arXiv subjects

Óscar Rivero

Publications and source records attributed to Óscar Rivero.

13 recordsLinked to original sources

arXiv2025

Eisenstein degeneration of Beilinson--Kato classes and circular units

The aim of this note is to explore the Euler system of Beilinson--Kato elements in families passing through the critical $p$-stabilization of an Eisenstein series attached to two Dirichlet characters $(ψ,τ)$. In this context, we establish an explicit connection with the system of circular units, utilizing suitable factorization formulas in a situation where several of the $p$-adic $L$-functions vanish. In that regard, our main results may be seen as an Euler system incarnation of the factorization formula of Bellaïche and Dasgupta. One of the most significant aspects is that, depending on the parity of $ψ$ and $τ$, different phenomena arise; while some can be addressed with our methods, others pose new questions. Finally, we discuss analogous results in the framework of Beilinson--Flach classes.

math.NT↗

arXiv2025

Wall-crossing and $p$-adic Artin formalism for ${\rm GSp}_4 \times {\rm GL}_2 \times {\rm GL}_2$

The goal of this article is to develop a $p$-adic Artin formalism in the context of $p$-adic families of automorphic forms on ${\rm GSp}_4 \times {\rm GL}_2 \times {\rm GL}_2$. Our treatment is guided by the (double) wall-crossing principle, emphasising an interplay between arithmetic GGP and $p$-adic explicit GGP formulae. Although the picture we present remains largely conjectural, we provide evidence in favour of our conjectures (a) in terms of algebraic $p$-adic $L$-functions, and (b) in endoscopic scenarios.

math.NT↗

arXiv2025

Anticyclotomic diagonal classes and Beilinson--Flach elements

We present a comparison between the anticyclotomic Euler system of diagonal cycles associated with the convolution of two modular forms and the cyclotomic Beilinson--Flach Euler system. This extends the seminal work of Bertolini, Darmon, and Venerucci, who established a link between (anticyclotomic) Heegner points and the Beilinson--Kato system. Our approach hinges on a detailed analysis of $p$-adic $L$-functions and Perrin-Riou maps and exploits the Eisenstein degeneration of diagonal cycles along Hida families, working with a CM family which specializes to an irregular Eisenstein series in weight one. We use these results to derive some arithmetic applications.

math.NT↗

arXiv2025

On $p$-adic $L$-functions for $\text{GSp}_4 \times \text{GL}_2$

We use higher Coleman theory to construct a new $p$-adic $L$-function for $\text{GSp}_4 \times \text{GL}_2$. While previous works by the first author, Pilloni, Skinner and Zerbes had considered the $p$-adic variation of classes in the $H^2$ of Shimura varieties for $\text{GSp}_4$, in this note we explore the interpolation of classes in the $H^1$, which allows us to access to a different range of weights. Further, we show an interpolation property in terms of complex $L$-values using the algebraicity results established in previous work by the authors.

math.NT↗

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

math.NT↗

arXiv2024

Eisenstein degeneration of Euler systems

We discuss the theory of Coleman families interpolating critical-slope Eisenstein series. We apply it to study degeneration phenomena at the level of Euler systems. In particular, this allows us to prove relations between Kato elements, Beilinson--Flach classes and diagonal cycles, and also between Heegner cycles and elliptic units. We expect that this method could be extended to construct new instances of Euler systems.

math.NT↗

arXiv2023

Eisenstein congruences among Euler systems

We investigate Eisenstein congruences between the so-called Euler systems of Garrett--Rankin--Selberg type. This includes the cohomology classes of Beilinson--Kato, Beilinson--Flach and diagonal cycles. The proofs crucially rely on different known versions of the Bloch--Kato conjecture, and are based on the study of the Perrin-Riou formalism and the comparison between the different $p$-adic $L$-functions.

math.NT↗

arXiv2023

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.

math.NT↗

arXiv2023

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.

math.NT↗

arXiv2023

Algebraicity of $L$-values for $\text{GSp}_4 \times \text{GL}_2$ and $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$

