arXiv subjects

Cristian D. Popescu

Publications and source records attributed to Cristian D. Popescu.

13 recordsLinked to original sources

arXiv2026

Fitting Ideals of Projective Limits of Modules over Non-Noetherian Iwasawa Algebras

In \cite{grku1}, Greither and Kurihara proved a theorem about the commutativity of projective limits and Fitting ideals for modules over the classical equivariant Iwasawa algebra $Λ_G=\mathbb{Z}_p[G][[T]]$, where $G$ is a finite, abelian group and $\Bbb Z_p$ is the ring of $p$--adic integers, for some prime $p$. In this paper, we generalize their result first to the Noetherian Iwasawa algebras $\mathcal O[[T_1, T_2, \dots, T_n]]$ and, most importantly, to non-Noetherian algebras $\mathcal O[[T_1, T_2, \dots, T_n, \dots]]$ of countably many generators, with more general rings of coefficients $\mathcal O$. The latter generalization is motivated by the recent work of Bley--Popescu on the Geometric Equivariant Iwasawa Conjecture for function fields, as well as by the emerging Iwasawa theory of Taelman class--modules associated to Drinfeld modules, where the Iwasawa algebras are not Noetherian, of the type described above. A sample application of our results to non--Noetherian geometric Iwasawa theory is given in Appendix B. Further number theoretic applications will be given in an upcoming paper.

math.AC↗

arXiv2025

Equivariant Iwasawa Theory for Ritter-Weiss Modules and Applications

We consider a finite, abelian, CM extension $H/F$ of a totally real number field $F$, and construct a $\mathbb{Z}_p[[G(H_\infty/F)]]-$module $\nabla_S^T(H_\infty)_p$, where $p>2$ is a prime and $H_\infty$ is the cyclotomic $\Bbb Z_p$-extension of $H$. This is the Iwasawa theoretic analogue of a module introduced by Ritter and Weiss in \cite{Ritter-Weiss} and studied further by Dasgupta and Kakde in \cite{Dasgupta-Kakde}. Our main result states that the $\Bbb Z_p[[G(H_\infty/F]]^-$-module $\nabla_S^T(H_\infty)_p$ is of projective dimension $1$, is quadratically presented, and that its Fitting ideal is principal, generated by an equivariant $p$-adic $L$-function $Θ_S^T(H_\infty/F)$. As a first application, we compute the Fitting ideal of an arithmetically interesting $\Bbb Z_p[[G(H_\infty/F)]]^-$-module $X_S^{T,-}$, which is a variant of the classical unramified Iwasawa module $X$ (the Galois group of the maximal abelian, unramified, pro-$p$ extension of $H_\infty$), extending earlier results of Greither-Kataoka-Kurihara \cite{Greither-Kataoka-Kurihara}. These are all instances of what is now called an Equivariant Main Conjecture in the Iwasawa theory of totally real number fields, and refine the classical main conjecture, proved by Wiles in \cite{wiles}. As a final application, we give a short, Iwasawa theoretic proof of the minus $p$-part of the far-reaching Equivariant Tamagawa Number Conjecture for the Artin motive $h_{H/F}$, for all primes $p>2$, a result also obtained, independently and with different (Euler system) methods, by Bullack-Burns-Daoud-Seo \cite{Bullach-Burns-Daoud-Seo} and Dasgupta-Kakde-Silliman \cite{Dasgupta-Kakde-Silliman-ETNC}.

math.NT↗

arXiv2024

Euler factors of equivariant $L$--functions of Drinfeld modules and beyond

In \cite{FGHP}, the first author and his collaborators proved an equivariant Tamagawa number formula for the special value at $s=0$ of a Goss--type $L$--function, equivariant with respect to a Galois group $G$, and associated to a Drinfeld module defined on $\Bbb F_q[t]$ and over a finite, integral extension of $\Bbb F_q[t]$. The formula in question was proved provided that the values at $0$ of the Euler factors of the equivariant $L$--function in question satisfy certain identities involving Fitting ideals of certain $G$--cohomologically trivial, finite $\Bbb F_q[t][G]$--modules associated to the Drinfeld module. In \cite{FGHP}, we prove these identities in the particular case of the Carlitz module. In this paper, we develop general techniques and prove the identities in question for arbitrary Drinfeld modules. Further, we indicate how these techniques can be extended to the more general case of higher dimensional abelian $t$--modules, which is relevant in the context of the proof of the equivariant Tamagawa number formula for abelian $t$--modules given by N. Green and the first author in \cite{Green-Popescu}. This paper is based on a lecture given by the first author at ICMAT Madrid in May 2023 and builds upon results obtained by the second author in his PhD thesis \cite{Ramachandran-thesis}.

