Place the olives in a food processor and process until a coarse paste forms.
Add in the pine nuts, basil, arugula, garlic, lemon juice, olive oil and process until a fine paste forms. This will be your potato salad dressing so you want it to to still have some texture, but to be thinned out enough that it can be easily distributed over the delicate potatoes and arugula. If it needs more thinning, add olive oil one tablespoon at a time.
Taste and, only if needed, add salt. Set aside.
Cut the potatoes into ½ inch chunks, trying to keep the pieces around the same size.
Place the potatoes in a large pot and cover by about a couple of inches with cold water. Bring the water to a boil and toss in a generous pinch of kosher salt.
Turn the heat to just above medium and boil gently. Check the potatoes every 5 minutes for doneness, about 10-5 minutes. You should be able to easily pierce them with a fork, but they shouldn’t fall apart. Poor the potatoes into a colander and run cold water over them to stop the cooking process. Set aside to cool.
When the potatoes have cooled, place them in a large bowl with the remaining arugula, tomatoes, and scallions or bunching onions.
Spoon in a few generous dollops of the pesto and gently toss the salad together with your hands using your fingers to spread the pesto around.