We prove algebraicity results for critical $L$-values attached to the group $\text{GSp}_4 \times \text{GL}_2$, and for Gan--Gross--Prasad periods which are conjecturally related to central $L$-values for $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$. Our result for $\text{GSp}_4 \times \text{GL}_2$ gives a new proof (by a very different method) of a recent result of Morimoto, and will be used in a sequel paper to construct a new $p$-adic $L$-function for $\text{GSp}_4 \times \text{GL}_2$. The results for Gross--Prasad periods appear to be new. A key aspect is the computation of certain archimedean zeta integrals, whose $p$-adic counterparts are also studied in this note.

math.NT↗

arXiv2021

Motivic congruences and Sharifi's conjecture

Let $f$ be a cuspidal eigenform of weight two and level $N$, let $p\nmid N$ be a prime at which $f$ is congruent to an Eisenstein series and let $V_f$ denote the $p$-adic Tate module of $f$. Beilinson constructed a class $κ_f\in H^1(\mathbb Q,V_f(1))$ arising from the cup-product of two Siegel units and proved a striking relationship with the first derivative $L'(f,0)$ at the near central point $s=0$ of the $L$-series of $f$, which led him to formulate his celebrated conjecture. In this note we prove two congruence formulae relating the "motivic part" of $L'(f,0) \,(\mathrm{mod} \, p)$ and $L''(f,0) \,(\mathrm{mod} \, p)$ with circular units. The proofs make use of delicate Galois properties satisfied by various integral lattices within $V_f$ and exploits Perrin-Riou's, Coleman's and Kato's work on the Euler systems of circular units and Beilinson--Kato elements and, most crucially, the work of Sharifi, Fukaya--Kato and Ohta.

math.NT↗

arXiv2020

The exceptional zero phenomenon for elliptic units

The exceptional zero phenomenon has been widely studied in the realm of $p$-adic $L$-functions, where the starting point lies in the foundational work of Mazur, Tate and Teitelbaum. This phenomenon also appears in the study of Euler systems, which comes as no surprise given the interaction between these two settings. When this occurs, one is led to study higher order derivatives of the Euler system in order to extract the arithmetic information which is usually encoded in the explicit reciprocity laws. In this work, we focus on the elliptic units of an imaginary quadratic field and study this exceptional zero phenomenon, proving an explicit formula relating the logarithm of a {\it derived} elliptic unit either to special values of Katz's two variable $p$-adic $L$-function or to its derivatives. Further, we interpret this fact in terms of an $\mathcal L$-invariant, and relate this result to other approaches to the exceptional zero phenomenon concerning Heegner points and Beilinson--Flach elements.

math.NT↗

arXiv2020

Derived Beilinson-Flach elements and the arithmetic of the adjoint of a modular form

Kings, Lei, Loeffler and Zerbes constructed a three-variable Euler system $κ({\bf g},{\bf h})$ of Beilinson-Flach elements associated to a pair of Hida families $({\bf g},{\bf h})$ and exploited it to obtain applications to the arithmetic of elliptic curves by specializing the Euler system to points of weights $(2,1,1)$. The aim of this article is showing that this Euler system also encodes arithmetic information at points of weights $(1,1,0)$, concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when ${\bf g}$ and ${\bf h}$ specialize in weight $1$ to $p$-stabilizations of eigenforms such that one is dual of another. We encounter an exceptional zero phenomenon which forces the specialization of $κ({\bf g}, {\bf h})$ at $(1,1,0)$ to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of a conjecture of Darmon, Lauder and Rotger on iterated integrals and another conjecture of Darmon and Rotger for Beilinson-Flach elements in the adjoint setting. The main point of this paper is that the methods of previous works, where the above conjectures are proved when the weight $1$ eigenforms have CM, do not apply to our setting and new ideas are required. Here, a factorization of $p$-adic $L$-functions is not available due to the lack of critical points. Instead we resort to the principle of improved Euler systems and $p$-adic $L$-functions to reduce our problems to questions which can be resolved using Galois deformation theory. We expect this approach may be adapted to prove other cases of the elliptic Stark conjecture and of its generalizations that are appearing in the literature.

math.NT↗