Bloch-Kato Conjecture Research Articles

Newform Eisenstein congruences of local origin

We give a general conjecture concerning the existence of Eisenstein congruences between weight k≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 3$$\\end{document} newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character χ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\chi $$\\end{document} of conductor N. The special cases M=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M=1$$\\end{document} and M=p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M=p$$\\end{document} prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof.

Refined unramified cohomology of schemes

We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch–Ogus, Colliot-Thélène–Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch–Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel–Jacobi invariant has coniveau $1$. This establishes a torsion version of a conjecture of Jannsen originally formulated $\otimes \mathbb {Q}$. We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel–Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove $\ell$-adic analogues of these results over any field $k$ which contains all $\ell$-power roots of unity.

THE DIAGONAL CYCLE EULER SYSTEM FOR

Abstract We construct an anticyclotomic Euler system for the Rankin–Selberg convolutions of two modular forms, using p-adic families of generalised Gross–Kudla–Schoen diagonal cycles. As applications of this construction, we prove new results on the Bloch–Kato conjecture in analytic ranks zero and one, and a divisibility towards an Iwasawa main conjecture.

From sum of two squares to arithmetic Siegel–Weil formulas

The main goal of this expository article is to survey recent progress on the arithmetic Siegel–Weil formula and its applications. We begin with the classical sum of two squares problem and put it in the context of the Siegel–Weil formula. We then motivate the geometric and arithmetic Siegel–Weil formula using the classical example of the product of modular curves. After explaining the recent result on the arithmetic Siegel–Weil formula for Shimura varieties of arbitrary dimension, we discuss some aspects of the proof and its application to the arithmetic inner product formula and the Beilinson–Bloch conjecture. Rather than being intended as a complete survey of this vast field, this article focuses more on examples and background to provide easier access to several recent works by the author with W. Zhang and Y. Liu.

Arithmetic level raising on triple product of Shimura curves and Gross–Schoen Diagonal cycles, I : Ramified case

In this article we study the Gross-Schoen diagonal cycle on a triple product of Shimura curves at a place of bad reduction. We relate the image of the diagonal cycle under the Abel-Jacobi map to certain period integral that governs the central critical value of the Garrett-Rankin type triple product $L$-function via level raising congruences. As an application we prove certain rank $0$ case of the Bloch-Kato conjecture for the symmetric cube motive of a weight $2$ modular form.

On the anticyclotomic Iwasawa main conjecture for Hilbert modular forms of parallel weights

In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove an one-sided divisibility result toward the Iwasawa main conjecture in this setting. The proof relies on the first and second reciprocity laws relating theta elements to Heegner point Euler systems on Shimura curves. As a by-product we also prove a result towards the rank 0 case of certain Bloch–Kato conjecture and a parity conjecture.

On Siegel eigenvarieties at Saito–Kurokawa points

We study the geometry of the $p$-adic Siegel eigenvariety $\mathcal{E}$ of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal new eigenform of $\mathrm{S}_{2k-2}(\Gamma_0(N))$ ordinary at a prime $p\nmid N$ with sign $\epsilon_f=-1$ and write $\alpha$ for the $p$-adic unit root of the Hecke polynomial of $f$ at $p$. Let $\pi_\alpha$ be the semi-ordinary $p$-stabilization of the Saito-Kurokawa lift of the cusp form $f$ to $\mathrm{GSp}(4)$ of weight $(k,k)$ and paramodular tame level. Under the assumption that the dimension of the Selmer group $H^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))$ attached to $f$ is at most one and some mild assumptions on the automorphic representation attached to $f$, we show that $\mathcal{E}$ is smooth at the point corresponding to $\pi_\alpha$, and that the irreducible component of $\mathcal{E}$ specializing to $\pi_\alpha$ is not globally endoscopic. Finally we give an application to the Bloch-Kato conjecture, by proving under some mild assumptions that the smoothness failure of $\mathcal{E}$ at $\pi_\alpha$ yields that $\dim H^1_{f,\mathrm{unr}}(\mathbb{Q},\rho_f(k-1))\geq 2$.

