Let f be a modular form of weight 2 and trivial character. Fix also an imaginary quadratic field K. We use work of Bertolini-Darmon and Vatsal to study the mu-invariant of the p-adic Selmer group of f over the anticyclotomic Zp-extension of K. In particular, we verify the mu-part of the main conjecture in this context. The proof of this result is based on an analysis of congruences of modular forms, leading to a conjectural quantitative version of level-lowering (which we verify in the case that Mazur's principle applies).
Under certain assumptions, we prove an anticyclotomic analogue of the "weak main conjecture" à la Mazur and Tate for modular forms over a large class of cyclic ring class extensions.
In this paper we study the exceptional zero phenomenon for Hilbert modular forms in the anticyclotomic setting. We prove a formula expressing the leading term of the p-adic L-functions via arithmetic L-invariants.
In this article, we generalize some works of Bertolini-Darmon and Vatsal on anticyclotomic L-functions attached to modular forms of weight two to higher weight case. We construct a class of anticyclotomic p-adic L-functions for ordinary modular forms and derive the functional equation and the interpolation formula at all critical specializations. Moreover, we prove results on the vanishing of mu-invariant of these p-adic L-functions and the non-vanishing of central L-values with anticyclotomic twists.
Following Bertolini and Darmon's method, with "Ihara's lemma" among other conditions Longo and Wang proved one divisibility of Iwasawa main conjecture for Hilbert modular forms of weight $2$ and general low parallel weight respectively. In this paper, we remove the "Ihara's lemma" condition in their results.
Let $K$ be an imaginary quadratic field where $p$ splits. We study signed Selmer groups for non-ordinary modular forms over the anticyclotomic $\mathbf{Z}_p$-extension of $K$, showing that their $μ$-invariants vanish. This generalizes and gives a new proof of a recent result of Matar on the vanishing of the $μ$-invariants of plus and minus signed Selmer groups for elliptic curves.
We study the variation of Iwasawa invariants of the anticyclotomic Selmer groups of congruent modular forms under the Heegner hypothesis. In particular, we show that even if the Selmer groups we study may have positive coranks, the mu-invariant vanishes for one modular form if and only if it vanishes for the other, and that their lambda-invariants are related by an explicit formula. This generalizes results of Greenberg-Vatsal for the cyclotomic extension, as well as results of Pollack-Weston and Castella-Kim-Longo for the anticyclotomic extension when the Selmer groups in question are cotorsi…
In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove a one sided divisibility result toward the Iwasawa main conjecture. The proof relies on the first and second reciprocity law relating theta elements to Heegner points Euler system. As a by-product we also prove certain Bloch-Kato type result in the rank $0$ case and a parity conjecture
This is the first in a series of articles where we will study the Iwasawa theory of an elliptic modular form f along the anticyclotomic Zp-tower of an imaginary quadratic field K where the prime p splits completely. Our goal in this portion is to prove the Iwasawa main conjecture for suitable twists of f assuming that f is p-ordinary, both in the definite and indefinite setups simultaneously, via an analysis of Beilinson-Flach elements.
We study the Selmer group associated to a $p$-ordinary newform $f \in S_{2r}(Γ_0(N))$ over the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K/\mathbb{Q}$. Under certain assumptions, we prove that this Selmer group has no proper $Λ$-submodules of finite index. This generalizes work of Bertolini in the elliptic curve case. We also offer both a correction and an improvement to an earlier result on Iwasawa invariants of congruent modular forms by the present authors.
In this article, we follow Hida's approach to study the mu-invariant of the anticyclotomic projection of p-adic Hecke L-functions for CM fields along a branch character. We prove a conjecture of Gillard on the vanishing of the mu-invariant and give an exact mu-invariant formula for self-dual branch characters.
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.
Let $E$ be an elliptic curve over an imaginary quadratic field $K$, and $p$ be an odd prime such that the residual representation $E[p]$ is reducible. The $μ$-invariant of the fine Selmer group of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ is studied. We do not impose the Heegner hypothesis on $E$, thus allowing certain primes of bad reduction to decompose infinitely in the anticyclotomic $\mathbb{Z}_p$-extension. It is shown that the fine $μ$-invariant vanishes if certain explicit conditions are satisfied. Further, a partial converse is proven.
Let A be a modular elliptic curve over a totally real field F, and let E/F be a totally imaginary quadratic extension. In the event of exceptional zero phenomenon, we prove a formula for the derivative of the multivariable anticyclotomic p-adic L-function attached to (A,E), in terms of the Hasse-Weil L-function and certain p-adic periods attached to the respective automorphic forms. Our methods are based on a new construction of the anticyclotomic p-adic L-function by means of the corresponding automorphic representation.
