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Francesc Castella

Publications and source records attributed to Francesc Castella.

At least 19 recordsLinked to original sources
arXiv2026

Kato's main conjecture for nonordinary modular forms

We prove Kato's main conjecture for modular forms at nonordinary primes for weights in the Fontaine-Laffaille range. The key ingredients in the proof are a reformulation of the conjecture in terms of signed Selmer groups due to Lei-Loeffler-Zerbes, certain $p$-adic families of Rankin-Eisenstein classes arising from the work of Lei-Loeffler-Zerbes and Kings-Loeffler-Zerbes, and the lower bound divisibility in an Iwasawa-Greenberg main conjecture for Rankin-Selberg $p$-adic $L$-functions obtained in our earlier work \cite{CLW}.

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arXiv2026

On refined nonvanishing conjectures by Kurihara and Kolyvagin

In this paper, we extend the results of \cite{BCGS} on refined conjectures by Kurihara and Kolyvagin, allowing primes of any reduction type in the case of Kurihara's conjectures, and inert primes in the underlying imaginary quadratic field in the case of Kolyvagin's. The key innovation is a new approach to the computation of the $p$-divisibility index of certain special elements in Galois cohomology (the bottom class of a $\Lambda$-adic Euler system twisted by a character sufficiently close to the trivial character) based on a reformulation of the Iwasawa Main Conjectures in terms of determinants of Selmer complexes.

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arXiv2025

A formula of Perrin-Riou and characteristic power series of signed Selmer groups

We prove a conjecture of Kundu--Ray, following from the $p$-adic Birch--Swinnerton-Dyer conjecture for supersingular primes by Bernardi--Perrin-Riou and Kato's Main Conjecture, predicting an expression for the leading term (up to a $p$-adic unit) of a characteristic power series of Kobayashi's signed Selmer groups attached to elliptic curves $E/\mathbb{Q}$ with supersingular reduction at a prime $p>2$ with $a_p=0$. The proof is deduced from a similar formula due to Perrin-Riou for a generator of her module of arithmetic $p$-adic $L$-functions with values in the Dieudonn\'{e} module of $E$.

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arXiv2025

An anticyclotomic Euler system of Hirzebruch--Zagier cycles I: Norm relations and $p$-adic interpolation

We construct an anticyclotomic Euler system for the Asai Galois representation associated to $p$-ordinary Hilbert modular forms over real quadratic fields. We also show that our Euler system classes vary in $p$-adic Hida families. The construction is based on the study of certain Hirzebruch--Zagier cycles obtained from modular curves of varying level diagonally emdedded into the product with a Hilbert modular surface. By Kolyvagin's methods, in the form developed by Jetchev--Nekov\'{a}\v{r}--Skinner in the anticyclotomic setting, the construction yields new applications to the Bloch--Kato conjecture and the Iwasawa Main Conjecture.

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arXiv2024

Exceptional zeros for Heegner points and $p$-converse to the theorem of Gross-Zagier and Kolyvagin

We prove a $p$-converse to the theorem of Gross-Zagier and Kolyvagin for elliptic curves $E/\mathbf{Q}$ at primes $p>3$ of multiplicative reduction. Two key ingredients in the argument are an extension to this setting of a $p$-adic formula of Bertolini-Darmon-Prasanna obtained in our earlier work, and an exceptional zero formula for Heegner points. By independent approaches different from ours, a similar $p$-converse theorem was obtained by Skinner--Zhang under additional ramification hypotheses on $E[p]$, and by Venerucci assuming finiteness of the $p$-primary part of the Tate-Shafarevich group.

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arXiv2024

Tamagawa number conjecture for CM modular forms and Rankin--Selberg convolutions

Let $E/F$ be an elliptic curve defined over a number field $F$ with complex multiplication by the ring of integers of an imaginary quadratic field $K$ such that the torsion points of $E$ generate over $F$ an abelian extension of $K$. In this paper we prove the $p$-part of the Birch--Swinnerton-Dyer formula for $E/F$ in analytic rank $1$ for primes $p>3$ split in $K$. This was previously known for $F=\mathbb{Q}$ by work of Rubin as a consequence of his proof of Mazur's Main Conjecture for rational CM elliptic curves, but the problem for $[F:\mathbb{Q}]>1$ remained wide open. The approach introduced in this paper also yields a proof of similar results for CM abelian varieties $A/K$ and for CM modular forms, as well as an analogue in this setting of Skinner's $p$-converse to the theorem of Gross--Zagier and Kolyvagin.