The universal p-adic Gross–Zagier formula

Let \(\mathrm {G}\) be the group \( (\mathrm {GL}_{2}\times \mathrm{GU}(1))/\mathrm {GL}_{1}\) over a totally real field F, and let \(\mathscr {X}\) be a Hida family for \(\mathrm {G}\). Revisiting a construction of Howard and Fouquet, we construct an explicit section \(\mathscr {P}\) of a sheaf of Selmer groups over \(\mathscr {X}\). We show, answering a question of Howard, that \(\mathscr {P}\) is a universal Heegner class, in the sense that it interpolates geometrically defined Heegner classes at all the relevant classical points of \(\mathscr {X}\). We also propose a ‘Bertolini–Darmon’ conjecture for the leading term of \(\mathscr {P}\) at classical points. We then prove that the p-adic height of \(\mathscr {P}\) is given by the cyclotomic derivative of a p-adic L-function. This formula over \(\mathscr {X}\) (which is an identity of functionals on some universal ordinary automorphic representations) specialises at classical points to all the Gross–Zagier formulas for \(\mathrm {G}\) that may be expected from representation-theoretic considerations. Combined with a result of Fouquet, the formula implies the p-adic analogue of the Beilinson–Bloch–Kato conjecture in analytic rank one, for the selfdual motives attached to Hilbert modular forms and their twists by CM Hecke characters. It also implies one half of the first example of a non-abelian Iwasawa main conjecture for derivatives, in \(2[F:\mathbf {Q}]\) variables. Other applications include two different generic non-vanishing results for Heegner classes and p-adic heights.

On the Bloch–Kato conjecture for the symmetric cube

We prove one inclusion in the Iwasawa main conjecture, and the Bloch–Kato conjecture in analytic rank 0, for the symmetric cube of a level 1 modular form.

Integral period relations and congruences

Under relatively mild and natural conditions, we establish an integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the {\it congruence number} controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch-Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modular form, the cotangent complex of the corresponding Hecke algebra and the archimedean regulator attached to some Beilinson-Flach element.

The Eisenstein ideal for weight $k$ and a Bloch–Kato conjecture for tame families

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 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 recall Kato’s formulation of this main conjecture in the case of a family of motives given by twists by characters of conductor N and p -power order and its relation to other formulations of the equivariant main conjecture.

Trace and Künneth formulas for singularity categories and applications

We present an $\ell$-adic trace formula for saturated and admissible dg-categories over a base monoidal differential graded (dg)-category. Moreover, we prove Künneth formulas for dg-category of singularities and for inertia-invariant vanishing cycles. As an application, we prove a categorical version of Bloch's conductor conjecture (originally stated by Spencer Bloch in 1985), under the additional hypothesis that the monodromy action of the inertia group is unipotent.

On the Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives

In this article, we study the Beilinson–Bloch–Kato conjecture for motives associated to Rankin–Selberg products 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 a certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanishing of the central critical first derivative of the Rankin–Selberg L-function, then the Bloch–Kato Selmer group is of rank one.

$1$-smooth pro-$p$ groups and Bloch–Kato pro-$p$ groups

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 - which states that the Rost-Voevodsky Theorem (a.k.a. Bloch-Kato Conjecture) follows from 1-smoothness - for the class of finitely generated $p$-adic analytic pro-$p$ groups.

Self-duality of the lattice of transfer systems via weak factorization systems

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" - which states that the Rost-Voevodsky Theorem (a.k.a. Bloch-Kato Conjecture) follows from 1-smoothness - for the class of finitely generated $p$-adic analytic pro-$p$ groups.

