Antonio Lei

Euler systems for Rankin-Selberg convolutions of modular forms

We construct a Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use these elements to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin–Selberg L-function is nonvanishing at s=1.

Iwasawa theory for modular forms at supersingular primes

We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p -adic L -functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms

This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic Zp-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson-Flach elements.

On the asymptotic growth of Bloch-Kato-Shafarevich-Tate groups of modular forms over cyclotomic extensions

We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.

Coleman maps and the p-adic regulator

We study the Coleman maps for a crystalline representation V with non-negative Hodge–Tate weights via Perrin-Riou’s p-adic “regulator” or “expanded logarithm” map ℒV . Denote by ℋ(Γ) the algebra of ℚp-valued distributions on Γ =Gal(ℚp(μp∞)∕ℚp). Our first result determines the ℋ(Γ)-elementary divisors of the quotient of Dcris(V ) ⊗ (Brig,ℚp+)ψ=0 by the ℋ(Γ)-submodule generated by (φ∗ℕ(V ))ψ=0, where ℕ(V ) is the Wach module of V . By comparing the determinant of this map with that of ℒV (which can be computed via Perrin-Riou’s explicit reciprocity law), we obtain a precise description of the images of the Coleman maps. In the case when V arises from a modular form, we get some stronger results about the integral Coleman maps, and we can remove many technical assumptions that were required in our previous work in order to reformulate Kato’s main conjecture in terms of cotorsion Selmer groups and bounded p-adic L-functions.

Critical slope p-adic L-functions of CM modular forms

For ordinary modular forms, there are two constructions of a p-adic L-function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the critical-slope L-function arising from Kato’s Euler system and comparing this with results of Bellaïche on the critical-slope L-function defined using overconvergent modular symbols.

Coleman Maps for Modular Forms at Supersingular Primes over Lubin-Tate Extensions

Abstract Text Given an elliptic curve with supersingular reduction at an odd prime p , Iovita and Pollack have generalised results of Kobayashi to define even and odd Coleman maps at p over Lubin–Tate extensions given by a formal group of height 1. We generalise this construction to modular forms of higher weights. Video For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=KQpsht0JaME .

NON-COMMUTATIVE p-ADIC L-FUNCTIONS FOR SUPERSINGULAR PRIMES

Let E/ℚ be an elliptic curve with good supersingular reduction at p with ap(E) = 0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the ℤp-cyclotomic extension of a finite Galois extension of ℚ where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.