The polynomial multidimensional Szemer,di Theorem along shifted primes

If : a"currency sign -> a"currency sign (a"") are polynomials with zero constant terms and E aS, a"currency sign (a"") has positive upper Banach density, then we show that the set E a (c) (E - (p - 1)) a (c) aEuro broken vertical bar a (c) (E - (p - 1)) is nonempty for some prime p. We also prove mean convergence for the associated averages along the prime numbers, conditional to analogous convergence results along the full integers. This generalizes earlier results of the authors, of Wooley and Ziegler, and of Bergelson, Leibman and Ziegler.