Families of coherent PEL automorphic forms

In this article we give a construction of eigenvarieties by geometrically interpolating coherent automorphic sheaves of (PEL) Shimura varieties and their global sections. The new feature is that we particularly study the case of an empty ordinary locus, and thus use a replacement of the canonical subgroup in this situation. We specifically take into account the case of small primes and use this particular construction at a specific endoscopic point to prove new cases of the Bloch-Kato conjecture for characters of an imaginary quadratic field.

Chow groups and L-derivatives of automorphic motives for unitary groups

In this article, we study the Chow group of the motive associated to a tempered global $L$-packet $\pi$ of unitary groups of even rank with respect to a CM extension, whose global root number is $-1$. We show that, under some restrictions on the ramification of $\pi$ if the central derivative $L'(1/2,\pi)$ is nonvanishing, then the $\pi$-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and L-functions, which generalizes the Birch and Swinnerton-Dyer conjecture. Moreover, assuming the modularity of Kudla's generating functions of special cycles, we explicitly construct elements in a certain $\pi$-nearly isotypic subspace of the Chow group by arithmetic theta lifting, and compute their heights in terms of the central derivative $L'(1/2,\pi)$ and local doubling zeta integrals. This confirms the conjectural arithmetic inner product formula proposed by one of us, which generalizes the Gross--Zagier formula to higher dimensional motives.

CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINORL-VALUES

AbstractFollowing Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinorL-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

Supersingular O’Grady Varieties of Dimension Six

Abstract O’Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O’Grady’s construction to fields of positive characteristic $p\neq 2$, called OG6 varieties. Assuming $p\geq 3$, we show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin–Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces.

Applications of the Bloch–Kato conjecture to cohomological invariants and symbol length

Let F be a field, $$n\ge 2$$ a positive integer not divisible by . Suppose the group of roots of unity of degree n is contained in $$F^*$$ . Let l, m be positive integers, $$m\ge 2$$ , n is divisible by $$m^2$$ , $$n=mk$$ . Assume that $$\xi _n\in F$$ if n is odd, and $$\xi _{2n}\in F$$ if n is even. Applying the Bloch–Kato conjecture and the divided power operations on $$K_2(F)/n$$ , we construct an invariant in $$H^{2l+2}(F,\mu _n)$$ for elements $$A\in {\text {Br}}(F)$$ such that mA is the sum of l symbol algebras of degree k. Namely, if $$A=\sum \nolimits _{i=1}^l D_i +\alpha =\sum \nolimits _{i=1}^l {{\widetilde{D}}}_i +{{\widetilde{\alpha }}}$$ , where $$D_i,{{\widetilde{D}}}_i$$ are symbol algebras of degree n, and , then $$\begin{aligned} kD_1\cup \cdots \cup kD_l\cup \alpha =k{\widetilde{D}}_1\cup \cdots \cup k{\widetilde{D}}_l\cup {{\widetilde{\alpha }}}\in H^{2l+2}(F,\mu _m). \end{aligned}$$ When $$\xi _{2n}\notin F$$ , but $$\xi _n\in F$$ we construct the above invariant in the case where $$l=1$$ , $$m=2$$ , and $$n\ge 4$$ is 2-primary, using Suslin’s theorem on the torsion in $$K_2(F)$$ and the divided power operation $$\gamma _2: K_2(F)/2\rightarrow K_4(F)/2$$ . In the second part of the paper we consider one problem on the symbol length for $$K_q(F)/n$$ . For $$n,q\ge 2$$ and $$u\in K_q(F)/n$$ denote by $$L^{(n)}(u)$$ the minimal number k such that u is the sum of k symbols in $$K_q(F)/n$$ . For any $$m\ge 1$$ we give a sufficient condition on u for the equality $$L^{(mn)}(mu)= L^{(n)}(u)$$ . Using the index reduction formula for central simple algebras, we also construct examples with $$L^{(mn)}(mu) < L^{(n)}(u)$$ for any noncoprime m and n. In the case we give an upper bound for $$ L^{(n)}(u)\over {L^{(mn)}(mu)}$$ .