About

Francesc Castella is an Associate Professor in the Department of Mathematics at the University of California, Santa Barbara. Prior to his current position, he served as an Instructor at Princeton University and as a Hedrick Assistant Professor at UCLA. His research is partially supported by the National Science Foundation and the American Mathematical Society Centennial Research Fellowship. He is actively involved in organizing the Seminar on Geometry and Arithmetic. Castella's research interests lie primarily in Number Theory and Arithmetic Geometry, with a particular focus on p-adic L-functions, Euler systems, and Iwasawa theory. His work contributes to advancing the understanding of these areas through both theoretical developments and collaborative research.

Research topics

• Combinatorics
• Mathematics
• Pure mathematics
• Discrete mathematics

Selected publications

• Nonvanishing of Generalised Kato Classes and Iwasawa Main Conjectures

  Springer proceedings in mathematics & statistics · 2026-01-01

  preprintOpen access1st authorCorresponding
  PublisherOA PDFDOI

• On refined nonvanishing conjectures by Kurihara and Kolyvagin

  arXiv (Cornell University) · 2026-01-20

  preprintOpen access1st authorCorresponding

  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 $Λ$-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.

  PublisherDOI

• On refined nonvanishing conjectures by Kurihara and Kolyvagin

  ArXiv.org · 2026-01-20

  articleOpen access1st authorCorresponding

  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 $Λ$-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.

  PublisherOA PDF

• Non-vanishing of Kolyvagin systems and Iwasawa theory

  Cambridge Journal of Mathematics · 2026-01-01

  article

  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.

  PublisherDOI

• Tamagawa number conjecture for CM modular forms and Rankin–Selberg convolutions

  Proceedings of the London Mathematical Society · 2025-10-01

  article1st authorCorresponding

  Abstract Let be an elliptic curve defined over a number field with complex multiplication by the ring of integers of an imaginary quadratic field such that the torsion points of generate over an abelian extension of . In this paper, we prove the ‐part of the Birch–Swinnerton‐Dyer formula for in analytic rank 1 for primes split in . This was previously known for by work of Rubin as a consequence of his proof of Mazur's Main Conjecture for rational CM elliptic curves, but the problem for remained wide open. The approach introduced in this paper also yields a proof of similar results for CM abelian varieties and for CM modular forms, as well as an analog in this setting of Skinner's ‐converse to the theorem of Gross–Zagier and Kolyvagin.

  PublisherDOI

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

  ArXiv.org · 2025-02-26

  preprintOpen access1st authorCorresponding

  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 Dieudonné module of $E$.

  PublisherOA PDFDOI

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

  ArXiv.org · 2025-01-25

  preprintOpen access

  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--Nekovář--Skinner in the anticyclotomic setting, the construction yields new applications to the Bloch--Kato conjecture and the Iwasawa Main Conjecture.

  PublisherOA PDFDOI

• Critical Λ-adic modular forms and biordinarycomplexes

  Tunisian Journal of Mathematics · 2025-09-12

  preprintOpen access1st authorCorresponding

  We produce a flat $Λ$-module of $Λ$-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 $Λ$. 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.

  PublisherOA PDFDOI

• Diagonal cycles and anticyclotomic Iwasawa theory of modular forms

  Journal of the European Mathematical Society · 2025-11-28

  articleOpen access1st authorCorresponding

  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–Nekovář–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.

  PublisherDOI

• Mazur’s main conjecture at Eisenstein primes

  Mathematische Annalen · 2025-09-26 · 3 citations

  article1st author
  PublisherDOI

Recent grants

• Euler Systems, p-adic Deformations, and the Birch-Swinnerton-Dyer Conjecture

  NSF · $138k · 2018–2019

• Euler Systems, p-adic Deformations, and the Birch-Swinnerton-Dyer Conjecture

  NSF · $88k · 2019–2022

• Elliptic Curves, p-adic Deformations, and Iwasawa Theory

  NSF · $180k · 2021–2024

Frequent coauthors

• Óscar Rivero

  Universidade de Santiago de Compostela

  28 shared

• Raúl Alonso

  Mathematical Sciences Research Institute

  26 shared

• Ming-Lun Hsieh

  9 shared

• Xin Wan

  Chinese Academy of Sciences

  9 shared

• Giada Grossi

  Sorbonne Université

  8 shared

• Chan-Ho Kim

  Korea Institute for Advanced Study

  6 shared

• Christopher Skinner

  6 shared

• Matteo Longo

  6 shared