In a 5- to 6-quart pan over high heat, bring about 4 quarts water to a boil. Add whole potatoes and cook until tender when pierced, 20 to 30 minutes. Drain and rinse in cold water until cool enough to handle; peel and grate (or press through a food mill or ricer).
In a large bowl, mix potatoes, corn, cheddar cheese, cottage cheese, eggs, onions, cilantro, jalapeños, cornmeal, cumin, salt, and pepper until well blended.
Pour 1 teaspoon oil into a 10- to 12-inch nonstick frying pan over medium-high heat; when hot, drop batter into pan in about 1/3-cup portions, and use a spoon to spread slightly into 3- to 4-inch cakes. Cook, turning once with a spatula, until cakes are browned on both sides and firm to the touch in the center, about 6 minutes total. Transfer to an ovenproof plate and keep warm in a 200° oven while you cook remaining cakes, adding more oil to pan as necessary.