Sunil Chebolu, Ján Mináč
In this paper we concentrate on the relations between the structure of small Galois groups, arithmetic of fields, Bloch-Kato conjecture, and Galois groups of maximal pro-$p$-quotients of absolute Galois groups.
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We prove a special case of the Bloch-Kato conjecture for adjoint motives associated to modular abelian surfaces.
Let $ρ$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $ρ$ satisfies an irreducibility condition, we prove the vanishing of the adjoint Bloch--Kato Selmer group attached to $ρ$. This generalizes previous work of the author and James Newton.
We study the adjoint Bloch--Kato Selmer groups attached to a classical point in the cuspidal eigenvariety associated with $\mathrm{GSp}_{2g}$. Our strategy is based on the study of families of Galois representations on the eigenvariety, which is inspired by the book of J. Bellaiche and G. Chenevier.
On the Bloch--Kato conjecture for the symmetric cube
本文针对1级模形式的对称三次方，证明了Iwasawa主猜想的一个包含关系及解析秩0下的Bloch-Kato猜想。
Adjoint Bloch--Kato Selmer groups of regular algebraic automorphic Galois representations
该研究证明了 CM 域上一般线性群正则代数自守表示相关伽罗瓦表示的伴随布洛赫 - 加藤塞尔默群消失，关键在于仅对 p 进伽罗瓦表示而非其模 p 剩余表示施加条件。
On the Beilinson-Bloch-Kato conjecture for polarized motives
本文研究极化动机的Beilinson-Bloch-Kato猜想，在共轭自对偶情形证明中心L值非零时Selmer群消失，并将自对偶解析秩零情形归结为内窥Rankin-Selberg猜想。
Bloch-Kato groups over perfectoid fields and Galois theory of $p$-adic periods
该研究将完美oid域上de Rham伽罗瓦表示的Bloch-Kato群结构与p进周期环B_dR⁺的伽罗瓦理论关联，解答了Coates和Greenberg的问题并推广相关结果，方法依赖Fargues-Fontaine曲线上向量丛的分类。
We use $p$-adic families of automorphic forms for an unitary group in three variables, containing some non-tempered forms constructed by Rogawski, to prove some cases of the Bloch-Kato conjectures.
We prove the Bloch--Kato conjecture for certain critical values of degree 8 $L$-functions associated to cusp forms on $\mathrm{GSp}_4 \times \mathrm{GL}_2$. We also construct a $p$-adic Eichler--Shimura isomorphism in Hida families for $\mathrm{GSp}_4$, relating $H^2$ of automorphic vector bundles with $\mathbf{D}_{\mathrm{cris}}$ of a subquotient of étale cohomology.
The aim of this paper is to prove inequalities towards instances of the Bloch-Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the $L$-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.
This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number for all odd primes (both ordinary and supersingular), using Kato's explicit reciprocity law for one dimensional Lubin-Tate formal group. I also study p-part of Shafarevich-Tate group for motives associated to Hecke characters, using Rubin's Main Conjecture in Iwasawa theory. Interestingly a congruence property between p-adic periods of elliptic curve and weig…
In this paper, we generalise the construction of the Bloch-Kato exponential map to complete discrete valuation fields of mixed characteristic (0,p) whose residue fields have a finite p-basis. As an application we prove an explicit reciprocity law, extending a result of Cherbonnier and Colmez in the classical case. This result relies on the calculation of the Galois cohomology of a p-adic representation V in terms of its (phi,G)-module.
We investigate the relations in Galois groups of maximal p-extensions of fields, the structure of their natural filtrations, and their relationship with the Bloch-Kato conjecture proved by Rost and Voevodsky with Weibel's patch. Our main focus is on the third degree, but we provide examples for all degrees.
On the Bloch-Kato conjecture for GSp(4)
研究通过欧拉系统证明 reciprocity 法律，得出Iwasawa主猜想和Bloch-Kato猜想在解析秩0时的包含关系。
A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic $\mathbb Z_p$-extensions at good ordinary primes $p$. We extend Greenberg's result to more general $p$-adic Galois representations, including a large subclass of those attached to $p$-ordinary modular forms of level $Γ_0(N)$ with $p\nmid N$.
