In this dissertation, we discuss mainly the corresponding geometric and representation theoretic aspects of relative $p$-adic Hodge theory and $p$-adic motives. To be more precise, we study the corresponding analytic geometry of the corresponding spaces over and attached to period rings in the relative $p$-adic Hodge theory, including derived topological de Rham complexes and derived topological logarithmic de Rham complexes after Bhatt, Gabber, Guo and Illusie which is in some sense equivalent to the derived prismatic cohomology of Bhatt-Scholze as shown in the work of Li-Liu, $\mathcal{O}\mathbb{B}_\mathrm{dR}$-sheaves after Scholze, $φ$-$\widetilde{C}_X$-sheaves and relative-$B$-pairs after Kedlaya-Liu, multidimensional rings after Carter-Kedlaya-Zábrádi and Pal-Zábrádi and many other possible general universal motivic rings or sheaves. Many contexts are expected to be sheafified, such as over Scholze's pro-étale sites of the considered analytic spaces by using perfectoids or the quasisyntomic sites by using quasiregular semiperfectoids as in the work of Bhatt-Morrow-Scholze and Bhatt-Scholze.
We discuss the deep relationship between $\infty$-categorical functional analysis and the anticipated theory of $p$-adic motives. The motivation fundamentally comes from applications essentially in arithmetics from very broad perspectives. The $\infty$-categorical homotopical aspects we considered here come mainly from Bambozzi-Ben-Bassat-Kremnizer, Ben-Bassat-Mukherjee, Bambozzi-Kremnizer, Clausen-Scholze and Kelly-Kremnizer-Mukherjee, based on deep robust foundation of $\infty$-categorical solids and $\infty$-categorical ind-Banach or bornological modules over general Banach rings.
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 rea…
We associate canonical virtual motives to definable sets over a field of characteristic zero. We use this construction to show that very general p-adic integrals are canonically interpolated by motivic ones.
We use the $p$-divisible group attached to a 1-motive to generalize the conjugate $p$-adic uniformization of Iovita--Morrow--Zaharescu to arbitrary $p$-adic formal semi-abelian schemes or $p$-divisible groups over the ring of integers in a $p$-adic field. This mirrors a mixed Hodge theory construction of the inverse uniformization map for complex semi-abelian varieties. We also highlight the geometric structure of the target of the conjugate uniformization map, which is an étale cover of a negative Banach--Colmez space in the sense of Fargues--Scholze.
For a number field, we have a Tannaka category of mixed Tate motives at our disposal. We construct p-adic points of the associated Tannaka group by using p-adic Hodge theory. Extensions of two Tate objects yield functions on the Tannaka group, and we show that evaluation at our p-adic points is essentially given by the inverse of the Bloch-Kato exponential map.
We introduce the categories of geometric mixed Hodge modules on algebraic varieties over a subfield $k\subset\mathbb C$, and for a prime number $p$, the categories of geometric $p$-adic mixed Hodge modules on algebraic varieties over a subfield $k\subset K\subset\mathbb C_p$ of a p-adic field $K$. We then give an Hodge realization functor on the de derived category of relative motives over $k\subset\mathbb C$, and for a prime number $p$ a $p$-adic Hodge realization functor on the derived category of relative motives over $k\subset K\subset\mathbb C_p$.
