Let $f$ and $g$ be two modular forms which are non-ordinary at $p$. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution $f \otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on non-vanishing of $p$-adic $L$-fuctions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei-Loeffler-Zerbes.
Furthermore, we define an explicit logarithmic matrix using the theory of Wach modules, and show that this describes the growth of the Euler systems and $p$-adic $L$-functions associated to $f \otimes g$ in the cyclotomic tower. This allows us to formulate "signed" Iwasawa main conjectures for $f\otimes g$ in the spirit of Kobayashi's $\pm$-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.
Let $f$ be a normalized cuspidal eigen-newform of level coprime to $p$ with $a_p(f)=0$. We formulate both integral signed Iwasawa main conjectures and analytic Iwasawa man conjectures attached to the symmetric square motive of $f$ twisted by an auxiliary Dirichlet character. We show that the Beilinson--Flach elements attached to the symmetric square motive factorize into integral signed Beilinson--Flach elements, giving evidence towards the existence a rank-two Euler system predicted by Perrin-Riou. We use these integral elements to prove one inclusion in the integral and analytic Iwasawa main…
We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated p-adic L-functions. It also has applications to cyclotomic main conjectures for Hilb…
We investigate the cases for which products of two quasimodular or nearly holomorphic eigenforms are eigenforms. We also genaralize the results of Ghate \cite{ghate1} to the case of Rankin-Cohen brackets.
We use Maeda's Conjecture to prove that the Rankin-Cohen bracket of an eigenform and any modular form is only an eigenform when forced to be because of the dimensions of the underlying spaces. We further determine when the Rankin-Cohen bracket of an eigenform and modular form is not forced to produce an eigenform and when it is determined by the injectivity of the operator itself. This can also be interpreted as using the Rankin-Cohen bracket operator of eigenforms to create evidence for Maeda's Conjecture.
In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin-Selberg product of a general weight two modular form which is unramified and non-ordinary at $p$, and an ordinary CM form of higher weight contains the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa-Greenberg main conjecture for the $p$-adic $L$-function. This generalizes an earlier work of the author to the non-ordinary case. The result of this paper plays a crucial role in the proof of Iwasawa main conjecture and refined Birch-Swinnerton-Dyer formula for supersi…
In this paper we prove that the p-adic L-function that interpolates the Rankin-Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for the p-adic L-function. We prove this conjecture using the congruences between Klingen Eisensteinseries and cusp forms on the group GU(3; 1), following the strategy of a recent work of C. Skinner and E. Urban. This theorem can be used to deduce a converse of Gross-Zagier-Kolyvagin theorem and the precise BSD formula in …
We investigate when the product of two Hecke eigenforms for Γ_1(N) is again a Hecke eigenform. In this paper we prove that the product of two normalized eigenforms for Γ_1(N), of weight greater than 1, is an eigenform only 61 times, and give a complete list. Duke [Duk99] and Ghate [Gha00] independently proved that with eigenforms for SL_2(Z), there are only 16 such product identities. Ghate [Gha02] also proved related results for Γ_1(N), with N square free. Emmons [Emm05] proved results for eigenforms away from the level, for prime level. The methods we use are elementary and effective, and do…
In this paper we prove an anticyclotomic Iwasawa main conjecture proposed by Perrin-Riou for Heegner points when the global sign is -1, using a recent work of the author on one divisibility of Iwasawa-Greenberg main conjecture for Rankin-Selberg p-adic L-functions. As a byproduct we also prove the equality for the above mentioned main conjecture under some local conditions.
For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f = sum(a_n q^n) be a normalized new modular eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f, we define two Coleman maps with values in the Iwasawa algebra of Zp^* (after extending scalars to some extension of Qp). Applying these maps to the Kato zeta elements …
We establish formulae for the Iwasawa invariants of Mazur--Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of "medium" weight, and our second deals with forms of small slope . We give examples illustrating the strange behavior which can occur in the high weight, high slope case.
