J. S. Milne
We prove that Grothendieck's Hodge standard conjecture holds for abelian varieties in arbitrary characteristic if the Hodge conjecture holds for complex abelian varieties of CM-type. For abelian varieties with no exotic algebraic classes, we prove the Hodge standard conjecture unconditionally.
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In this short note, we will give the key point of the section conjecture of Grothendieck, that is reformulated by monodromy actions. Here, we will also give the result of the section conjecture for algebraic schemes over a number field.
Using Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck-Katz $p$-Curvature Conjecture for certain locally symmetric varieties, including the moduli space of abelian varieties ${\cal A}_g$ when $g > 1.$
The Absolute Grothendieck Conjecture is false for Fargues-Fontaine Curves
本文证明绝对格罗滕迪克猜想对法尔格-方丹曲线不成立，给出该猜想在相关对象上的反例。
In this paper, we prove a prime-to-p version of Grothendieck's anabelian conjecture for hyperbolic curves over finite fields of characteristic p>0, whose original (full profinite) version was proved by Tamagawa in the affine case and by Mochizuki in the proper case.
Grothendieck gave two forms of his "main conjecture of anabelian geometry", i.e. the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck's conjecture (equivalent in the case of curves) and prove that Grothendieck's statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If $X$ is a DM stack over a field $k\subseteq\mathbb{C}$ finitely generated over $\mathbb{Q}$, we prove that whethe…
We prove that a certain class of open homomorphisms between Galois groups of function fields of curves over finite fields arise from embeddings between the function fields.
We discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham-Betti realization of algebraic varieties over number fields, of the classical conjectures of Hodge and Tate. These results give new evidence towards the conjectures of Grothendieck and Kontsevich-Zagier concerning transcendence properties of the torsors of periods of varieties over number fields. We notably establish that the Grothendieck period conjecture holds in degree 1 for products of curves, of abelian varieties, and of K3 surfaces, and that it holds in degre…
Using the identification of sections of the Galois group of the ground field into the arithmetic fundamental group with neutral fiber functors of the category of finite connections, we define the "packets" in Grothendieck's section conjecture and show their properties predicted by him.
The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved earlier by the author.
This paper is around the topics I discussed in the lecture I gave at the Isaac Newton Institute in Cambridge, July 2009, in the Introductory Workshop. This paper can be read as a companion to my paper [Saï di], where detailed proofs can be found.
In this paper we prove a refined version of Uchida's theorem on isomorphisms between absolute Galois groups of global fields in positive characteristics, where one "ignores" the information provided by a "small" set of primes.
Grothendieck-Katz conjecture
本文证明仅含代数解的线性微分方程在模p^k下m-曲率为零（除有限素数外），从而以更强假设重新表述了Grothendieck-Katz猜想。
For a given arithmetic scheme, in this paper we will introduce and discuss the monodromy action on a universal cover of the étale fundamental group and the monodromy action on an \emph{sp}-completion constructed by the graph functor, respectively; then by these results we will give a proof of the section conjecture of Grothendieck for arithmetic schemes.
In this paper we prove a refined version of a theorem by Tamagawa and Mochizuki on isomorphisms between (tame) arithmetic fundamental groups of hyperbolic curves over finite fields, where one "ignores" the information provided by a "small" set of primes.
Proving the Grothendieck--Teichmüller Conjecture for Profinite Spaces & The Galois Grothendieck Path Integral
本文在有限空间情形下证明格罗滕迪克-泰希缪勒猜想，通过推广路径积分构造新算术不变量，并提出立方套娃算法编码GT与Gal路径，揭示二者同构与对称性对应。
On Grothendieck's section conjecture for curves of index $1$
本文证明了具有非循环 $p$-群忠实作用的双曲曲线存在满足 Grothendieck 截面猜想的指标为 $1$ 的扭曲形式，并证明了在 $\mathbb{Q}$ 上有限生成域上的双曲曲线经有限扩张与有限平展覆盖后可满足该猜想。
Let $k$ be a base field and $G$ be an algebraic group over $k$. J.-P. Serre defined $G$ to be special if every $G$-torsor $T \to X$ is locally trivial in the Zariski topology for every reduced algebraic variety $X$ defined over $k$. In recent papers an a priori weaker condition is used: $G$ is called special if every $G$-torsor $T \to Spec(K)$ is split for every field $K$ containing $k$. We show that these two definitions are equivalent. We also generalize this fact and propose a strengthened version of the Grothendieck-Serre conjecture based on the notion of essential dimension.
We show that Grothendieck's standard conjectures are implied by either of two other motivic conjectures: (a) by that of the existence of the motivic t-structure, and (b) by (a weak form of) Suslin's Lawson homology conjecture.