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanis…
The purpose of this article is to give formulas for Bloch-Kato's exponential map and its dual for an absolutely crystalline p-adic representation V, in terms of the (phi,Gamma)-module associated to that representation. As a corollary of these computations, we can give a very simple (and slightly improved) description of Perrin-Riou's exponential map (which interpolates Bloch-Kato's exponentials for V(k)). This new description directly implies Perrin-Riou's reciprocity formula.
We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.
Let f be a newform of weight 2k-2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypothesis if a prime divides the algebraic part of L(k,f), then the prime divides the order of the Selmer group associated to f. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel eigenform that is not a Saito-Kurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a non-trivial p-torsion element in the Selmer gro…
In this article we construct a $p$-adic three dimensional Eigenvariety for the group $U(2,1)(E)$, where $E$ is a quadratic imaginary field and $p$ is inert in $E$. The Eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta-Iovita-Pilloni by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch-Kato conjecture for some Galois characters of $E$, exte…
Following Ribet's seminal 1976 paper there have been many results employing congruences between stable cuspforms and lifted forms to construct non-split extensions of Galois representations. We show how this strategy can be extended to construct elements in the Bloch-Kato Selmer groups of +/--Asai (or tensor induction) representations associated to Bianchi modular forms. We prove, in particular, how the Galois representation associated to a suitable low weight Siegel modular form produces elements in the Selmer group for exactly the Asai representation (+ or -) that is critical in the sense of…
In this paper we define a $p$-adic analogue of the Borel regulator for the $K$-theory of $p$-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this $p$-adic regulator to the Bloch-Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups.
This is a contribution to the ICM 2002. We explain the relation between the (equivariant) Bloch-Kato conjecture for special values of L-functions and the Main Conjecture of (non-abelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how "twisting" in the non-abelian case would allow to reduce the general conjecture to the case of number fields. This is one the main motivations for a non-abelian Main Conjecture.
The depth-weight compatibility on the motivic fundamental Lie algebra and the Bloch-Kato conjecture for modular forms
本文证明动机基本李代数上的深度-权重相容性可推出与全水平 Hecke 本征尖形式相关的 $p$-进 Galois 表示的 Bloch-Kato 猜想，该猜想关联 $L$-函数秩与 Selmer 群秩。
Theta cycles and the Beilinson--Bloch--Kato conjectures
本文研究了Theta循环与Beilinson-Bloch-Kato猜想之间的关系，通过构造Selmer群中的标准类，并探讨其与特殊循环及Heegner点的联系。
For an elliptic curve $E$ over $\mathbb{Q}$, putting $K=\mathbb{Q}(E[p])$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group $\mathrm{Cl}_K$ of $K$ as a $\mathrm{Gal}(K/\mathbb{Q})$-module. More precisely, for any $j$ with $1\leqslant j \leqslant p-2$, we give a condition that $\mathrm{Cl}_K\otimes \mathbb{F}_p$ has the symmetric power $\mathrm{Sym}^j E[p]$ of $E[p]$ as its quotient $\mathrm{Gal}(K/\mathbb{Q})$-module, in terms of Bloch-Kato's Tate-Shafarevich group of $\mathrm{Sym}^j V_p E$. Here $V_p E$ denotes the rational $p$-adic Tate module of …
A Bloch-Kato pro-p group G is a pro-p group with the property that the F_p-cohomology ring of every closed subgroup of G is quadratic. It is shown that either such a pro-p group G contains no closed free pro-p groups of infinite rank, or there exists an orientation $θ\colon G\rightarrow \Z_p^\times$ such that G is theta-abelian. In case that G is also finitely generated, this implies that G is powerful, p-adic analytic with d(G)=cd(G), and its \F_p-cohomology ring is an exterior algebra. These results will be obtained by studying locally powerful groups (see Theorem A). There are certain Galoi…
