Arithmetic of Critical 𝑝-Adic 𝐿-Functions

Our objective in the present work is to develop a fairly complete arithmetic theory of critical p p -adic L L -functions on the eigencurve. To this end, we carry out the following tasks: We give an “étale” construction of Bellaïche’s p p -adic L L -functions at a θ \theta -critical point on the cuspidal Coleman–Mazur–Buzzard eigencurve. 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 L \mathscr L -invariant L I w c r \mathscr {L}^{cr}_{Iw} . We formulate the (punctual) critical main conjectures, and study its relationship with its slope-zero counterpart. Along the way, we also develop descent theory (paralleling Perrin-Riou’s work). We introduce what we call thick (Iwasawa theoretic) fundamental line and the thick Selmer complex to counter Bellaïche’s secondary p p -adic L 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. We establish an O X \mathcal {O}_\mathcal {X} -adic leading term formula for the two-variable p p -adic L L -function over the affinoid neighborhood X = S p m ( O X