The algebraic pp-adic LL-function and isogeny between families of Galois representations

In this paper, we give a formula to compare the algebraic pp-adic LL-functions for two different lattices of a given family of Galois representations over a deformation ring RR. This generalizes a classical comparison formula by Schneider [P. Schneider, The mu-invariant of isogenies, J. Indian Math. Soc. 52 (1987) 159–170] and Perrin-Riou [B. Perrin-Riou, Variation de la fonction LpLp-adique par isogénie, Algebraic number theory, Adv. Stud. Pure Math. 17 (1989) 347–358], for which RR is the cyclotomic Iwasawa algebra.Recall that we studied the two-variable Iwasawa theory for residually irreducible nearly ordinary Hida deformations in [T. Ochiai, A generalization of the Coleman map for Hida deformations, The Amer. J. Math. 125 (2003) 849–892; T. Ochiai, Euler system for Galois deformation, Ann. Inst. Fourier 55 (2005) 113–146; T. Ochiai, On the two-variable Iwasawa Main Conjecture, Compos. Math. 142 (2006) 1157–1200] and we acquired sufficient understanding through these works. By applying our formula to Hida’s nearly ordinary deformations, we understand better the two-variable Iwasawa theory for residually reducible cases, where the choice of lattices is not unique anymore.