Kazım Büyükboduk will speak on

Arithmetic of $\theta$-critical \(p\)-adic \(L\)-functions

Abstract: In joint work with Denis Benois, we give an étale construction of Bellaïche’s \(p\)-adic \(L\)-functions about $\theta$-critical points on the Coleman–Mazur eigencurve. I will discuss applications of this construction towards leading term formulae in terms of \(p\)-adic regulators on what we call the thick Selmer groups, which come attached to the infinitesimal deformation at the said $\theta$-critical point along the eigencurve. Besides our interpolation of the Beilinson–Kato elements about this point (which rests upon the overconvergent étale cohomology of Andreatta–Iovita–Stevens), the key input to prove the interpolative properties of this \(p\)-adic \(L\)-function is a new \(p\)-adic Hodge-theoretic “eigenspace-transition via differentiation” principle.