Partial Bloch--Kato Selmer groups of $B$-pairs as delta functors
该研究将丁引入的1阶部分Selmer群扩展到更高上同调次数，证明其在部分de Rham正B-对子范畴上构成满足特定性质的上同调δ函子。
We study the Eisenstein ideal for modular forms of even weight $k>2$ and prime level $N$. We pay special attention to the phenomenon of $\mathit{extra \ reducibility}$: the Eisenstein ideal is strictly larger than the ideal cutting out reducible Galois representations. We prove a modularity theorem for these extra reducible representations. As consequences, we relate the derivative of a Mazur-Tate $L$-function to the rank of the Hecke algebra, generalizing a theorem of Merel, and give a new proof of a special case of an equivariant main conjecture of Kato. In the second half of the paper, we r…
Asai-Flach classes, p-adic L-functions and the Bloch-Kato conjecture for GO(4)
本文针对二次域上p常Hilbert模形式的Asai L函数，证明临界值的Bloch-Kato猜想及Iwasawa主猜想的单侧包含，并建立Hida族的p进Eichler-Shimura比较同构。
On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers
本文研究模形式在反 cyclotomic Z_p 塔中的 Bloch-Kato 柯西群结构，证明在广义 Heegner 假设下，柯西群的 Pontryagin 对偶为自由模，并使 Bloch-Kato-Shafarevich-Tate 群消失。
For certain algebraic Hecke characters chi of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL_2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-1}. This uses the work of R. Taylor et al. on a…
We construct a variant of Karoubi's relative Chern character for smooth, separated schemes over the ring of integers in a p-adic field and prove a comparison with the rigid syntomic regulator. For smooth projective schemes we further relate the relative Chern character to the etale p-adic regulator via the Bloch-Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers and generalizes it to all smooth projective schemes as above.
A difference formula of $p$-adic height pairings via the Bloch-Kato logarithm map
本文研究$p$-adic高度配对对局部分裂的依赖性，通过Bloch-Kato对数给出差分公式，并证明在特定条件下至少一个分圆高度配对非平凡。
We prove the "divisible case" of the Milnor-Bloch-Kato conjecture (which is the first step of Voevodsky's proof of this conjecture for arbitrary prime l) in a rather clear and elementary way. Assuming this conjecture, we construct a 6-term exact sequence of Galois cohomology with cyclotomic coefficients for any finite extension of fields whose Galois group has an exact quadruple of permutational representations over it. Examples include cyclic groups, dihedral groups, the biquadratic group Z/2\times Z/2, and the symmetric group S_4. Several exact sequences conjectured by Bloch-Kato, Merkurjev-…
2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves II
该研究提出精细化方法证明带单有理魏尔斯特拉斯点的超椭圆曲线的Chabauty--Coleman--Kim集合有限性，给出模2 Bloch--Kato塞尔默群的2进与实进塞尔默条件计算方法，验证了多数LMFDB中秩2亏格2曲线的有限性，还得到超椭圆曲线雅可比簇二次下降的域论描述。
2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves I
本文提出了一种通过 Galois 协同论方法确定超椭圆曲线 Jacobian 的 Bloch-Kato Selmer 组秩的方法，并利用该方法回答了关于有理点的问题。
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
Let $F$ be a cuspidal eigenform of even weight and trivial nebentypus, let $p$ be a prime not dividing the level of $F$, and let $ρ_F$ be the $p$-adic Galois representation attached to $F$. Assume that the $L$-function attached to the symmetric cube of $ρ_F$ vanishes to odd order at its central point. Then under some mild hypotheses, and conditional on certain consequences of Arthur's conjectures, we construct a nontrivial element in the Bloch--Kato Selmer group of an appropriate twist of the symmetric cube of $ρ_F$, in accordance with the Bloch--Kato conjectures. Our technique is based on the…
Refined effective bounds for Bloch-Kato Selmer groups associated to hyperelliptic curves
本文推广 Dogra 的显式 $2$-下降方法，在无有理 Weierstrass 点及 $2$ 处好通常约化条件下给出 Bloch–Kato Selmer 群维数的有效界，并据此建立深度 $2$ Chabauty–Kim 集有限性的新判定准则。
The Bloch--Kato conjecture, decomposing fields, and generating cohomology in degree one
本文研究了Bloch-Kato猜想下生成一维同调的群结构，探讨了有限群的性质，并定义了分解域以改进该猜想。
