Leopoldt's Conjecture Research Articles

Derived Hecke action at p and the ordinary p -adic cohomology of arithmetic manifolds

Abstract: We study the derived Hecke action at $p$ on the ordinary $p$-adic cohomology of arithmetic subgroups of reductive groups ${\rm G}(\Bbb{Q})$. This is the analog at $\ell=p$ of derived Hecke actions studied by Venkatesh in the tame case, and is the derived analog of Hida's theory for ordinary Hecke algebras. We show that properties of the derived Hecke action at $p$ are related to deep conjectures in Galois cohomology which are higher analogs of the classical Leopoldt conjecture.

The vanishing of Iwasawa's $\mu$-invariant implies the weak Leopoldt conjecture

Let K denote a number field containing a primitive p -th root of unity; if p=2 , then we assume K to be totally imaginary. If K_\infty/K is a \mathbb{Z}_p -extension such that no prime above p splits completely in K_\infty/K , then the vanishing of Iwasawa's invariant \mu(K_\infty/K) implies that the weak Leopoldt Conjecture holds for K_\infty/K . This is actually known due to a result of Ueda, which appears to have been forgotten. We present an elementary proof which is based on a reflection formula from class field theory. In the second part of the article, we prove a generalisation in the context of non-commutative Iwasawa theory: we consider admissible p -adic Lie extensions of number fields, and we derive a variant for fine Selmer groups of Galois representations over admissible p -adic Lie extensions.

On the Weak Leopoldt Conjecture and Coranks of Selmer Groups of Supersingular Abelian Varieties in $p$-adic Lie Extensions

Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not neccesasily containing the cyclotomic $\Zp$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that the aforementioned conclusions carry over to the Selmer group of a non-ordinary cuspidal modular form.

On the p‐adic Stark conjecture at s=1 and applications

Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F. We prove this conjecture unconditionally when E/Q is abelian. We also show that for certain non-abelian extensions E/F the p-adic Stark conjecture at s=1 is implied by Leopoldt's conjecture for E at p. Moreover, we prove that for a fixed prime p, the p-adic Stark conjecture at s=1 for E/F implies Stark's conjecture at s=1 for E/F. This leads to a `prime-by-prime' descent theorem for the `equivariant Tamagawa number conjecture' (ETNC) for Tate motives at s=1. As an application of these results, we provide strong new evidence for special cases of the ETNC for Tate motives and the closely related `leading term conjectures' at s=0 and s=1.

Iwasawa theory for class groups of CM fields with p = 2

In this paper, we study Iwasawa theory for p=2. First of all, we show that the classical Iwasawa main conjecture holds true even for p=2 over a totally real field k assuming μ=0 and Leopoldt's conjecture. Using the Iwasawa main conjecture, we study the 2-component of the ideal class group of a CM-field K of finite degree as a Galois module. More precisely, for a CM-field K which is cyclic over the base field k, we determine the Fitting ideal of the minus quotient of the 2-component of the ideal class group. In particular, when k=Q and K/Q is imaginary and cyclic, we prove that the Fitting ideal coincides with the Stickelberger ideal.

Composantes isotypiques de pro-$p$-extensions de corps de nombres et $p$-rationalite

Soit p un nombre premier et soit K/k une extension galoisienne finie de corps de nombres de groupe de Galois ∆ d’ordre etranger a p. Soit S un ensemble fini de places finies de k contenant l’ensemble S p des places p-adiques et soit KS la pro-p-extension maximale de K non-ramifiee en dehors de S. Soi tG:“GS{Hun quotient de GS:“Gal KS/K) sur lequel ∆ agit trivialement. Posons X:“H{rH, Hs. Dans ce travail,nous etudions la φ-composante X φ de X pour les caracteres Qp-irreductibles φ de ∆ et nous montrons entre autres, sous la conjecture de Leopoldt et pour tout caractere non trivial φ, que X φ est ZpvGw-libre si et seulement si la φ-composante de la Zp-torsion de Gs /{Gs, Gs est triviale. Nous faisons egalement une etude numerique sur la liberte de Xφ dans des extensions cycliques K{Qde degre 3 et de degre 4 (`a partir de familles de polynomes donnees par Balady, Lecacheux et plus recemment par Balady et Washington),et ainsi que dans des extensions diedrales de degre 6 de Q. Les resultats de nos calculs renforcent une recente conjecture de Gras

On the main conjecture of Iwasawa theory for certain non‐cyclotomic Zp‐extensions

Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, with ring of integers $\mathcal{O}$ and Hilbert class field $H$. Suppose $p\nmid [H:K]$ is a prime number which splits in $K$, say $p\mathcal{O}=\mathfrak{p}\mathfrak{p}^*$. Let $H_\infty=HK_\infty$ where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. Write $M(H_\infty)$ for the maximal abelian $p$-extension of $H_\infty$ unramified outside the primes above $\mathfrak{p}$, and set $X(H_\infty)=\mathrm{Gal}(M(H_\infty)/H_\infty)$. In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_\infty)$. As a consequence, we have that if $X(H_\infty)=0$, the relevant $L$-values are $\mathfrak{p}$-adic units. In addition, the main conjecture for $X(H_\infty)$ has implications toward (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak $\mathfrak{p}$-adic Leopoldt conjecture.

Heuristics and conjectures in the direction of a $p$-adic Brauer–Siegel Theorem

Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K and let T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We conjecture the existence of a constant C\_p>0 such that log(\#T\_K) $\le$ C\_p log(\sqrt(D\_K)) when K varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a p-adic analogue, of the classical Brauer--Siegel Theorem, wearing here on the valuation of the residue at s=1 (essentially equal to \#T\_K) of the p-adic zeta-function zeta\_p(s) of K.We shall use a different definition that of Washington, given in the 1980's, and approach this question via the arithmetical study of T\_K since p-adic analysis seems to fail because of possible abundant "Siegel zeros" of zeta\_p(s), contrary to the classical framework.We give extensive numerical verifications for quadratic and cubic fields (cyclic or not) and publish the PARI/GP programs directly usable by the reader for numerical improvements. Such a conjecture (if exact) reinforces our conjecture that any fixed number field K is p-rational (i.e., T\_K=1) for all p >> 0 .

T-ranks of Iwasawa modules

Let p be a rational prime. We study Galois (co-)invariants of Iwasawa modules attached to Zp-extensions of number fields, by encoding the corresponding orders in so-called T-ranks. We show that the growth of T-ranks at small layers of a Zp-extension bounds the over-all growth. This allows for effective algorithms for checking whether the T-ranks of the layers in a Zp-extension remain bounded. We apply this theoretical tool for checking the boundedness of T-ranks in order to prove a conjecture of Gross for all number fields containing exactly two primes above p, and we verify this conjecture by computations for many cubic non-normal number fields with three such primes. Moreover, our method can also be used to verify numerically Leopoldt's Conjecture for K and p, provided that p is totally split in K. Finally, we study the problem of semi-simplicity.

The p-adic Kummer–Leopoldt constant: Normalized p-adic regulator

The [Formula: see text]-adic Kummer–Leopoldt constant [Formula: see text] of a number field [Formula: see text] is (assuming the Leopoldt conjecture) the least integer [Formula: see text] such that for all [Formula: see text], any global unit of [Formula: see text], which is locally a [Formula: see text]th power at the [Formula: see text]-places, is necessarily the [Formula: see text]th power of a global unit of [Formula: see text]. This constant has been computed by Assim and Nguyen Quang Do using Iwasawa’s techniques, after intricate studies and calculations by many authors. We give an elementary [Formula: see text]-adic proof and an improvement of these results, then a class field theory interpretation of [Formula: see text]. We give some applications (including generalizations of Kummer’s lemma on regular [Formula: see text]th cyclotomic fields) and a natural definition of the normalized [Formula: see text]-adic regulator for any [Formula: see text] and any [Formula: see text]. This is done without analytical computations, using only class field theory and especially the properties of the so-called [Formula: see text]-torsion group [Formula: see text] of Abelian [Formula: see text]-ramification theory over [Formula: see text].

Potential Automorphy and the Leopoldt conjecture

We study in this paper Hida's $p$-adic Hecke algebra for ${\rm GL}_n$ over a CM field $F$. Hida has made a conjecture about the dimension of these Hecke algebras, which he calls the non-abelian Leopoldt conjecture, and shown that his conjecture in the case $F={\Bbb Q}$ implies the classical Leopoldt conjecture for a number field $K$ of degree $n$ over ${\Bbb Q}$, if one assumes further the existence of automorphic induction of characters for the extension $K/{\Bbb Q}$. We study Hida's conjecture using the automorphy lifting techniques adapted to the ${\rm GL}_n$ setting by Calegari-Geraghty. We prove an automorphy lifting result in this setting, conditional on existence and local-global compatibility of Galois representations arising from torsion classes in the cohomology of the corresponding symmetric manifolds. Under the same conditions we show that one can deduce the classical (abelian) Leopoldt conjectures for a totally real number field $K$ and a prime $p$ using Hida's non-abelian Leopoldt conjecture for $p$-adic Hecke algebra for ${\rm GL}_n$ over CM fields without needing to assume automorphic induction of characters for the extension $K/{\Bbb Q}$. For this methods of potential automorphy results are used.