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arXiv2024

Critical Lambda-adic modular forms and bi-ordinary complexes

We produce a flat $\Lambda$-module of $\Lambda$-adic critical slope overconvergent modular forms, producing a Hida-type theory that interpolates such forms over $p$-adically varying integer weights. This provides a Hida-theoretic explanation for an observation of Coleman that the rank of such forms is locally constant in the weight. The key to the interpolation is to use Coleman's presentation of de Rham cohomology in terms of overconvergent forms to link critical slope overconvergent modular forms with the part of the first coherent cohomology of modular curves interpolated by Boxer-Pilloni's higher Hida theory. The novelty is that we interpolate a critical period in cohomology using modular forms, complementing the classical Hida-theoretic interpolation of an ordinary period. Using this interpolation, we also interpolate bi-ordinary complexes in various weights into a perfect and self-dual complex of length 1 over $\Lambda$. By design, the cohomology of the bi-ordinary complex supports 2-dimensional $p$-adic representations of ${\rm Gal}(\bar{\bf Q}/{\bf Q})$ that become reducible and decomposable upon restriction to a decomposition group at $p$. As applications and motivations for the above constructions, we prove "$R = T$" theorems for the critical and bi-ordinary Hecke algebras, produce a degree-shifting Hecke action on the co-homology of bi-ordinary complexes, and specialize this degree-shifting action to weight 1 to produce, under a supplemental assumption, an action of a Stark unit on the part of weight 1 coherent cohomology over ${\bf Z}_p$ that is isotypic for an ordinary eigenform with complex multiplication.

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arXiv2024

Base change and Iwasawa Main Conjectures for ${\rm GL}_2$

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ of conductor $N$, $p$ an odd prime of good ordinary reduction such that $E[p]$ is an irreducible Galois module, and $K$ an imaginary quadratic field with all primes dividing $Np$ split. We prove Iwasawa Main Conjectures for the $\mathbb{Z}_p$-cyclotomic and $\mathbb{Z}_p$-anticyclotomic deformations of $E$ over $\mathbb{Q}$ and $K$ respectively, dispensing with any of the ramification hypotheses on $E[p]$ in previous works. The strategy employs base change and the two-variable zeta element associated to $E$ over $K$, via which the sought after main conjectures are deduced from Wan's divisibility towards a three-variable main conjecture for $E$ over a quartic CM field containing $K$ and certain Euler system divisibilities. As an application, we prove cases of the two-variable main conjecture for $E$ over $K$. The aforementioned one-variable main conjectures imply the $p$-part of the conjectural Birch and Swinnerton-Dyer formula for $E$ if ${\rm ord}_{s=1}L(E,s)\leq 1$. They are also an ingredient in the proof of Kolyvagin's conjecture and its cyclotomic variant in our joint work with Grossi.

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arXiv2023

Non-vanishing of Kolyvagin systems and Iwasawa theory

Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime. In 1991 Kolyvagin conjectured that the system of cohomology classes for torsion quotients of the $p$-adic Tate module of $E$ derived from Heegner points over ring class fields of a suitable imaginary quadratic field $K$ (i.e., the Heegner point Kolyvagin system of $E/K$) is non-trivial. In this paper we prove Kolyvagin's conjecture when $p$ is a prime of good ordinary reduction for $E$ that splits in $K$. In particular, our results cover many cases where $p$ is an Eisenstein prime for $E$, complementing Wei Zhang's earlier results on the conjecture by a different approach. Our methods also yield a proof of a refinement of Kolyvagin's conjecture expressing the divisibility index of the Heegner point Kolyvagin system in terms of the Tamagawa numbers of $E$, as conjectured by Wei Zhang in 2014, as well as proofs of analogous results for the Kolyvagin system obtained from Kato's Euler system.

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arXiv2023

Nonvanishing of generalised Kato classes and Iwasawa main conjectures

A construction due to Darmon--Rotger gives rise to generalised Kato classes $\kappa_p(E)$ in the $p$-adic Selmer group ${\rm Sel}(\mathbf{Q},V_pE)$ of elliptic curves $E/\mathbf{Q}$ of positive even analytic rank, where $p>3$ is any prime of good ordinary reduction for $E$. In some cases, their conjectured that $\kappa_p(E)\neq 0$ precisely when ${\rm Sel}(\mathbf{Q},V_pE)$ is two-dimensional. The first cases of this conjecture were obtained in a joint work of the author with M.-L. Hsieh. In this note we give a new proof of the implication \[ \kappa_p(E)\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2 \] established in op. cit., and show that the converse implication holds if \emph{and only if} the restriction map ${\rm loc}_p:{\rm Sel}(\mathbf{Q},V_pE)\rightarrow E(\mathbf{Q}_p)\hat\otimes\mathbf{Q}_p$ is nonzero. The present approach is an adaptation to the non-CM case of the method introduced by the author in the case of CM elliptic curves.