Let k be a positive integer divisible by 4, l>k a prime, and f an elliptic cuspidal eigenform of weight k-1, level 4, and non-trivial character. Let ρ_f be the l-adic Galois representation attached to f. In this paper we provide evidence for the Bloch-Kato conjecture for a twist of the adjoint motif of ρ_f in the following way. Let L(f,s) denote the symmetric square L-function of f. We prove that (under certain conditions) the l-adic valuation of the algebraic part of L(f, k) is no greater than the l-adic valuation of the order of S, where S is (the Pontryagin dual of) the Selmer group attache…
On the Bloch-Kato conjecture for some four-dimensional symplectic Galois representations
本文证明了对GSp_4自守表示对应的四维辛Galois表示，当中心L值非零时Selmer群消失，并利用提升同余构造上同调类，同时处理秩一情形，推进了Bloch-Kato猜想。
Let $p$ be a prime. A pro-$p$ group $G$ is said to be 1-smooth if it can be endowed with a homomorphism of pro-$p$ groups $G\to1+p\mathbb{Z}_p$ satisfying a formal version of Hilbert 90. By Kummer theory, maximal pro-$p$ Galois groups of fields containing a root of 1 of order $p$, together with the cyclotomic character, are 1-smooth. We prove that a finitely generated $p$-adic analytic pro-$p$ group is 1-smooth if, and only if, it occurs as the maximal pro-$p$ Galois group of a field containing a root of 1 of order $p$. This gives a positive answer to De Clerq-Florence's "Smoothness Conjecture…
In this paper we develop a new method to study Iwasawa theory and Eisenstein families for unitary groups $\mathrm{U}(r,s)$ of general signature over a totally real field $F$. As a consequence we prove that for a motive corresponding to a regular algebraic cuspidal automorphic representation $π$ on $\mathrm{U}(r,s)_{/F}$ which is ordinary at $p$, twisted by a Hecke character, if its Selmer group has rank $0$, then the corresponding central $L$-value is nonzero. This generalizes a result of Skinner-Urban in their ICM 2006 report in the special case when $F=\mathbb{Q}$ and the motive is conjugate…
We show an Iwasawa functional equation for a two dimensional $p$-adic representation of the absolute Galois group of $\mathbf{Q}_p$. This allows us to complete Nakamura's proof of Kato's local $ε$-conjecture in dimension $2$.
Let K be an imaginary quadratic field of discriminant -D_K<0. We introduce a notion of an adelic Maass space S_{k, -k/2}^M for automorphic forms on the quasi-split unitary group U(2,2) associated with K and prove that it is stable under the action of all Hecke operators. When D_K is prime we obtain a Hecke-equivariant descent from S_{k,-k/2}^M to the space of elliptic cusp forms S_{k-1}(D_K, χ_K), where χ_K is the quadratic character of K. For a given ϕ\in S_{k-1}(D_K, χ_K), a prime l >k, we then construct (mod l) congruences between the Maass form corresponding to ϕand hermitian modular forms…
For a prime number $p$, we show that if two certain canonical finite quotients of a finitely generated Bloch-Kato pro-$p$ group $G$ coincide, then $G$ has a very simple structure, i.e., $G$ is a $p$-adic analytic pro-$p$ group. This result has a remarkable Galois-theoretic consequence: if the two corresponding canonical finite extensions $F^{(3)}/F$ and $F^{\{3\}}/F$ of a field $F$ -- with $F$ containing a primitive $p$-th root of unity -- coincide, then $F$ is $p$-rigid. The proof relies only on group-theoretic tools, and on certain properties of Bloch-Kato pro-$p$ groups.
The 2-part of the Bloch-Kato conjecture, and indivisibility results, for $K_2$ of some elliptic curves
本文研究特定整数u的2-部分Bloch-Kato猜想，分析Selmer群的2- torsion，并构造K₂的不可除元素，通过Merkur'ev-Suslin同构证明其不可除性，同时探讨2-可除性界及数值验证。
The conjectures of Deligne, Be\uılinson, and Bloch-Kato assert that there should be relations between the arithmetic of algebro-geometric objects and the special values of their $L$-functions. We make a numerical study for symmetric power $L$-functions of elliptic curves, obtaining data about the validity of their functional equations, frequency of vanishing of central values, and divisibility of Bloch-Kato quotients.