As already noted by Niels Borne and Michel Emsalem, there is a natural generalization of the section conjecture for proper orbicurves. Combined with the reformulation by Niels Borne and Angelo Vistoli of the conjecture in terms of the étale fundamental gerbe, this suggests an even stronger conjecture for orbicurves, asking an equivalence of categories instead of a mere bijection. We prove that the three versions of the conjecture are in fact equivalent, and that "injectivity" (i.e. full faithfulness) holds in the case of orbicurves. As a byproduct, we obtain a new proof of the fact that the se…
Using morphic cohomology, we produce a sequence of conjectures, called morphic conjectures, which terminates at the Grothendieck standard conjecture A. A refinement of Hodge structures is given, and with the assumption of morphic conjectures, we prove a Hodge index theorem. We answer a question of Friedlander and Lawson by assuming the Grothendieck standard conjecture B and prove that the topological filtration from morphic cohomology is equal to the Grothendieck arithmetic filtration for some cases.
Functional Transcendence of Periods and the Geometric André--Grothendieck Period Conjecture
本文通过证明代数簇族中代数形式积分的函数超越性，给出混合 Hodge 结构变分的 Ax–Schanuel 结果，并据此建立几何形式的 André–Grothendieck 周期猜想。
We prove here some supplementary statements that appeared without proof in I. Panin, A. Stavrova, N. Vavilov, On Grothendieck--Serre's conjecture concerning principal $G$-bundles over reductive group schemes:I, arXiv:0905.1418
Grothendieck's standard conjecture of Lefschetz type has two main forms: the weak form $C$ and the strong form $B$. The weak form is known for varieties over finite fields as a consequence of the proof of the Weil conjectures. This suggests that the strong form of the conjecture in the same setting may be the most accessible of the standard conjectures. Here, as an advertisement for the conjecture, we explain some of its remarkable consequences.
The geometrically m-step solvable Grothendieck conjecture for genus 0 curves over finitely generated fields
研究远阿贝尔几何中的几何 m 步可解 Grothendieck 猜想，证明任意特征素域的有限生成扩域上零亏格曲线满足几何 3 步可解情形。
The unramified Grothendieck--Serre conjecture
本文证明非分歧正则局部环上约化群概形的一般平凡主齐性空间必为平凡，解决了该情形下的 Grothendieck--Serre 猜想。
We prove some new cases of the Grothendieck-Serre conjecture for classical groups. This is based on a new construction of the Gersten-Witt complex for Witt groups of Azumaya algebras with involution on regular semilocal rings, with explicit second residue maps; the complex is shown to be exact when the ring is of dimension $\le 2$ (or $\le 4$, with additional hypotheses on the algebra with involution). Note that we do not assume that the ring contains a field.
In three preprints [Pan2],[Pan3] and the present one we prove Grothendieck-Serre's conjecture concerning principal G-bundles over regular semi-local domains R containing a finite field (here G is a reductive group scheme). The present preprint contains main geometric presentation theorems which are necessary for that. The preprint [Pan2] contains reduction of the Grothendieck-Serre's conjecture to the case of a simple simply-connected group scheme. The preprint [Pan3] contains a proof of Grothendieck-Serre's conjecture for regular semi-local domains R containing a finite field. One of the main…
The Grothendieck-Katz Conjecture for privileged local systems
本文定义了光滑拟射影复簇上具有固定边界拟单值单性的特权局部系统，推广了Katz在射影直线上的物理刚性概念，并给出避免中间卷积的新证明，从而在高维上验证了p-曲率猜想。
In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and better structured proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field. The outline of the proof is the same as in [P1],[P2],[P3]. If the semi-local regular ring contains an infinite field, then the conjecture is proved in [FP]. Thus the conjecture is true for regular local rings containing a field. A proof of Grothendieck--Serre conjecture on principal bundles over a semi-local regular ring containing an arbitrary field is given in [Pan3]. That proof is heavily based on …
The metabelian Grothendieck conjecture for genus zero curves over finitely generated fields
研究有限生成域上零亏格曲线的亚交换格罗滕迪克猜想，通过证明两条双曲零亏格曲线在基域上同构（正特征下经弗罗贝尼乌斯扭转）的充要条件是其几何极大亚交换驯基本群在基域绝对伽罗瓦群上同构来得出结论。
The geometrically m-step solvable Grothendieck conjecture for affine hyperbolic curves over finitely generated fields
本文在非交换几何中针对素域上有限生成域上的仿射双曲曲线，通过先证有限域情形、再证Oda-Tamagawa准则的几何m步可解版本，最终证明了（弱与强双非交换的）几何m步可解Grothendieck猜想。
In three preprints [Pan1], [Pan3] and the present one we prove Grothendieck-Serre's conjecture concerning principal G-bundles over regular semi-local domains R containing a finite field (here $G$ is a reductive group scheme). The preprint [Pan1] contains main geometric presentation theorems which are necessary for that. The present preprint contains reduction of the Grothendieck--Serre's conjecture to the case of semi-simple simply-connected group schemes (see Theorem 1.0.1). The preprint [Pan3] contains a proof of that conjecture for regular semi-local domains R containing a finite field. The…
The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. The mixed characteristic case of the conjecture is widely open. We consider the following setup. Let $A$ be a mixed characteristic DVR, $G$ a reductive group scheme over $A$, $X$ an irreducible smooth projective $A$-scheme, $\mathcal G$ a principal $G$-bundle over $X$. Suppose $\mathcal G$ is generically trivial. We prove that in this case $\mathcal G$ is Zariski locally trivial. This result confirms the conjecture.
