Adrian Iovita

Iwasawa theory of elliptic curves at supersingular primes over ℤ p -extensions of number fields

Abstract In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary ℤp-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), no. 1, 1–36.] and Perrin-Riou [B. Perrin-Riou, Arithmétique des courbes elliptiques á réduction supersingulière en p, Experiment. Math. 12 (2003), no. 2, 155–186.], we define restricted Selmer groups and λ±, μ±-invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms of these invariants. To be able to work with non-cyclotomic ℤ p -extensions, a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime along any ramified ℤ p -extension of ℚ p .

THE ANTICYCLOTOMIC MAIN CONJECTURE FOR ELLIPTIC CURVES AT SUPERSINGULAR PRIMES

The Main Conjecture of Iwasawa theory for an elliptic curve E over Q and the anticyclotomic Zp-extension of an imaginary quadratic field K was studied in [BD2], in the case where p is a prime of ordinary reduction for E. Analogous results are formulated, and proved, in the case where p is a prime of supersingular reduction. The foundational study of supersingular main conjectures carried out by Perrin-Riou [PR2], [PR4], Pollack [Po1], Kurihara [Ku], Kobayashi [Kob], and Iovita-Pollack [IP] are required to handle this case in which many of the simplifying features of the ordinary setting break down.

Overconvergent Eichler–Shimura isomorphisms

We provide a geometric Hodge-Tate map giving generic description of the overconvergent modular symbols of some p-adic (accessible) weight k, base-changed to C_p, in terms of overconvergent modular forms of weight k+2.

Logarithmic differential forms on p-adic symmetric spaces

We give an explicit description in terms of logarithmic differential forms of the isomorphism of P. Schneider and U. Stuhler relating de Rham cohomology of p-adic symmetric spaces to boundary distributions. As an application we prove a Hodgetype decomposition for the de Rham cohomology of varieties over p-adic fields which admit a uniformization by a p-adic symmetric space.

A

3 Universal unipotent objects 9 3.1 The Kummer etale site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 The etale category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 The de Rham category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 The crystalline category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.5 Axiomatic characterization and properties of the universal unipotent objects . . 11 3.6 Existence of universal projective systems . . . . . . . . . . . . . . . . . . . . . . 13 3.7 Fundamental groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

p

2 Fontaine sheaves 9 2.1 Faltings’ topos; the algebraic setting . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Faltings’ topos; the formal setting . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Continuous functors. Localization functors . . . . . . . . . . . . . . . . . . . . . 15 2.4 The sheaf A+inf,M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 The sheaf Acris,M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.6 The sheaf Acris,M. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.7 Further properties of Acris,M and Acris,M. . . . . . . . . . . . . . . . . . . . . . . . 35 2.8 The ind-sheaves Bcris and Bcris. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.9 The fundamental exact sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

-adic nonabelian criterion for good reduction of curves

For a smooth proper scheme over a local field of mixed characteristics which has semistable reduction we define the category of its semistable etale sheaves and under certain hypothesis we prove the appropriate semistable comparison isomorphisms.

Comparison isomorphisms for smooth formal schemes

Let K0 be an unramified, complete discrete valuation field of mixed characteristics (0, p) with perfect residue field. We consider two finite, free Zp-representations of GK0 , T1 and T2, such that Ti ⊗Zp Qp, for i = 1, 2, are crystalline representations with Hodge-Tate weights between 0 and r ≤ p− 2. Let K be a totally ramified extension of degree e of K0. Supposing that p ≥ 3 and e(r − 1) ≤ p − 1, we prove that for every integer n ≥ 1 and i = 1, 2, the inclusion H fin(K,Ti)/p H fin(K,Ti) ↪→ H(K,Ti/pTi) of the finite Bloch-Kato cohomology into the Galois cohomology is functorial with respect to morphisms as Z/pZ[GK0 ]-modules from T1/pT1 to T2/pT2. In the appendix we give a related result for p = 2.

Semistable Sheaves and Comparison Isomorphisms in the Semistable Case

We consider the Galois representation associated with a finite slope

On the continuity of the finite Bloch–Kato cohomology

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