math.NT↗

arXiv2023

An unconditional main conjecture in Iwasawa theory and applications

We improve upon the recent keystone result of Dasgupta-Kakde on the $\Bbb Z[G(H/F)]^-$-Fitting ideals of certain Selmer modules $Sel_S^T(H)^-$ associated to an abelian, CM extension $H/F$ of a totally real number field $F$ and use this to compute the $\Bbb Z_p[[G(H_\infty/F)]]^-$-Fitting ideal of the Iwasawa module analogues $Sel_S^T(H_\infty)_p^-$ of these Selmer modules, where $H_\infty$ is the cyclotomic $\Bbb Z_p$-extension of $H$, for an odd prime $p$. Our main Iwasawa theoretic result states that the $\Bbb Z_p[[G(H_\infty/F]]^-$-module $Sel_S^T(H_\infty)_p^-$ is of projective dimension $1$, is quadratically presented, and that its Fitting ideal is principal, generated by an equivariant $p$-adic $L$-function $Θ_S^T(H_\infty/F)$. Further, we establish a perfect duality pairing between $Sel_S^T(H_\infty)_p^-$ and a certain $\Bbb Z_p[[G(H_\infty/F)]]^-$-module $\mathcal M_S^T(H_\infty)^-$, essentially introduced earlier by Greither-Popescu. As a consequence, we recover the Equivariant Main Conjecture for the Tate module $T_p(\mathcal M_S^T(H_\infty))^-$, proved by Greither-Popescu under the hypothesis that the classical Iwasawa $μ$-invariant associated to $H$ and $p$ vanishes. As a further consequence, we give an unconditional proof of the refined Coates-Sinnott Conjecture, proved by Greither-Popescu under the same $μ=0$ hypothesis, and also recently proved unconditionally but with different methods by Johnston-Nickel, regarding the $\Bbb Z[G(H/F)]$-Fitting ideals of the higher Quillen $K$-groups $K_{2n-2}(\mathcal O_{H,S})$, for all $n\geq 2$. Finally, we combine the techniques developed in the process with the method of ''Taylor-Wiles primes'' to strengthen further the keystone result of Dasgupta-Kakde and prove, as a consequence, a conjecture of Burns-Kurihara-Sano on Fitting ideals of Selmer groups of CM number fields.

math.NT↗

arXiv2022

Geometric Main Conjectures in Function Fields

We prove an Equivariant Main Conjecture in Iwasawa Theory along any rank one, sign-normalized Drinfeld modular, split at infinity Iwasawa tower of a general function field of characteristic p, for the Iwasawa modules recently considered by Greither and Popescu, in their proof of the classical Equivariant Main Conjecture along the (arithmetic) cyclotomic Iwasawa tower.

math.NT↗

arXiv2020

An Equivariant Tamagawa Number Formula for Drinfeld Modules and Applications

We fix data $(K/F, E)$ consisting of a Galois extension $K/F$ of characteristic $p$ global fields with arbitrary abelian Galois group $G$ and a Drinfeld module $E$ defined over a certain Dedekind subring of $F$. For this data, we define a $G$-equivariant $L$-function $Θ_{K/F}^E$ and prove an equivariant Tamagawa number formula for certain Euler-completed versions of its special value $Θ_{K/F}^E(0)$. This generalizes Taelman's class number formula for the value $ζ_F^E(0)$ of the Goss zeta function $ζ_F^E$ associated to the pair $(F, E)$. Taelman's result is obtained from our result by setting $K=F$. As a consequence, we prove a perfect Drinfeld module analogue of the classical (number field) refined Brumer--Stark conjecture, relating a certain $G$-Fitting ideal of Taelman's class group $H(E/K)$ to the special value $Θ_{K/F}^E(0)$ in question.

math.NT↗

arXiv2017

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. that the new Iwasawa modules satisfy an equivariant main conjecture. In this paper we link the new modules to the $\ell$--adified Tate canonical class, defined by Tate in 1960 and give an explicit construction of (the minus part of) $\ell$--adic Tate sequences for any Galois CM extension $K/k$ of an arbitrary totally real number field $k$. These explicit constructions are significant and useful in their own right but also due to their applications (via our previous results on the Equivariant Main Conjecture in Iwasawa theory) to a proof of the minus part of the far reaching Equivariant Tamagawa Number Conjecture for the Artin motive associated to the Galois extension $K/k$.