Constant case of the Grothendieck-Serre conjecture in mixed characteristic
本文证明了混合特征DVR下约化D-群概形的Grothendieck-Serre猜想，并在DVR背景下证明了Lindel-Ojanguren-Gabber几何表示引理的一个版本。
In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and better structured proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field. The outline of the proof is the same as in [P1],[P2],[P3]. If the semi-local regular ring contains an infinite field, then the conjecture is proved in [FP]. Thus the conjecture is true for regular local rings containing a field. The present paper is the one [Pan1] in that new series. Theorem 1.1 is one of the main result of the paper. It is also one of the key steps in the proof of the Grothendieck--S…
Grothendieck conjectured in the sixties that the even Kunneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's original conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's conjectures in the case of intersections of quadrics, line…
Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. If the regular local ring R contains an infinite field this result is proved in [FP]. Thus the conjecture is true for regular local rings containing a field.
We prove the Grothendieck-Serre conjecture for quasi-split reductive groups schemes. Our method involves reducing to the Borel subgroup in order to conclude the result from purity for tori and the structure theorem for unipotent radicals of parabolic subgroups in a reductive group.
Grothendieck's period conjecture for Kummer surfaces of self-product CM type
研究自积CM型椭圆曲线的库默表面，证明格罗滕迪克周期猜想成立，周期同构在动机周期torsor中稠密，且表面幂上的动机类为代数类。
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. Th…
We consider p-divisible groups (also called Barsotti-Tate groups) in characteristic p, their deformations, and we draw some conclusions. For such a group we can define its Newton polygon (abbreviated NP). This is invariant under isogeny. For an abelian variety (in characteristic p) the Newton polygon of its p-divisible group is ``symmetric''. In 1963 Manin conjectured that conversely any symmetric Newton polygon is ``algebroid''; i.e., it is the Newton polygon of an abelian variety. This conjecture was shown to be true and was proved with the help of the ``Honda-Serre-Tate theory''. We give an…
For a reductive group scheme $G$ over a semilocal Dedekind ring $R$ with total ring of fractions $K$, we prove that no nontrivial $G$-torsor trivializes over $K$. This generalizes a result of Nisnevich-Tits, who settled the case when $R$ is local. Their result, in turn, is a special case of a conjecture of Grothendieck-Serre that predicts the same over any regular local ring. With a patching technique and weak approximation in the style of Harder, we reduce to the case when $R$ is a complete discrete valuation ring. Afterwards, we consider Levi subgroups to reduce to the case when $G$ is semis…
Let R be a regular local ring, containing an infinite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R.
Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split.
Grothendieck--Serre for constant reductive group schemes
本文针对常数可约群方案在普雷弗基底上的格罗滕迪克-塞雷猜想进行了推广，解决了诺特情况下的猜想，并受最近的Cesnaviç文章启发，证明了若干版本的尼斯涅维奇猜想。
Let R be a semi-local Dedekind domain and let K be the field of fractions of R. Let G be a reductive semisimple simply connected R-group scheme such that every semisimple normal R-subgroup scheme of G contains a split R-torus G_m. We prove that the kernel of the map H^1_et(R,G)-> H^1_et(K,G) induced by the inclusion of R into K, is trivial. This result partially extends a theorem of Nisnevich.
