred space manually, like that.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{align*}
F_1 = &\sum_{i=1}^{r^2+1} a_{i1}x_i^r \equiv 0 \pmod{p},\\
F_2 = &\sum_{i=1}^{r^2+1} a_{i2}x_i^r \equiv 0 \pmod{p},\\
      &\;\;\;\vdots \\
F_r = &\sum_{i=1}^r a_{ir}x_i^r \equiv 0 \pmod{p}.
\end{align*}
\end{document}