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.

A note on Rubio de Francia's extrapolation in tent spaces and applications

The Rubio de Francia extrapolation theorem is a very powerful result which states that in order to show that certain operators satisfy weighted norm inequalities with Muckenhoupt weights it suffices to see that the corresponding inequalities hold for some fixed exponent, for instance p=2. In this paper we extend this result and show that this extrapolation principle allows one to obtain weighted estimates in tent spaces. From our extrapolation result we automatically derive new estimates (and reprove some other) concerning Calderón-Zygmund operators, operators associated with the Kato conjecture, or fractional operators.

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.

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.

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.

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.

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.

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.

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 $$ L^{(n)}(u)\over {L^{(mn)}(mu)}$$ .

Stark–Heegner cycles attached to Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch--Kato conjecture predicts that this should force the existence of non-trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Barrera Salazar and the second author to construct certain P-adic Abel--Jacobi maps, which we use to propose a construction of such classes via "Stark--Heegner cycles". This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms.

CONGRUENCE PRIMES FOR SIEGEL MODULAR FORMS OF PARAMODULAR LEVEL AND APPLICATIONS TO THE BLOCH–KATO CONJECTURE

AbstractIt has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel–Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito–Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch–Kato conjecture for elliptic newforms of square-free level and odd functional equation.

Supersingular locus of Hilbert modular varieties, arithmetic level raising and Selmer groups

This article has three goals. First, we generalize the result of Deuring and Serre on the characterization of supersingular locus of modular curves to all Shimura varieties given by totally indefinite quaternion algebras over totally real number fields. Second, we generalize the result of Ribet on arithmetic level raising to such Shimura varieties in the inert case. Third, as an application to number theory, we use the previous results to study the Selmer group of certain triple product motive of an elliptic curve, in the context of the Bloch--Kato conjecture.

Elliptic Functional Differential Equations with Degenerations

This review is devoted to differential-difference equations with degeneration in a bounded domain $$Q\subset\mathbb{R}^{n}$$ and applications (Kato conjecture, nonlocal boundary value problem). We consider differential-difference operators with degeneration of the second order and generalization to $$2m$$ -order and the case where differential-difference operator contains several degenerate difference operators with degeneration. Generalized solutions of such equations may not belong even to the Sobolev space $$W^{1}_{2}(Q)$$ .

Riesz transform and fractional integral operators generated by nondegenerate elliptic differential operators

‎‎‎The Morrey boundedness is proved for the Riesz transform and the inverse operator of the nondegenerate elliptic differential operator of divergence form generated by a vector-function in $(L^\infty)^{n^2}$‎, ‎and for the inverse operator of the Schrodinger operators whose nonnegative potentials satisfy a certain integrability condition‎. ‎In this note‎, ‎our result is not obtained directly from the estimates of integral formula‎, ‎which reflects the fact that the solution of the Kato conjecture did not use any integral expression of the operators‎. ‎One of the important tools in the proof is the decomposition of the functions in Morrey spaces based on the elliptic differential operators in question‎. ‎In some special cases where the integral kernel comes into play‎, ‎the boundedness property of the Littlewood-Paley operator was already obtained by Gong‎. ‎So‎, ‎the main novelties of this paper are the decomposition results associated with elliptic differential operators and the result in the case where the explicit formula of the integral kernel of the heat semigroup is unavailable‎.

Families of Picard modular forms and an application to the Bloch–Kato conjecture

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 and Stevens [$p$-adic families of Siegel modular cuspforms Ann. of Math. (2) 181, (2015), 623–697] 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 $E$, extending the results of Bellaiche and Chenevier to the case of a positive sign.

On a class of functional–differential operators satisfying the Kato conjecture

Second order elliptic differential-difference operators with degeneration in a cylinder are considered. It is proved that these operators satisfy the Kato conjecture about the square root of an operator.

PARALLEL WEIGHT 2 POINTS ON HILBERT MODULAR EIGENVARIETIES AND THE PARITY CONJECTURE

Let$F$be a totally real field and let$p$be an odd prime which is totally split in$F$. We define and study one-dimensional ‘partial’ eigenvarieties interpolating Hilbert modular forms over$F$with weight varying only at a single place$v$above$p$. For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch–Kato conjecture for finite slope Hilbert modular forms with trivial central character (with a technical assumption if$[F:\mathbb{Q}]$is odd), by reducing to the case of parallel weight$2$. As another consequence of our results on partial eigenvarieties, we show, still under the assumption that$p$is totally split in$F$, that the ‘full’ (dimension$1+[F:\mathbb{Q}]$) cuspidal Hilbert modular eigenvariety has the property that many (all, if$[F:\mathbb{Q}]$is even) irreducible components contain a classical point with noncritical slopes and parallel weight$2$(with some character at$p$whose conductor can be explicitly bounded), or any other algebraic weight.

Kurokawa–Mizumoto congruences and degree-8 L-values

Let f be a Hecke eigenform of weight k, level 1, genus 1. Let $$E^k_{2,1}(f)$$ be its genus-2 Klingen–Eisenstein series. Let F be a genus-2 cusp form whose Hecke eigenvalues are congruent modulo $${\mathfrak {q}}$$ to those of $$E^k_{2,1}(f)$$ , where $${\mathfrak {q}}$$ is a “large” prime divisor of the algebraic part of the rightmost critical value of the symmetric square L-function of f. We explain how the Bloch–Kato conjecture leads one to believe that $${\mathfrak {q}}$$ should also appear in the denominator of the “algebraic part” of the rightmost critical value of the tensor product L-function $$L(s,f\otimes F)$$ , i.e. in an algebraic ratio obtained from the quotient of this with another critical value. Using pullback of a genus-5 Siegel–Eisenstein series, we prove this, under weak conditions.