Approche p-adique de la conjecture de Greenberg pour les corps totalement réels

Let k be a totally real number field ant let k$\infty$ be its cyclotomic Zp-extension for a prime p\textgreater{}2. We give (Theorem 3.2) a sufficient condition of nullity of the Iwasawa invariants lambda, mu, when p totally splits in k, and we obtain important tables of quadratic fields and p for which we can conclude that lambda = mu=0.We show that the number of ambiguous p-classes of kn (nth stage in k$\infty$) is equal to the order of the torsion group T, of the Galois group of the maximal Abelian p-ramified pro-p-extension of k (Theorem 4.2), for all n \textgreater{}\textgreater{} e, where p^e is the exponent of U*/ adh(E) (in terms of local and global units of k). Then we establish analogs of Chevalley's formula using a family (Lambda\_i^n)\_{0$\le$i$\le$m\_n} of subgroups of k* containing E, in which any x is norm of an ideal of kn. This family is attached to the classical filtration of the p-class group of kn defining the algorithm of computation of its order in m\_n steps. From this, we prove (Theorem 6.1) that m\_n $\ge$ (lambda.n + mu.p^n + nu)/v\_p(T\_k) and that the condition m\_n = O(1) (i.e., lambda = mu=0) essentially depends on the P-adic valuations of the (x^(p-1)-1)/p, x in Lambda\_i^n, for P I p, so that Greenberg's conjecture is strongly related to "Fermat quotients" in k*. Heuristics and statistical analysis of these Fermat quotients (Sections 6, 7, 8) show that they follow natural probabilities, linked to T\_k whatever n, suggesting that lambda = mu=0 (Heuristics 7.1, 7.2, 7.3). This would imply that, for a proof of Greenberg's conjecture, some deep p-adic results (probably out of reach now), having some analogy with Leopoldt's conjecture, are necessary before referring to the sole algebraic Iwasawa theory.

Sur le module de Bertrandias–Payan dans une p-extension – Noyau de capitulation

For a number field K and a prime number p we denote by BP\_K the compositum of the cyclic p-extensions of K embeddable in a cyclic p-extension of arbitrary large degree. Then BP\_K is p-ramified (= unramified outside p) and is a finite extension of the compositum K~ of the Z\_p-extensions of K.We study the transfer map j\_(L/K) (as a capitulation map of ideal classes) for the Bertrandias-Payan module bp\_K:=Gal(BP\_K/K~) in a p-extension L/K (p\textgreater{}2, assuming the Leopoldt conjecture). In the cyclic case of degree p, j\_(L/K) is injective except if L/K is kummerian, p-ramified, non globally cyclotomic but locally cyclotomic at p (Theorem 3.1). We then intend to characterize the condition \#bp\_K divides \#bp\_L^G (fixed points). So we study bp\_L^G when j\_(L/K) is not injective and show that it depends on the Galois group (over K~) of the maximal Abelian p-ramified pro-p-extension of K.We give complete proofs in an elementary way using ideal approach of global class field theory.

Level raising for p-adic Hilbert modular forms

This paper generalises previous work of the author to the setting of overconvergent p-adic automorphic forms for a definite quaternion algebra over a totally real field. We prove results which are analogues of classical ‘level raising’ results in the theory of mod p modular forms. Roughly speaking, we show that an overconvergent eigenform of finite slope whose associated local Galois representation at some auxiliary prime l - p is (a twist of) a direct sum of trivial and cyclotomic characters lies in a family of eigenforms whose local Galois representation at l is generically (a twist of) a ramified extension of trivial by cyclotomic. We give some explicit examples of p-adic automorphic forms to which our results apply, and give a general family of examples whose existence would follow from counterexamples to the Leopoldt conjecture for totally real fields. These results also play a technical role in other work of the author on the problem of local–global compatibility at Steinberg places for Hilbert modular forms of partial weight one.

On the rank one abelian Gross–Stark conjecture

Let F be a totally real number field, p a rational prime, and \chi a finite order totally odd abelian character of Gal (\overline{F}/F) such that \chi(\mathfrak{p})=1 for some \mathfrak{p}|p . Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the p -adic L -function associated to \chi at its exceptional zero and the \mathfrak{p} -adic logarithm of a p -unit in the \chi component of F_\chi^\times . In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture in the rank one setting assuming two conditions: that Leopoldt's conjecture holds for F and p , and that if there is only one prime of F lying above p , a certain relation holds between the \mathscr{L} -invariants of \chi and \chi^{-1} . The main result of this paper removes both of these conditions, thus giving an unconditional proof of the rank one conjecture.

