Sarah Livia Zerbes

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.

Rankin–Eisenstein classes and explicit reciprocity laws

We construct three-variable

Rankin–Eisenstein classes in Coleman families

p

Local epsilon isomorphisms

-adic families of Galois cohomology classes attached to Rankin convolutions of modular forms, and prove an explicit reciprocity law relating these classes to critical values of L-functions. As a consequence, we prove finiteness results for the Selmer group of an elliptic curve twisted by a 2-dimensional odd irreducible Artin representation when the associated

Selmer Groups over p-Adic Lie Extensions I

L

Wach modules and critical slope p-adic L-functions

-value does not vanish.

Finite polynomial cohomology for general varieties

We show that the Euler system associated with Rankin–Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in p-adic Coleman families. We prove an explicit reciprocity law for these families and use this to prove cases of the Bloch–Kato conjecture for Rankin–Selberg convolutions.

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

In this paper, we prove the “local e-isomorphism” conjecture of Fukaya and Kato for a particular class of Galois modules, obtained by interpolating the twists of a fixed crystalline representation of GQp by a family of characters of GQp . This can be regarded as a local analogue of the Iwasawa main conjecture for abelian p-adic Lie extensions of Q p , extending earlier work of Kato for rank one modules, and of Benois and Berger for the cyclotomic extension. We show that such an e-isomorphism can be constructed using the the 2-variable version of the Perrin-Riou regulator map constructed by the first and third authors.

Coleman maps and the p-adic regulator

Let be an elliptic curve defined over a number field . The paper concerns the structure of the -Selmer group of over -adic Lie extensions of which are obtained by adjoining to the -division points of an abelian variety defined over . The main focus of the paper is the calculation of the Gal-Euler characteristic of the -Selmer group of . The main theory is illustrated with the example of an elliptic curve of conductor 294.

Euler systems for modular forms over imaginary quadratic fields

Abstract We study Kato and Perrin-Rious critical slope p-adic L-function attached to an ordinary modular form using the methods of A. Lei, D. Loeffler and S. L. Zerbes, Wach modules and Iwasawa theory for modular forms, Asian J. Math. 14 (2010), 475–528. We show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p. We use this decomposition to prove results on the zeros of the p-adic L-function, and we show that our results match the behaviour observed in examples calculated by Pollack and Stevens in “Overconvergent modular symbols and p-adic L-functions”, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 1, 1–42.