We reinterpret the explicit construction of Gross points given by Chida-Hsieh as a non-Archimedian analogue of the standard geodesic cycle from zero to the infinity on the Poincare upper half plane. This analogy allows us to consider certain distributions, which can be regarded as anticyclotomic p-adic L-functions for modular forms of non-critical slope following the overconvergent strategy a la Stevens. We also give a geometric interpretation of their Gross points for the case of weight two forms. Our construction generalizes those of Bertolini-Darmon, Bertolini-Darmon-Iovita-Spiess, and Chid…
Iwasawa theory of modular forms over anticyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields has been studied by several authors, starting from the works of Bertolini-Darmon and Iovita-Spiess, under the crucial assumption that the prime $p$ is unramified in $K$. We start in this article the systematic study of anticyclotomic $p$-adic $L$-functions when $p$ is ramified in $K$. In particular, when $f$ is a weight $2$ modular form attached to an elliptic curve $E/\mathbb{Q}$ having multiplicative reduction at $p$, and $p$ is ramified in $K$, we show an analogue of the exceptional z…
The purpose of this article is to generalize some results of Vatsal on studying the special values of Rankin-Selberg L-functions in an anticyclotomic $\mathbb{Z}_{p}$-extension. Let $g$ be a cuspidal Hilbert modular form of parallel weight (2,...,2) and level $\mathcal{N}$ over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant $\mathcal{D}$. We study the $l$-adic valuation of the special values $L(g,χ,\frac{1}{2})$ as χvaries over the ring class characters of K of $\mathcal{P}$-power conductor, for some fixed prime ideal $\mathcal{P}$. We p…
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.
Let $E/\mathbb{Q}$ be an elliptic curve and let $K$ be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for $E$ using $K$-CM points and conjectured they did not all vanish. Conditional on this conjecture, he described the Selmer rank of $E$ using his system of classes. We extend work of Wei Zhang to prove new cases of Kolyvagin's conjecture by considering congruences of modular forms modulo large powers of $p $. Additionally, we prove an analogous result, and give a description of the Selmer rank, in a complementary "definite" case (usi…
When the branch character has root number -1, the corresponding anticyclotomic Katz p-adic L-function identically vanishes. In this case, we study the $μ$-invariant of the cyclotomic derivative of Katz p-adic L-function. As an application, this proves the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with the root number -1. The result also plays a crucial role in the recent work of Hsieh on the Eisenstein ideal approach to a one-sided divisibility of the CM main conjecture.
We construct the three-variable p-adic triple product L-functions attached to Hida families of ellptic newforms and prove the explicit interpolation formulae at all critical specializations by establishing explicit Ichino's formulae for the trilinear period integrals of automorphic forms. Our formulae perfectly fit the conjectural shape of p-adic L-functions predicted by Coates and Perrin-Riou. As an application, we prove the factorization of certain unbalanced p-adic triple product L-functions into a product of anticyclotomic p-adic L-functions for modular forms. By this factorization, we giv…
In this paper, we prove under mild hypotheses the Iwasawa main conjectures of Lei--Loeffler--Zerbes for modular forms of weight $2$ at non-ordinary primes. Our proof is based on the study of the two-variable analogues of these conjectures formulated by Büyükboduk--Lei for imaginary quadratic fields in which $p$ splits, and on anticyclotomic Iwasawa theory. As application of our results, we deduce the $p$-part of the Birch and Swinnerton-Dyer formula in analytic ranks $0$ or $1$ for abelian varieties over $\mathbb{Q}$ of ${\rm GL}_2$-type for non-ordinary primes $p>2$.