For smooth linear group schemes over $\bbZ$ we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori the cohomological and the motivic Tamagawa measures coincide, which reproves the Bloch-Kato conjecture for motives associated to tori.
Let $E$ be an elliptic curve over $\mathbb{Q}$ and $A$ be another elliptic curve over a real quadratic number field. We construct a $\mathbb{Q}$-motive of rank $8$, together with a distinguished class in the associated Bloch-Kato Selmer group, using Hirzebruch-Zagier cycles, that is, graphs of Hirzebruch-Zagier morphisms. We show that, under certain assumptions on $E$ and $A$, the non-vanishing of the central critical value of the (twisted) triple product $L$-function attached to $(E,A)$ implies that the dimension of the associated Bloch-Kato Selmer group of the motive is $0$; and the non-vani…
The aim of this article is to generalize Kato's (commutative) p-adic local epsilon-conjecture [Ka93b] for families of (phi,Gamma)-modules over the Robba ring. In particular, we prove the generalized local epsilon-conjecture for rank one (phi,Gamma)-modules, which is a generalization of Kato's theorem [Ka93b] for rank one Galois representations. The key ingredients are the recent results of Kedlaya-Pottharst-Xiao [KPX12] on the finiteness of cohomology of (phi,Gamma)-modules and the theory of Bloch-Kato's exponential map for (phi,Gamma)-modules developed in [Na13].
Profinite groups with a cyclotomic $p$-orientation are introduced and studied. The special interest in this class of groups arises from the fact that any absolute Galois group $G_{K}$ of a field $K$ is indeed a profinite group with a cyclotomic $p$-orientation $θ_{K,p}\colon G_{K}\to\mathbb{Z}_p^\times$ which is even Bloch-Kato. The same is true for its maximal pro-$p$ quotient $G_{K}(p)$ provided the field $K$ contains a primitive $p^{th}$-root of unity. The class of cyclotomically $p$-oriented profinite groups (resp. pro-$p$ groups) which are Bloch-Kato is closed with respect to inverse limi…
The aim of this article is to study Bloch-Kato's exponential map and Perrin-Riou's "big" exponential map purely in terms of (ϕ,Γ)-modules over the Robba ring. We first generalize the definition of Bloch-Kato's exponential map for all the (ϕ,Γ)-modules without using Fontaine's rings B_{cris}, B_{dR} of p-adic periods and then we generalize the construction of Perrin-Riou's "big" exponential map for all the de Rham (ϕ,Γ)-modules and prove that this map interpolates our Bloch-Kato's exponential map and the dual exponential map. Finally, we prove a theorem concerning to the determinant of our "big…
Assuming the Bloch-Kato Conjecture, we determine precise conditions under which Hilbert 90 is valid for Milnor k-theory and Galois cohomology. In particular, Hilbert 90 holds for degree n when the cohomological dimension of the Galois group of the maximal p-extension of F is at most n.
Extended differential symbol and the Kato homology groups
该研究基于前期工作，构造Bloch-Gabber-Kato定理微分符号的类似物，建立域上代数簇高阶Chow群与第0个Kato同调群的对应，刻画了算术域上该同调群的结构并以互反层表述结果。
The purpose of this article is to give a proof of the $C_{EP,F}(V)$ conjecture for some semi-stable representations and of the $δ_{\Zp}(V)$ conjecture for some crystalline representations. There are two major ingredients: first, the computation of Tamagawa numbers using a variation on results of Fontaine-Laffaille and of Bloch-Kato, and second some continuity results for Bloch-Kato's exponential which were proved by Perrin-Riou.
This paper proves local-global principles for Galois cohomology groups over function fields $F$ of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for $H^n(F, Z/mZ(n-1))$, for all $n>1$. This is motivated by work of Kato and others, where such principles were shown in related cases for $n=3$. Using our results in combination with cohomological invariants, we obtain local-global principles for torsors and related algebraic structures over $F$. Our arguments rely on ideas from patching as well as the Bloch-Kato conjecture.
Deformations of reducible Galois representations with large Selmer $p$-rank
本文推广 Hamblen–Ramakrishna 形变提升技术，对任意 $p\geq5$ 构造与模形式相关的 Galois 表示，证明其 Bloch-Kato Selmer 群可具有任意大的 $p$-秩。
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