math.NT↗

arXiv2014

Hecke characters and the $K$-theory of totally real and CM number fields

Let $F/K$ be an abelian extension of number fields with $F$ either CM or totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke characters for $F$ with infinite type equal to a special value of certain $G(F/K)$--equivariant $L$-functions. Using results of Greither-Popescu on the Brumer-Stark conjecture we construct $l$-adic imprimitive versions of these characters, for primes $l> 2$. Further, the special values of these $l$-adic Hecke characters are used to construct $G(F/K)$-equivariant Stickelberger-splitting maps in the $l$-primary Quillen localization sequence for $F$, extending the results obtained in 1990 by Banaszak for $K = \Bbb Q$. We also apply the Stickelberger-splitting maps to construct special elements in the $l$-primary piece $K_{2n}(F)_l$ of $K_{2n}(F)$ and analyze the Galois module structure of the group $D(n)_l$ of divisible elements in $K_{2n}(F)_l$, for all $n>0$. If $n$ is odd and coprime to $l$ and $F = K$ is a fairly general totally real number field, we study the cyclicity of $D(n)_l$ in relation to the classical conjecture of Iwasawa on class groups of cyclotomic fields and its potential generalization to a wider class of number fields. Finally, if $F$ is CM, special values of our $l$-adic Hecke characters are used to construct Euler systems in the odd $K$-groups with coefficients $K_{2n+1}(F, \Bbb Z/l^k)$, for all $n>0$. These are vast generalizations of Kolyvagin's Euler system of Gauss sums and of the $K$-theoretic Euler systems constructed in Banaszak-Gajda when $K = \Bbb Q$.

math.NT↗

arXiv2011

The Stickelberger splitting map and Euler systems in the $K$--theory of number fields

For a CM abelian extension $F/K$ of an arbitrary totally real number field $K$, we construct the Stickelberger splitting maps (in the sense of \cite{Ba1}) for both the étale and the Quillen $K$--theory of $F$ and we use these maps to construct Euler systems in the even Quillen $K$--theory of $F$. The Stickelberger splitting maps give an immediate proof of the annihilation of the groups of divisible elements $div K_{2n}(F)_l$ of the even $K$--theory of the top field by higher Stickelberger elements, for all odd primes $l$. This generalizes the results of \cite{Ba1}, which only deals with CM abelian extensions of $\Bbb Q$. The techniques involved in constructing our Euler systems at this level of generality are quite different from those used in \cite{BG1}, where an Euler system in the odd $K$--theory with finite coefficients of abelian CM extensions of $\Bbb Q$ was given. We work under the assumption that the Iwasawa $μ$--invariant conjecture holds. This permits us to make use of the recent results of Greither-Popescu \cite{GP} on the étale Coates-Sinnott conjecture for arbitrary abelian extensions of totally real number fields, which are conditional upon this assumption. In upcoming work, we will use the Euler systems constructed in this paper to obtain information on the groups of divisible elements $div K_{2n}(F)_l$, for all $n>0$ and odd $l$. It is known that the structure of these groups is intimately related to some of the deepest unsolved problems in algebraic number theory, e.g. the Kummer-Vandiver and Iwasawa conjectures on class groups of cyclotomic fields. We make these connections explicit in the introduction.

math.NT↗

arXiv2011

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 Galois module structure point of view in our recent work. We prove that the new Iwasawa modules are of projective dimension 1 over the appropriate profinite group rings. In the abelian case, we prove an Equivariant Main Conjecture, identifying the first Fitting ideal of the Iwasawa module in question over the appropriate profinite group ring with the principal ideal generated by a certain equivariant p-adic L-function. This is an integral, equivariant refinement of the classical Main Conjecture over totally real number fields proved by Wiles in 1990. Finally, we use these results and Iwasawa co-descent to prove refinements of the (imprimitive) Brumer-Stark Conjecture and the Coates-Sinnott Conjecture, away from their 2-primary components, in the most general number field setting. All of the above is achieved under the assumption that the relevant prime p is odd and that the appropriate classical Iwasawa mu-invariants vanish (as conjectured by Iwasawa.)

math.NT↗

arXiv2010

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

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 defined over a finite field and if G is abelian, we compute the first Fitting ideal of each l-adic realization over a certain profinite l-adic group algebra in terms of an equivariant Artin L-function. This is a refinement of earlier work of Deligne and Tate on Picard 1-motives associated to global function fields. As a consequence, we prove refined versions of the Brumer-Stark and the (étale) Coates-Sinnott Conjectures for global function fields. In upcoming work, we use these results to prove several other conjectures on special values of global L-functions in characteristic p. Also, we prove analogous results in characteristic 0, where the l-adic realizations of Picard 1-motives will be replaced by a new class of Iwasawa modules.

math.NT↗

arXiv1997

On a Refined Stark Conjecture for Function Fields

We prove that a refinement of Stark's Conjecture formulated by Rubin is true up to primes dividing the order of the Galois group, for finite, abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions a statement stronger than Rubin's holds true.

math.NT↗

arXiv1997

Gras-Type Conjectures for Function Fields

Based on results obtained in a companion paper [MSRI preprint 1997-002], we construct groups of special $S$--units for function fields of characteristic $p>0$, and show that they satisfy Gras--type Conjectures. We use these results in order to give a new proof of Chinburg's $Ω_3$--Conjecture on the Galois module structure of the group of $S$--units, for cyclic extensions of prime degree of function fields.

math.NT↗