In this article we introduce the notion of a quasi-compatible system of Galois representations. The quasi-compatibility condition is a slight relaxation of the classical compatibility condition in the sense of Serre. The main theorem that we prove is the following: Let $M$ be an abelian motive, in the sense of Yves André. Then the $\ell$-adic realisations of $M$ form a quasi-compatible system of Galois representations. (In fact, we actually prove something stronger. See theorem 5.1.) As an application, we deduce that the absolute rank of the $\ell$-adic monodromy groups of $M$ does not depend …
Assuming specific instances of two general conjectures in arithmetic algebraic geometry (bijectivity of $p$-adic regulator maps, injectivity of $p$-adic Abel-Jacobi maps), we prove several cases of the $p$-part of the Tamagawa number conjecture ($p$-TNC) of Bloch-Kato and Fontaine-Perrin-Riou for (homological) motives of modular forms of even weight $\geq4$ in analytic rank $1$. More precisely, we prove our results for a large class of newforms $f$ and prime numbers $p$ that are ordinary for $f$ and such that the weight of $f$ is congruent to $2$ modulo $2(p-1)$. Inspired by work of W. Zhang i…
In this paper, we prove that for suitably chosen Dwork motives, the local Galois representation arising from middle cohomology of fibers over a point with $p$-adic valuation $<0$ on the base is regular and ordinary. The result will be crucial in the forthcoming work Potential Automorphy for $GL_n$. The proof consists of the construction of a semistable blowup and a use of Hyodo-Kato's log crystalline cohomology theory.
This is my habilitation thesis. As the tradition wants, I tried to give an introduction of my field of research. I post it on the ArXiv with the hope it can be useful to young researchers looking for a short and friendly text on cohomologies of algebraic varieties, periods, algebraic cycles and motives. I might one day find the energy to expand these notes and maybe translate them in English. In the meantime, please feel free to ask questions. The first sections of this text propose an introduction to the theory, I tried to present points of view and examples which are not always stressed in t…
We construct motivic $\ell$-adic representations of $\GQ$ into exceptional groups of type $E_7,E_8$ and $G_2$ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and uses the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\FF_{\ell})$ are Galois groups over $\QQ$ for large enough primes $\ell$.
We show that the l-adic realization functor is conservative when restricted to the Chow motives of abelian type over a finite field. A weak version of this conservativity result extends to mixed motives of abelian type.
The purpose of this work is to generalize, in the context of 1-motives, the $p$-adic height pairings constructed by B. Mazur and J. Tate on abelian varieties. Following their approach, we define a global pairing between the rational points of a 1-motive and its dual. We also provide local pairings between zero-cycles and divisors on a curve, which is done by considering its Picard and Albanese 1-motives.
In the present article we investigate properties of the category of the integral Grothendieck-Chow motives over a field. We discuss the Krull-Schmidt principle for integral motives, provide a complete list of the generalized Severi-Brauer varieties with indecomposable integral motive, and exploit a relation between the category of motives of twisted flag varieties and integral $p$-adic representations of finite groups.
This paper is concerned with an interpretation of f-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields, number fields and number rings, we show that the two extant definitions of f-cohomology of mixed motives $M_η$ over F--one via ramification conditions on $\ell$-adic realizations, another one via the K-theory of proper regular models--both agree with motivic cohomology of $η_{!*} M_η[1]$. Here $η_{!*}$ is constructed by a limiting process in terms of intermediate…
We prove potential automorphy results for a single Galois representation $G_F \rightarrow GL_n(\overline{\mathbb{Q}}_l)$ where $F$ is a CM number field. The strategy is to use the $p,q$ switch trick and modify the Dwork motives employed in \cite{HSBT} to break self-duality of the motives, but not the Hodge-Tate weights. Another key result to prove is the ordinarity of certain $p$-adic representations, which follows from log geometry techniques. One input is the automorphy lifting theorem in \cite{tap}.