Building on the construction of big Heegner points in the quaternionic setting, and their relation to special values of Rankin-Selberg $L$-functions, we obtain anticyclotomic analogues of the results of Emerton-Pollack-Weston on the variation of Iwasawa invariants in Hida families. In particular, combined with the known cases of the anticyclotomic Iwasawa main conjecture in weight $2$, our results yield a proof of the main conjecture for $p$-ordinary newforms of higher weights and trivial nebentypus.
In this paper we propose and prove an anticyclotomic Iwasawa main conjecture for Heegner points on supersingular elliptic curves with $a_p=0$. The result has a "$\pm$" nature in the sense of Kobayashi. The proof uses a recent work of the second author on one divisibility in the Iwasawa--Greenberg main conjecture for Rankin-Selberg $p$-adic $L$-functions, together with an argument of Howard (adapted to our "$\pm$"-situation). As a byproduct, we also obtain an improvement of Skinner's result on a converse to the Gross--Zagier--Kolyvagin theorem.
For the anticyclotomic p-adic Rankin--Selberg L-function attached to a fixed Hecke eigenform and an imaginary quadratic field we introduce the second p-adic variable by considering Hida families of Hecke eigenforms parametrized by the weight.
We address the problem of identifying a Hecke-Maass cusp form $f$ of full level from the central values of the Rankin-Selberg $L$-functions $L(1/2,f\otimes h)$ where $h$ runs through the set of Hecke-Maass eigenforms of full level. We prove a quantitative result in this direction.
In this paper, we obtain a subconvexity result for the Rankin-Selberg L-function in both levels. The new feature in this result is applying an amplification method of Duke-Friedlander-Iwaniec to a double Petersson-Kuznetsov trace formula. As the trace formula ranges over all the $GL_2$ spectrum, this subconvexity result is unconditional of which Hecke eigenforms are chosen in the Rankin-Selberg L-function.
Congruences of $p$-adic $L$-functions of modular forms at non-ordinary primes
Institute of Mathematics University of the Philippines Diliman（菲律宾大学达曼数学研究所）；Department of Mathematics and Statistics University of Ottawa（渥太华大学数学与统计系）
In a 2005 paper, Yang constructed families of Hilbert Eisenstein series, which when restricted to the diagonal are conjectured to span the underlying space of elliptic modular forms. One approach to these conjectures is to show the non-vanishing of an inner product of elliptic eigenforms with the restrictions of Eisenstein series. In this paper, we compute this inner product locally by using explicit values of new vectors in the Waldspurger model.
In this paper we prove one side divisibility of the Iwasawa-Greenberg main conjecture for Rankin-Selberg product of a weight two cusp form and an ordinary CM form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm{GU}(3,1)$, generalizing earlier result of the third-named author to allow non-ordinary cusp forms. The main result is a key input in the third author's proof for Kobayashi's $\pm$-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semi-ordinary Hida theory along an appropriate smaller weight space, …
In this work, we establish a zero density result for the Rankin-Selberg $L$-functions. As an application, we apply it to distinguish the holomorphic Hecke eigenforms for $\operatorname{SL}_2(\mathbb{Z}).$
We prove the functional equation for the twisted spinor L-series of a cuspidal, holomorphic Siegel eigenform for the full modular group of genus 2. It follows from a more general functional equation, valid for Rankin convolutions of paramodular cuspforms. A non-vanishing result for Fourier-Jacabi coefficients of the eigenforms in question is the central pillar of the deduction of the former from the latter functional equation.
Let $p$ be an odd prime. We prove the cyclotomic Iwasawa Main Conjecture of K.Kato for the motive attached to an eigencuspform $f\in S_{k}(Γ_{0}(N))$ with arbitrary reduction type at $p$ under mild assumptions on the residual Galois representation $\barρ_{f}$. Under the same hypotheses, we also prove the generalized Iwasawa Main Conjecture for $p$-adic families of modular forms. The Iwasawa Main Conjecture for $f$ is deduced by a limit argument involving fundamental lines from a universal Iwasawa Main Conjecture over the universal deformation space of $\barρ_{f}$, which itself follows from the…
In this article, we prove an explicit Waldspurger formula for the toric Hilbert modular forms. As an application, we construct a class of anticyclotomic p-adic Rankin-Selberg L-functions for Hilbert modular forms, generalizing the construction of Bertolini, Darmon and Prasanna in the elliptic case. Moreover, building on works of Hida, we give a necessary and sufficient condition when the Iwasawa mu-invariant of this p-adic L-function vanishes and prove a result on the non-vanishing modulo $p$ of central Rankin-Selberg L-values with anticyclotomic twists.
