On local Galois representations associated to ordinary Hilbert modular forms

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) .

We describe what is known about the local splitting behaviour of Galois representations attached to ordinary cuspidal eigenforms. We relate this to a question of Coleman concerning the existence of non-CM eigenforms of weight k ≥ 2 in the image of the (k − 1)-st power of the theta derivation.

We develop a new strategy for studying low weight specializations of p-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate–Vatsal which states that a Hida family contains infinitely many classical eigenforms of weight one if and only if it has complex multiplication. Our strategy is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image, and that classical weight one eigenforms satisfy the Ramanujan conjecture. We indicate how this strategy might be used to prove similar statements in the case of partial weight one Hilbert modular forms, given a suitable development of Hida theory in that setting.

In a previous paper the authors showed that, under some technical conditions, the local Galois representations attached to the members of a non-CM family of ordinary cusp forms are indecomposable for all except possibly finitely many members of the family. In this paper we use deformation theoretic methods to give examples of non-CM families for which every classical member of weight at least two has a locally indecomposable Galois representation.

In this paper, we study the universal lifting spaces of local Galois representations valued in arbitrary reductive group schemes when $\ell \neq p$ . In particular, under certain technical conditions applicable to any root datum, we construct a canonical smooth component in such spaces, generalizing the minimally ramified deformation condition previously studied for classical groups. Our methods involve extending the notion of isotypic decomposition for a $\operatorname {\mathrm {GL}}_n$ -valued representation to general reductive group schemes. To deal with certain scheme-theoretic issues coming from this notion, we are led to a detailed study of certain families of disconnected reductive groups, which we call weakly reductive group schemes. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $\mathrm {G}_2$ -valued representations.

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of n-dimensional mod p Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for n-dimensional automorphic Galois representations.

Let [Formula: see text] be a prime, and let [Formula: see text] be a cuspidal eigenform of weight at least [Formula: see text] and level coprime to [Formula: see text] of finite slope [Formula: see text]. Let [Formula: see text] denote the mod [Formula: see text] Galois representation associated with [Formula: see text] and [Formula: see text] the mod [Formula: see text] cyclotomic character. Under an assumption on the weight of [Formula: see text], we prove that there exists a cuspidal eigenform [Formula: see text] of weight at least [Formula: see text] and level coprime to [Formula: see text] of slope [Formula: see text] such that [Formula: see text] up to semisimplification. The proof uses Hida–Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions of the local Galois representations associated to cusp forms with slopes in the interval [Formula: see text] were determined by Deligne, Buzzard and Gee and for slopes in [Formula: see text] by Bhattacharya, Ganguli, Ghate, Rai and Rozensztajn. We show that these reductions, in spite of their somewhat complicated behavior, are compatible with the displayed equation above. Moreover, the displayed equation above allows us to predict the shape of the reductions of a class of Galois representations attached to eigenforms of slope larger than [Formula: see text]. Finally, the methods of this paper allow us to obtain upper bounds on the radii of certain Coleman families.

The purpose of this course is to give an introduction to the theory of p-adic integration with values in spaces of modular forms (elliptic modular forms, Siegel modular forms, . . .). We show that very general p-adic families of modular forms can be constructed as moments of certain p-adic measures on a profinite group Y = lim ←− Yi with values in a formal q-expansion ring like Zp[[q ]] where B is an additive semi-group, and q = {q |ξ ∈ B} the corresponding formally written multiplicative semi-group (for example B = Bn = {ξ = ξ ∈ Mn(Q)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms). We discuss some applications of this theory to the construction of certain new p-adic families of modular forms (families of Klingen-Eisenstein series, families of theta-series with spherical polynomials. . .). Main sources of this theory are: • Serre’s theory of p-adic forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, LNM 350 (1973) 191-268) [Se73]. • Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi93]). • Construction of p-adic Siegel-Eisenstein series by the author, see [PaSE]. As an application, we describe a solution of a problem of Coleman-Mazur in [PaTV], using the RankinSelberg method and the p-adic integration in a Banach algebra A. An introductory cours given on November 29 in POSTECH (Pohang, Korea) 0 Introduction Let p be a prime number (we often assume p≥ 5). There are two different ways of introducing p-adic modular forms: the first approach uses formal q-expansions with coefficients in a p-adic ring [Se73], and the second approach is the p-adic interpolation of Galois representations attached to classical automorphic forms. The first approach was extensively developped by Katz [Ka78] for the group G = GL2 over a totally real number field, in order to construct p-adic L-functions for CM-fields using p-adic Hilbert-Eisenstein series. In general, in this q-expansion method a typical p-adic family φ of modular (automorphic) forms is an element of the Serre ring: φ ∈ Λ[[q]] where Λ = Zp[[T ]] is the Iwasawa algebra. In the second approach one considers Λ-adic Galois representations of type ρ : Gal(Q/Q) → GLm(Λ) (“Big Galois representations”, see [Hi86], [Til-U]). These two theories are essentially equivalent if we start from holomorphic automorphic forms on the group G = GL2 over a totally real field, but in other cases there is no direct link between φ and ρ. On the other hand there exist interesting examples of p-adic L-functions Lφ,p and Lρ,p attached to φ and to ρ. In general Lφ,p and Lρ,p should belong to the quotient field L = QuotΛ or to its finite extensions. If ρ interpolates a p-adic family of motives then there are conjectural general definitions of Lρ,p (see [Co-PeRi], [Colm98], [PaAdm]). It would be very interesting to formulate a general Langlands-type conjecture relating Λ-adic automorphic forms and Λ-adic Galois representations. As an application, we describe a solution of a problem of Coleman-Mazur, using the Rankin-Selberg method and the theory of p-adic integration with values in a p-adic algebra A. This problem was stated in "The Eigencurve" (1998), R.Coleman and B.Mazur stated the following as follows: Given a prime p and Coleman’s family {fk′} of cusp eigenforms of a fixed positive slope σ = ordp(αp(k )) > 0, to construct a two variable p-adic L-function interpolating on k the Amice-Velu p-adic L-functions Lp(fk′ ). Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. Such measures come from Eisenstein distributions with values in certain Banach A-modules M = M(N ;A) of families of overconvergent forms over A.