In this paper we prove that a pure, regular, totally odd, polarizable weakly compatible system of $l$-adic representations is potentially automorphic. The innovation is that we make no irreducibility assumption, but we make a purity assumption instead. For compatible systems coming from geometry, purity is often easier to check than irreducibility. We use Katz's theory of rigid local systems to construct many examples of motives to which our theorem applies. We also show that if $F$ is a CM or totally real field and if $π$ is a polarizable, regular algebraic, cuspidal automorphic representatio…
We will discuss $\infty$-categorical perverse $p$-adic differential equations over stacks. On one hand, we are going to study some $p$-adic analogous results of the Drinfeld's original lemma about the étale fundamental groups in the étale setting, in the context of $F$-isocrystals closely after Kedlaya and Kedlaya-Xu. We expect similar things could also be considered for diamonds after Scholze, in the context of Kedlaya-Liu's work namely the derived category of pseudocoherent Frobenius sheaves, which will induce some categorical form of Drinfeld's lemma for diamonds motivated by work of Carter…
The Grothendieck ring of varieties has well-known realization maps to, say, mixed Hodge structures or compactly supported $\ell$-adic cohomology. Zakharevich and\ Campbell have developed {a spectral refinement} of the Grothendieck ring of varieties. We develop a realization map to Voevodsky mixed motives, and this lifts the standard realizations of motives to this setting, at least over perfect fields which have resolution of singularities.
With representation-theoretic applications in mind, we construct a formalism of reduced motives with integral coefficients. These are motivic sheaves from which the higher motivic cohomology of the base scheme has been removed. We show that reduced stratified Tate motives satisfy favorable properties including weight and t-structures. We also prove that reduced motives on cellular (ind-)schemes unify various approaches to mixed sheaves in representation theory, such as Soergel-Wendt's semisimplified Hodge motives, Achar-Riche's complexes of parity sheaves, as well as Ho-Li's recent category of…
Let $G$ be a split semisimple algebraic group over a field and let $A^*$ be an oriented cohomology theory in the sense of Levine--Morel. We provide a uniform approach to the $A^*$-motives of geometrically cellular smooth projective $G$-varieties based on the Hopf algebra structure of $A^*(G)$. Using this approach we provide various applications to the structure of motives of twisted flag varieties.
We study the multiplicities of pure motives modulo numerical equivalence, which are defined as scalars comparing the tannakian trace with the ring-theoretic trace. Our general set-up is that of a rigid semi-simple tensor category such that End(1) is a field of characteristic 0. The main result is that, due to the existence of a Weil cohomology theory (to be defined appropriately in the general set-up), the multiplicities are integers. This property is sufficient for the rationality (and functional equation) of the zeta function of an (invertible) endomorphism. We also show that the classical e…
We establish the group-theoretic classification of Sato-Tate groups of self-dual motives of weight 3 with rational coefficients and Hodge numbers h^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motives that realize some of these Sato-Tate groups, and provide numerical evidence supporting equidistribution. One of these families arises in the middle cohomology of certain Calabi-Yau threefolds appearing in the Dwork quintic pencil; for motives in this family, our evidence suggests that the Sato-Tate group is always equal to the full unitary symplectic group USp(4).
This note explores the consequences of Koenigsmann's model theoretic argument from the proof of the birational p-adic section conjecture for curves in the context of higher dimensional varieties over p-adic local fields.
This paper is dedicated to the study of the kernel of the "compact motivization" functor $M_{k}^c:SH^c(k)\to DM^c(k)$ (i.e., we try to describe those compact objects of $SH(k)$ whose associated motives vanish. Moreover, we study the question when the $m$-connectivity of $M^c_{k}(E)$ ensures the $m$-connectivity of $E$ itself (with respect to the corresponding homotopy t-structures). We prove that the kernel of $M_{k}^c$ vanishes and the corresponding "connectivity detection" statement is also valid if and only if $k$ is a non-orderable field; this is an easy consequence of the corresponding re…
This paper is originally designed as a part of revision of the author's preprint math.AG/9908174 "P-adic Schwarzian triangle groups of Mumford type". Recently, Yves Andr'e pointed out a flaw in that preprint; more precisely, Proposition II there was not correct, and consequently, as he pointed out, some triangle groups are missing (but only in p=2,3,5). I apologize to the readers, and will release the other part soon by the original title. In this paper, we concentrate on the constructive part, but more systematically and in more general situation. More precisely, this paper presents a complet…