The purpose of this paper is to build on results in {\it{higher codimension Iwasawa theory}}. The setting of our results involves Galois representations arising as cyclotomic twist deformations associated to (i) the tensor product of two cuspidal Hida families $F$ and $G$, and (ii) the tensor product of three cuspidal Hida families $F$, $G$ and $H$. On the analytic side, we consider (i) a pair of $3$-variable Rankin-Selberg $p$-adic $L$-functions constructed by Hida and (ii) a balanced $4$-variable $p$-adic $L$-function (due to Hsieh and Yamana) and an unbalanced $4$-variable $p$-adic $L$-func…
We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.
Let M be an imaginary quadratic field, f a Hecke eigenform on GL2(Q) and πthe unitary base-change to M of the automorphic representation associated to f. Take a unitary arithmetic Hecke character χof M inducing the inverse of the central character of f. The celebrated formula of Waldspurger relates the square of an integral L(f) of f and χover the idele class group of M to the central critical value L(1/2,π\otimesχ). In this paper, we present a construction of a new p-adic L-function that interpolates L(f) over arithmetic characters for a cusp form f in the spirit of the landmark result of Kat…
For two Hecke eigenforms $h_1$ and $h_2$ in the Kohnen plus space of half-integral weight, let $I_n(h_1)$ and $I_n(h_2)$ be the Duke-Imamoglu-Ikeda lift of $h_1$ and $h_2$, respectively, which are Siegel cusp forms with respect to $Sp_n(\ZZ)$. Moreover, let $E_{n/2+1/2}$ be the Cohen Eisenstein series of weight $n/2+1/2$. We then express the Rankin-Selberg convolution $R(s,I_n(h_1),I_n(h_2))$ of $I_n(h_1)$ and $I_n(h_2)$ in terms of a certain Dirichlet series $D(s,h_1,h_2,E_{n/2+1/2})$, which is similar to the triple convolution product of $h_1, h_2$ and $E_{n/2+1/2}$. We apply our formula to …
In this paper we show how one can combine the p-adic Rankin-Selberg product construction of Hida with freeness results of Hecke modules of Wiles to establish interesting congruences between special values of L-functions. These congruences is a part of some deep conjectural congruences that follow from the work of Kato on the non-commutative Iwasawa theory of the false Tate curve extension.
The Rankin-Cohen bracket of two Eisenstein series provides a kernel yielding products of the periods of Hecke eigenforms at critical values. Extending this idea leads to a new type of Eisenstein series built with a double sum. We develop the properties of these series and their non-holomorphic analogs and show their connection to values of L-functions outside the critical strip.
Let $π$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times \operatorname{GL}_2$ Rankin-Selberg $L$-functions associated to $π$ times representations induced from Hecke characters of $K$ which are ramified only at a given prime ideal $\mathfrak{p}$ of $F$. More specifically, we use spectral decompositions of shifted convolution sums and relations to Fourier-Whittaker coefficients of genuine and non-genuine metaplectic forms…
In this article, we present the first half of our project on the Iwasawa theory of higher rank Galois deformations over deformations rings of arbitrary dimension. We develop a theory of Coleman maps for a very general class of coefficient rings, devise a dimension reduction procedure for locally restricted Euler systems and finally, put these into use in order to prove a divisibility in a 3-variable main conjecture for nearly ordinary families of Rankin-Selberg convolutions.
Kato's main conjecture for nonordinary modular forms
University of California, Santa Barbara（加州大学圣塔芭芭拉分校）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
