In a pot of well-salted water, cook the spaghetti according to package directions. Preserve 1/3 cup of the pasta water before you drain the spaghetti.
While the spaghetti is cooking, make the arugula pesto. Put the arugula, cheese, walnuts, garlic, and salt in a food processor and process to combine. With the machine running, add in the olive oil until you have a smooth pesto. Set aside.
In a pot or skillet, heat the reserved pasta water along with the tomatoes and peas over medium heat until they are hot, about 3 minutes. Add the pasta and pesto to the pan and toss until everything is evenly coated and hot. Enjoy!