On the second Tate–Shafarevich group of a 1-motive

We prove finiteness results for Tate--Shafarevich groups in degree 2 associated with 1--motives, rely them to Leopoldt's conjecture, and present an example of a semiabelian variety with an infinite Tate--Shafarevich group in degree 2. We also establish an arithmetic duality theorem for 1--motives over number fields which complements earlier results of Harari and Szamuely in this direction.

Ramification in Iwasawa Theory and Splitting Conjectures

We make a reciprocity conjecture that extends Iwasawa's analogy of direct limits of class groups along the cyclotomic tower of a totally real number field F to torsion points of Jacobians of curves over finite fields. The extension is to generalized class groups and generalized Jacobians. We state some splitting conjectures that are equivalent to Leopoldt's conjecture.

Algebraische Zahlentheorie

Please note the corrected of U. Hartl. The conference brought together researchers from Europe, the US, and Japan who reported on various recent and ongoing developments in algebraic number theory and related fields. As at previous meetings, organized by Deninger, Schneider and Scholl, one of the clearest themes was the prevalence of p -adic methods across a range of areas. A notable difference with previous years was the number of younger people both as speakers and participants. Colmez reported on his work relating unitary admissible \mathrm{GL}_2(\mathbb Q_p) -representations to local Galois representations. This realizes a program of Breuil and stands at the crossroads of p -adic Hodge theory, representations of p -adic reductive groups and explicit reciprocity laws, as well as having applications to modularity of global Galois representations. Related talks were given by Schneider who explained on going work with Vigneras, attempting to generalize some of Colmez' constructions to higher rank, as well as Orlik who discussed the construction of locally analytic representations from equivariant vector bundles on symmetric spaces. L. Berger reported on an extension of his earlier work on classification of local Galois representations. Hartl explained how these ideas could be used to give a description of the image of the Rapoport–Zink period morphism. This was a satisfying complement to his talk at the previous meeting where he had sketched some of these ideas. There were several talks related to Iwasawa theory and reciprocity laws. Zerbes reported on her work on reciprocity laws for higher dimensional local fields. Fukaya reported on joint work with Coates, Kato, Sujatha and Venjakob in non-abelian Iwasawa theory, and Ochiai discussed the Iwasawa theory of ordinary Hida families. The talk by Sharifi was also related to this area. It described a fascinating relation between Galois cohomology and modular symbols, which seems to be closely related to the Main conjecture of Iwasawa theory. There were a number of talks dealing with congruences between automorphic forms, and applications. The most exciting of these was by Fujiwara who outlined how Taylor–Wiles systems could be used, in certain circumstances, to prove the Leopoldt conjecture for totally real fields. Sorensen discussed his work on level raising for \mathrm{GSp}_4 and some applications to Selmer groups. T. Berger explained how to construct Galois representations attached to cusp forms on \mathrm{GL}_2 over an imaginary field. These had been constructed by Taylor about 15 years ago, but were previously known to have the correct L -factors only at a set of primes of density 1 . Berger also explained ongoing work on modularity lifting theorems in this situation. This would be an exciting advance since such theorems are currently available only over totally real fields. There were two talks on polylogarithms. Blottiere explained his results on the Eisenstein classes on Hilbert modular varieties and applications to special values of L -functions. Bannai discussed the crystalline realization of the elliptic polylogarithm. Somewhat related to this was the talk of Huber on the p -adic Borel regulator. Other talks were given by Yoshida who explained a computation of vanishing cycles on Shimura varieties, realizing the local Langlands and Jacquet–Langlands correspondences, Görtz who spoke on affine Deligne–Lusztig varieties, Saito who outlined his construction of the characteristic cycle of an l -adic sheaf, and Schmidt who discussed his work on integer rings of type K(π,1) .

Nearly ordinary Galois deformations over arbitrary number fields

AbstractLet K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

Weak Leopoldt's conjecture for Hecke characters of imaginary quadratic fields

We give a proof of the weak Leopoldt's conjecture à la Perrin-Riou, under some technical condition, for the p-adic realizations of the motive associated to Hecke characters over an imaginary quadratic field K of class number 1, where p is a prime >3 and where the CM elliptic curve associated to the Hecke character has good reduction at the primes above p in K. This proof makes use of the 2-variable Iwasawa main conjecture proved by Rubin. Thus we prove the Jannsen conjecture for the above p-adic realizations for almost all Tate twists.