Ultra Short Period Planets in K2 III: Neighbors are Common with 12 New Multi-Planet Systems and 26 Newly Validated Planets in Campaigns 0-8, 10

Using the EVEREST photometry pipeline, we have identified 74 candidate ultra-short-period planets (orbital period P<1 d) in the first half of the K2 data (Campaigns 0-8 and 10). Of these, 33 candidates have not previously been reported. A systematic search for additional transiting planets found 12 new multi-planet systems, double the current number and representing a third (31%) of USPs. The companions have periods from 1.4 to 31 days (median 5.5 d). Half (50 of 102) of the candidate USPs and companions have been statistically validated or confirmed, 22 for the first time in this work, including 13 USPs. Almost all candidates, and all validated planets, are small (radii Rp<=3 R_E) with a median radius of R_p=1.1 R_E; the validated and confirmed candidates have radii between 0.4 R_E and 2.4 R_E and periods from P=0.18 to 0.96 d. The lack of candidate (a) ultra-hot-Jupiters (R_p>10) and (b) short-period desert (3<=Rp<=10) planets suggests that both populations are rare. These results also provide strong evidence that we have not reached a lower planetary radius limit for planets at close proximity to a star, and suggest that additional improvements in photometry techniques would yield yet more ultra-short-period planets. The large fraction of USPs in known multi-planet systems supports origins models that involve dynamical interactions with exterior planets coupled to tidal decay of the USP orbits.