Let R be a semi-local regular ring containing an infinite perfect field, and let K be the field of fractions of R. Let H be a simple algebraic group of type F_4 over R such that H_K is the automorphism group of a 27-dimensional Jordan algebra which is a first Tits construction. If char K is not 2, this means precisely that the f_3 invariant of H_K is trivial. We prove that if an H-torsor is rationally trivial, then it is trivial over R. This result is a particular case of the Grothendieck-Serre conjecture. It continues the recent series of papers by I. Panin, N.Vavilov and the authors, and com…
In our PH.D. thesis we have showed that the Generalized Grothendieck's Conjecture of Periods applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers. In this paper we investigate the Generalized Grothendieck's Conjecture of Periods in the case of 1-motives whose underlying semi-abelian variety is a non trivial extension of a product of elliptic curves by a torus. This will imply the introduction of elliptic in…
Let k be an infinite field. Let R be the semi-local ring of a finite family of closed points on a k-smooth affine irreducible variety, let K be the fraction field of R, and let G be a reductive simple simply connected R-group scheme isotropic over R. We prove that for any Noetherian k-algebra A, the map of etale cohomology sets H^1(A\otimes_k R,G)-> H^1(A\otimes_ k K,G), induced by the inclusion of R into K, has trivial kernel. This implies the Serre-Grothendieck conjecture for such groups G. The main theorem for A=k and some other results of the present paper are used significantly in arXiv:1…
In the present paper we investigate the question about the injectivity of the map F(R) --> F(K) induced by the canonical inclusion of a local regular ring of geometric type R to its field of fractions K for a homotopy invariant functor F with transfers satisfying some additional properties. As an application we get the original proof of Special Unitary Case of Grothendieck's conjecture about principal homogeneous spaces and some other interesting examples.
Making use of topological periodic cyclic homology, we extend Grothendieck's standard conjectures of type C and D (with respect to crystalline cohomology theory) from smooth projective schemes to smooth proper dg categories in the sense of Kontsevich. As a first application, we prove Grothendieck's original conjectures in the new cases of linear sections of determinantal varieties. As a second application, we prove Grothendieck's (generalized) conjectures in the new cases of "low-dimensional" orbifolds. Finally, as a third application, we establish a far-reaching noncommutative generalization …
Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of pointed sets H^1_{et}(R,G) \to H^1_{et}(K,G), induced by the inclusion of R into K, has a trivial kernel. Certain arguments used in the present preprint do not work if the ring R contains a characteristic zero field. In that case and, more generally, in the case when the regular local ring R contains an infinite field this result is…
A conjecture in Schanuel style for 1-motives
本文提出一个Schanuel式猜想，用于半椭圆指数函数，涵盖Weierstrass函数和Serre函数的值，证明了该猜想与Grothendieck-André周期猜想的关系。
The Grothendieck--Serre conjecture over valuation rings
本文证明了赋值环上的 Grothendieck-Serre 猜想，通过 flasque resolution、局部上同调归纳及代数化技术克服了非离散赋值环的困难，并证明了 Nisnevich 线性猜想变体。
Assume that R is a semi-local regular ring containing an infinite perfect field, or that R is a semi-local ring of several points on a smooth scheme over an infinite field. Let K be the field of fractions of R. Let H be a strongly inner adjoint simple algebraic group of type E_6 or E_7 over R, or any twisted form of one of the split groups of classical type O^+_{n,R}, n>=4; PGO_{n,R}, n>=4; PSp_{2n,R}, n>=2; PGL_{n,R}, n>=2. We prove that the kernel of the map H^1_{et}(R,H)-> H^1_{et}(K,H) induced by the inclusion of R into K is trivial. This continues the recent series of papers by the author…
We introduce the category of structures and interpretations which allows us to discuss some issues of Grothendieck's anabelian geometry in model-theory terms. Our main result is a formulation in terms of pure stability theory of a problem closely related to Grothendieck's section conjecture
Quantum K-theory is a K-theoretic version of quantum cohomology, which was recently defined by Y.-P. Lee. Based on a presentation for the quantum K-theory of the classical flag variety Fl_n, we define and study quantum Grothendieck polynomials. We conjecture that they represent Schubert classes (i.e., the natural basis elements) in the quantum K-theory of Fl_n, and present strong evidence for this conjecture. We describe an efficient algorithm which, if the conjecture is true, computes the quantum K-invariants of Gromov-Witten type for Fl_n. Two explicit constructions for quantum Grothendieck …
Algebraization Techniques and Rigid-Analytic Artin-Grothendieck Vanishing
本文证明一般诺特环上 rig-光滑代数的代数化，并建立非阿基米德几何中沿一个方向的部分代数化结果；由此解决 affinoid 代数的 Artin-Grothendieck 消没猜想剩余情形，在特征 0 得到刚性解析消没猜想的更强版本，在正特征得到较弱版本。
Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W\times_kU$ is trivial if it is trivial over the generic fiber of the projection $W\times_kU\to U$. We also simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is t…
We reformulate a conjecture of Beauville on algebraic cycles on an abelian variety in terms of certain compatibility and vanishings of some naturally defined filtrations on the Grothendieck group of the abelian variety.
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