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arXiv2023

Derived $p$-adic heights and the leading coefficient of the Bertolini--Darmon--Prasanna $p$-adic $L$-function

Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be an odd prime of good reduction for $E$. Let $K$ be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which $p$ splits. The goal of this paper is two-fold: (1) We formulate a $p$-adic BSD conjecture for the $p$-adic $L$-function $L_{\mathfrak{p}}^{\rm BDP}$ introduced by Bertolini--Darmon--Prasanna. (2) For an algebraic analogue $F_{{\mathfrak{p}}}^{\rm BDP}$ of $L_{\mathfrak{p}}^{\rm BDP}$, we show that the ``leading coefficient'' part of our conjecture holds, and that the ``order of vanishing'' part follows from the expected ``maximal non-degeneracy'' of an anticyclotomic $p$-adic height. In particular, when the Iwasawa--Greenberg Main Conjecture $(F_{{\mathfrak{p}}}^{\rm BDP})=(L_{\mathfrak{p}}^{\rm BDP})$ is known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\rm BDP}$ at $T=0$ up to a $p$-adic unit. Moreover, by adapting the approach of Burungale--Castella--Kim, we prove the main conjecture for supersingular primes $p$ under mild hypotheses. In the $p$-ordinary case, and under some additional hypotheses, similar results were obtained by Agboola--Castella, but our method is new and completely independent from theirs, and apply to all good primes.

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arXiv2023

Diagonal cycles and anticyclotomic Iwasawa theory of modular forms

We construct a new Euler system for the Galois representation $V_{f,\chi}$ attached to a newform $f$ of weight $2r\geq 2$ twisted by an anticyclotomic Hecke character $\chi$. The Euler system is anticyclotomic in the sense of Jetchev-Nekovar-Skinner. We then show some arithmetic applications of the constructed Euler system, including new results on the Bloch-Kato conjecture in ranks zero and one, and a divisibility towards the Iwasawa-Greenberg main conjecture for $V_{f,\chi}$. In particular, in the case where the base-change of $f$ to our imaginary quadratic field has root number $+1$ and $\chi$ has higher weight (which implies that the complex $L$-function $L(V_{f,\chi},s)$ vanishes at the center), our results show that the Bloch-Kato Selmer group of $V_{f,\chi}$ is nonzero, as predicted by the Bloch-Kato conjecture; and if in addition a certain distinguished class $\kappa_{\,f,\chi}$ is nonzero, then the Selmer group is one-dimensional. Such applications to the Bloch-Kato conjecture for $V_{f,\chi}$ were left wide open by the earlier approaches using Heegner cycles and/or Beilinson-Flach elements. Our construction is based instead on a generalization of the Gross-Kudla-Schoen diagonal cycles.

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arXiv2023

Mazur's main conjecture at Eisenstein primes

Let $E/\mathbb{Q}$ be an elliptic curve, let $p>2$ be a prime of good reduction for $E$, and assume that $E$ admits a rational $p$-isogeny with kernel $\mathbb{F}_p(\phi)$. In this paper we prove the cyclotomic Iwasawa main conjecture for $E$, as formulated by Mazur in 1972, when $\phi\vert_{G_p}\neq 1,\omega$, where $G_p$ is a decomposition group at $p$ and $\omega$ is the Teichm\"uller character. Our proof is based on a study of the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field $K$ in which $p$ splits, and a congruence argument exploiting the cyclotomic Euler system of Beilinson--Flach classes.

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arXiv2022

Generalised Kato classes on CM elliptic curves of rank 2

Let $E/\mathbf{Q}$ be a CM elliptic curve and let $p\geq 5$ be a prime of good ordinary reduction for $E$. Suppose that $L(E,s)$ vanishes at $s=1$ and has sign $+1$ in its functional equation, so in particular ${\rm ord}_{s=1}L(E,s)\geq 2$. In this paper we slightly modify a construction of Darmon--Rotger to define a generalised Kato class $\kappa_p\in{\rm Sel}(\mathbf{Q},V_pE)$, and prove the following rank two analogue of Kolyvagin's result: \[ \kappa_p\neq 0\quad\Longrightarrow\quad{\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2. \] Conversely, when ${\rm dim}_{\mathbf{Q}_p}{\rm Sel}(\mathbf{Q},V_pE)=2$ we show that $\kappa_p\neq 0$ if and only if the restriction map \[ {\rm Sel}(\mathbf{Q},V_pE)\rightarrow E(\mathbf{Q}_p)\hat{\otimes}\mathbf{Q}_p \] is nonzero. The proof of these results, which extend and strenghten similar results of the author with Hsieh in the non-CM case, exploit a new link between the nonvanishing of generalised Kato classes and a main conjecture in anticyclotomic Iwasawa theory.

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arXiv2022

An anticyclotomic Euler system for adjoint modular Galois representations

Let $K$ be an imaginary quadratic field and $p$ a prime split in $K$. In this paper we construct an anticyclotomic Euler system for the adjoint representation attached to elliptic modular forms base changed to $K$. We also relate our Euler system to a $p$-adic $L$-function deduced from the construction by Eischen-Wan and Eischen-Harris-Li-Skinner of $p$-adic $L$-functions for unitary groups. This allows us to derive new cases of the Bloch-Kato conjecture in rank zero, and a divisibility towards an Iwasawa main conjecture.

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arXiv2021

Iwasawa-Greenberg main conjecture for non-ordinary modular forms and Eisenstein congruences on $\mathrm{GU}(3,1)$

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, and a study of the semi-ordinary Eisenstein family.

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arXiv2021

The diagonal cycle Euler system for ${\rm GL}_2\times{\rm GL}_2$

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

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