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We give a new method of producing relations in the tautological ring R(M_{g,1}), using the sl_2-action on the Chow ring of the universal Jacobian. Withthese relations, we prove that R(M_{g, 1}) is generated by \kappa_i for i nogreater than g/3, together with \psi. Our computation shows that Faber'sconjectures for M_{g, 1} are true for g up to 19. Further, by pushing relationsforward to M_g, we obtain a new proof of Faber's conjectures (for M_g) for g upto 23. For g = 24, our method recovers all the Faber-Zagier relations. We alsogive an algebraic proof of an identity of Morita.

On the tautological rings of M_{g, 1} and its universal Jacobian