The Galois module structure of l-adic realizations of Picard 1-motives and applications

@article{Greither2010TheGM,
  title={The Galois module structure of l-adic realizations of Picard 1-motives and applications},
  author={Cornelius Greither and Cristian D. Popescu},
  journal={arXiv: Number Theory},
  year={2010}
}
We show that the l-adic realizations of certain Picard 1-motives associated to a G-Galois cover of smooth, projective curves defined over an algebraically closed field are G-cohomologically trivial, for all primes l. In the process, we generalize a well-known theorem of Nakajima on the Galois module structure of certain spaces of Kahler differentials associated to the top curve of the cover, assuming that the field of definition is of characteristic p. If the cover and the Picard 1-motive are… 
Fitting ideals of ℓ-adic realizations of Picard 1-motives and class groups of global function fields
TLDR
This paper deduces results on the equivariant Fitting ideal of the Tate modules of the Jacobian of the top curve, and on theEquivariants Fitting Ideal of the (dual of the) degree zero class group of theTop curve.
An Equivariant Main Conjecture in Iwasawa theory and applications
We construct a new class of Iwasawa modules, which are the number field analogues of the p-adic realizations of the Picard 1-motives constructed by Deligne in the 1970s and studied extensively from a
Picard 1-motives and Tate sequences for function fields
We use our previous work [4] on the Galois module structure of `–adic realizations of Picard 1–motives to construct explicit representatives in the `–adified Tate class (i.e. explicit `–adic Tate
Abstract $\ell$-adic $1$-motives and Tate's class
In a previous paper we constructed a new class of Iwasawa modules as $\ell$--adic realizations of what we called abstract $\ell$--adic $1$--motives in the number field setting. We proved in loc. cit.
Stickelberger series and Main Conjecture for function fields
Let F be a global function field of characteristic p with ring of integers A and let \Phi be a Hayes module on the Hilbert class field H(A) of F. We prove an Iwasawa Main Conjecture for the
Integral and p-adic Reflnements of the Abelian Stark Conjecture
We give a formulation of the abelian case of Stark's Main Conjecture in terms of determinants of projective modules and brie∞y show how this formulation leads naturally to its Equivariant Tamagawa
On a Noncommutative Iwasawa Main Conjecture for Function Fields
We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for $\ell$-adic representations of the Galois group of a function field of characteristic $p$. We also prove a
...
...

References

SHOWING 1-10 OF 39 REFERENCES
Fitting ideals of ℓ-adic realizations of Picard 1-motives and class groups of global function fields
TLDR
This paper deduces results on the equivariant Fitting ideal of the Tate modules of the Jacobian of the top curve, and on theEquivariants Fitting Ideal of the (dual of the) degree zero class group of theTop curve.
Equivariant form of the Deuring-Šafarevič formula for Hasse-Witt invariants
Let zc: X ~ Y be a finite Galois covering of connected complete non-singular algebraic curves over an algebraically closed field k. We assume that char k = p > 0 and G = G a l ( X / Y ) is a p-group.
The p-rank of Artin-Schreier curves
The groundfield k is algebraically closed and of characteristic p ≠ O. The p-rank of an abelian variety A/k is σA if there are σA copies of Z/pZ in the group of points of order p in A(k). The p-rank
The Iwasawa conjecture for totally real fields
Let F be a totally real number field. Let p be a prime number and for any integer n let Fun denote the group of nth roots of unity. Let 41 be a p-adic valued Artin character for F and let F,, be the
The K-theory of fields in characteristic p
Abstract.We show that for a field k of characteristic p, Hi(k,ℤ(n)) is uniquely p-divisible for i≠n (we use higher Chow groups as our definition of motivic cohomology). This implies that the natural
Toward equivariant Iwasawa theory
Abstract. Let l be an odd prime number, K/k a finite Galois extension of totally real number fields, and G∞, X∞ the Galois groups of K∞/k and M∞/K∞, respectively, where K∞ is the cyclotomic
On the Coates–Sinnott Conjecture
In [5], Coates and Sinnott formulated a far reaching conjecture linking the values ΘF/k,S (1 — n) for even integers n ≥ 2 of an S ‐imprimitive, Galois‐equivariant L ‐function ΘF/k,S associated to an
On a Refined Stark Conjecture for Function Fields
We prove that a refinement of Stark's Conjecture formulated by Rubin in Ann. Inst Fourier 4 (1996) is true up to primes dividing the order of the Galois group, for finite, Abelian extensions of
Algebraic Number Theory
I: Algebraic Integers.- II: The Theory of Valuations.- III: Riemann-Roch Theory.- IV: Abstract Class Field Theory.- V: Local Class Field Theory.- VI: Global Class Field Theory.- VII: Zeta Functions
...
...