Let f be a primitive cusp form of weight at least 2, and let ρ f be the p -adic Galois representation attached to f . If f is p -ordinary, then it is known that the restriction of ρ f to a decomposition group at p is ``upper triangular’’. If in addition f has CM, then this representation is even ``diagonal’’. In this paper we provide evidence for the converse. More precisely, we show that the local Galois representation is not diagonal, for all except possibly finitely many of the arithmetic members of a non-CM family of p -ordinary forms. We assume p is odd, and work under some technical conditions on the residual representation. We also settle the analogous question for p -ordinary Λ -adic forms, under similar conditions.

We study the intimate interactions between the theory of p-adic Galois representations and the structure of pro-p Galois groups. In particular, information passes in both directions. Algebraic geometry, for instance in the guise of elliptic curves and modular forms, yields naturally occurring Galois representations, whereas on the other side, co-homological techniques and variants on class field theory tell us about the generators and relations of the pro-p Galois groups. In the case of pro-p extensions ramified at (primes above) p, this combination works together rather well to elucidate the structure of the set of Galois representations. In the case of pro-p extensions unramified at p,both sides are poorly understood, but there is the fundamental conjecture of Fontaine—Mazur claiming that such representations should have finite image (since algebraic geometry can produce no others). This has very interesting consequences for the corresponding pro-p Galois groups, possibly producing a new family of just-infinite pro-p groups.

We prove that the p p -adic local Langlands correspondence for GL 2 ⁡ ( Q p ) \operatorname {GL}_2(\mathbb {Q}_p) appears in the étale cohomology of the Lubin-Tate tower at infinity. We use global methods using recent results of Emerton on the local-global compatibility, and hence our proof applies to local Galois representations which come via a restriction from global pro-modular Galois representations.

We prove that the nonordinary component is connected in the moduli spaces of finite flat models of two-dimensional local Galois representations over finite fields. This was conjectured by Kisin. As an application to global Galois representations, we prove a theorem on the modularity comparing a deformation ring and a Hecke ring.

A generalization of Serre’s Conjecture asserts that if $F$ is a totally real field, then certain characteristic $p$ representations of Galois groups over $F$ arise from Hilbert modular forms. Moreover, it predicts the set of weights of such forms in terms of the local behaviour of the Galois representation at primes over $p$. This characterization of the weights, which is formulated using $p$-adic Hodge theory, is known under mild technical hypotheses if $p>2$. In this paper we give, under the assumption that $p$ is unramified in $F$, a conjectural alternative description for the set of weights. Our approach is to use the Artin–Hasse exponential and local class field theory to construct bases for local Galois cohomology spaces in terms of which we identify subspaces that should correspond to ones defined using $p$-adic Hodge theory. The resulting conjecture amounts to an explicit description of wild ramification in reductions of certain crystalline Galois representations. It enables the direct computation of the set of Serre weights of a Galois representation, which we illustrate with numerical examples. A proof of this conjecture has been announced by Calegari, Emerton, Gee and Mavrides.

We consider certain [Formula: see text]-ordinary non-CM Hida families with full residual Galois representation and give mild conditions under which every arithmetic point in these families is locally indecomposable when [Formula: see text]. The proof uses methods from deformation theory and mostly works for any odd prime [Formula: see text], but ultimately relies on the existence of a weight [Formula: see text] form in an auxiliary family which is available only for [Formula: see text]. We end by giving several non-trivial examples of [Formula: see text]-ordinary non-CM locally indecomposable modular forms of small level with full residual Galois representation.

Zero-shot learning (ZSL) endeavors to extend knowledge to novel classes by capitalizing on the semantic overlap across different categories. Contemporary ZSL approaches concentrate on isolating image-specific local features pertinent to attributes and using these to align class semantic vectors. However, existing methods often overlook the integration of global and local representation, a synthesis that could significantly enhance zero-shot recognition accuracy. This paper introduces an innovative ZSL technique, termed ZSL based on the fusion of global representation and local representation (ZGLR). Our approach incorporates a Transformer encoder constructed upon attribute prototypes to extract attribute-level features, which are then mapped to local representation to align with class semantic vectors. Concurrently, we introduce a discriminative semantic-visual mapping network that embeds class semantic vectors into the visual domain, thereby aligning global representation. During training, both global and local representation are optimized in tandem, while in the testing phase, the outcomes from both representational levels are consolidated to bolster classification precision. Practical results from three benchmark ZSL datasets demonstrate the superiority of our put-forward ZGLR solution.