Let $E/\mathbb{Q}$ be an elliptic curve with ordinary reduction at a prime $p$, and let $K$ be an imaginary quadratic field. The anticyclotomic Iwasawa main conjecture, depending upon the sign of the functional equation of $L(E/K,s)$, predicts the behavior of Selmer group of $E/\mathbb{Q}$ along the anticyclotomic tower of $K$. Some of the crucial ideas of Bertolini and Darmon on this conjecture have been abstracted by Howard into an axiomatic set-up through a notion of Bipartite Euler systems, assuming that $E[p]$ is an irreducible representation of $G_{K}$. We generalize this work by assumin…
We prove analogues of the major algebraic results of Greenberg-Vatsal for Selmer groups of $p$-ordinary newforms over $\mathbf{Z}_p$-extensions which may be neither cyclotomic nor anticyclotomic, under a number of technical hypotheses, including a cotorsion assumption on the Selmer groups. The main complication which arises in our work is the possible presence of finite primes which can split completely in the $\mathbf{Z}_p$-extension being considered, resulting in the local cohomology groups that appear in the definition of the Selmer groups being significantly larger than they are in the cas…
We formulate integral Iwasawa main conjectures for suitable twists of a newform $f$ that is non-ordinary at $p$, over the cyclotomic $\mathbb{Z}_p$-extension, the anticyclotomic $\mathbb{Z}_p$-extensions (in both the definite and the indefinite cases) as well as the $\mathbb{Z}_p^2$-extension of an imaginary quadratic field $K$ where $p$ splits. In order to do so, we define Kobayashi-Sprung-style signed Coleman maps, which we use to introduce doubly-signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson-Flach elements (out of the collection of unbounded Beilinson-Fl…
In our first article in this series ("Modular Invariant of Quantum Tori I: Definitions Nonstandard and Standard" arXiv:0909.0143) a modular invariant of quantum tori was defined. In this paper, we consider the case of the quantum torus associated to the golden mean. We show that the modular invariant is approximately 9538.249655644 by producing an explicit formula for it involving weighted versions of the Rogers-Ramanujan functions.
In this paper, we obtain two effective bounds for the $j$-invariant of integral points on certain modular curves which has positive genus and less than three cusps.
In this paper, we give some effective bounds for the $j$-invariant of integral points on arbitrary modular curves over arbitrary number fields assuming that the number of cusps is not less than 3.
In this paper we show that under certain condition the Fontaine--Mazur $L$-invariant for a Hilbert eigenform coincides with its Teitelbaum type $L$-invariant, and thus prove a conjecture of Chida, Mok and Park.
Inspired by prior work of Bruinier and Ono and Mertens and Rolen, we study class polynomials for non-holomorphic modular functions arising from modular forms of negative weight. In particular, we give general conditions for the irreducibility of class polynomials. This allows us to easily generate infintely many new class invariants.
We give a construction of a new $p$-adic Maass-Shimura operator defined on an affinoid subdomain of the preperfectoid $p$-adic universal cover $\mathcal{Y}$ of a modular curve $Y$. We define a new notion of $p$-adic modular forms as sections of a certain sheaf $\mathcal{O}_Δ$ of "nearly rigid functions" which transform under the action of subgroups of the Galois group $\mathrm{Gal}(\mathcal{Y}/Y)$ by $\mathcal{O}_Δ^{\times}$-valued weight characters. This extends Katz's notion of $p$-adic modular forms as functions on the Igusa tower $Y^{\mathrm{Ig}}$; indeed we may recover Katz's theory by re…
In this paper we describe a method for computing a basis for the space of weight $2$ cusp forms invariant under a non-split Cartan subgroup of prime level $p$. As an application we compute, for certain small values of $p$, explicit equations over $\bf Q$ for the canonical embeddings of the associated modular curves.
Given a cuspidal Hilbert modular eigenform $π$ of parallel weight 2 and a nonarchimedian place $\mathfrak p$ of the underlying totally real field such that the local component of $π$ at $\mathfrak p$ is the Steinberg representation, one can associate two types of $\mathscr L$-invariants, one defined in terms of the cohomology of arithmetic groups and the other in terms of the Galois representation associated to $π$. We show that the $\mathscr L$-invariants are the same.
The modular invariant of rank 1 Drinfeld modules is introduced and used to formulate and prove an exact analog of the Weber-Fueter theorem for global function fields. The main ingredient in the proof is a version of Shimura's Main Theorem of Complex Multiplication for global function fields, which is also proved here.
We provide explicit bounds on the difference of heights of the $j$-invariants of isogenous elliptic curves defined over $\overline{\mathbb{Q}}$. The first one is reminiscent of a classical estimate for the Faltings height of isogenous abelian varieties, which is indeed used in the proof. We also use an explicit version of Silverman's inequality and isogeny estimates by Gaudron and Rémond. We give applications to the study of Vélu's formulas and of modular polynomials.
The purpose of this paper is to prove the equality between the algebraic Iwasawa $λ$-invariant and the analytic Iwasawa $λ$-invariant for a Hilbert cusp form of parallel weight $2$ at an ordinary prime $p$ when the associated residual Galois representation is reducible. This is a generalization of a result of R. Greenberg and V. Vatsal.
Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number. We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with level prime to $p$, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-re…
