Arithmetic of critical $p$-adic $L$-functions

Abstract

Our objective in the present work is to develop a fairly complete arithmetic theory of critical $p$-adic $L$-functions on the eigencurve. To this end, we carry out the following tasks: a) We give an "\'etale" construction of Bella\"iche's $p$-adic $L$-functions at a $\theta$-critical point on the cuspidal eigencurve. b) We introduce the algebraic counterparts of these objects (which arise as appropriately defined Selmer complexes) and develop Iwasawa theory in this context, including a definition of an Iwasawa theoretic $\mathscr L$-invariant $\mathscr{L}^{\rm cr}_{\rm Iw}$. c) We formulate the (punctual) critical main conjecture and study its relationship with its slope-zero counterparts. Along the way, we also develop descent theory (paralleling Perrin-Riou's work). d) We introduce what we call thick (Iwasawa theoretic) fundamental line and the thick Selmer complex to counter Bella\"iche's secondary $p$-adic $L$-functions. This allows us to formulate an infinitesimal thickening of the Iwasawa main conjecture, and we observe that it implies both slope-zero and punctual critical main conjectures, but it seems stronger than both. e) We establish an $\mathcal{O}_{\mathcal{X}}$-adic leading term formula for the two-variable $p$-adic $L$-function over the affinoid neighbourhood $\mathcal{X}={\rm Spm}(\mathcal{O}_{\mathcal{X}})$ in the eigencurve about a $\theta$-critical point. Using this formula we prove, when the Hecke $L$-function of $f$ vanishes to order one at the central critical point, that the derivative of the secondary $p$-adic $L$-function can be computed in terms of the second order derivative of an $\mathcal{O}_{\mathcal{X}}$-adic regulator (rather than a regulator itself).