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Antonio Lei

Publications and source records attributed to Antonio Lei.

At least 37 records · Page 2Linked to original sources
arXiv2024

On a conjecture of Mazur predicting the growth of Mordell--Weil ranks in $\mathbb{Z}_p$-extensions

Let $p$ be an odd prime. We study Mazur's conjecture on the growth of the Mordell--Weil ranks of an elliptic curve $E/\mathbb{Q}$ over $\mathbb{Z}_p$-extensions of an imaginary quadratic field, where $p$ is a prime of good reduction for $E$. In particular, we obtain criteria that may be checked through explicit calculation, thus allowing for the verification of Mazur's conjecture in specific examples.

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arXiv2024

On towers of Isogeny graphs with full level structure

Let $p,q,l$ be three distinct prime numbers and let $N$ be a positive integer coprime to $pql$. For an integer $n\ge 0$, we define the directed graph $X_l^q(p^nN)$ whose vertices are given by isomorphism classes of elliptic curves over a finite field of characteristic $q$ equipped with a level $p^nN$ structure. The edges of $X_l^q(p^nN)$ are given by $l$-isogenies. We are interested in when the connected components of $X_l^q(p^nN)$ give rise to a tower of Galois covers as $n$ varies. We show that only in the supersingular case we do get a tower of Galois covers. We also study similar towers of isogeny graphs given by oriented supersingular curves, as introduced by Colò-Kohel, enhanced with a level structure.

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arXiv2024

On ordinary isogeny graphs with level structure

Let $l$ and $p$ be two distinct prime numbers. We study $l$-isogeny graphs of ordinary elliptic curves defined over a finite field of characteristic $p$, together with a level structure. Firstly, we show that as the level varies over all $p$-powers, the graphs form an Iwasawa-theoretic abelian $p$-tower, which can be regarded as a graph-theoretical analogue of the Igusa tower of modular curves. Secondly, we study the structure of the crater of these graphs, generalizing previous results on volcano graphs. Finally, we solve an inverse problem of graphs arising from the crater of $l$-isogeny graphs with level structures, partially generalizing a recent result of Bambury, Campagna and Pazuki.

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arXiv2024

Lambda-invariants of Mazur--Tate elements attached to Ramanujan's tau function and congruences with Eisenstein series

Let $p\in\{3,5,7\}$ and let $Δ$ denote the weight twelve modular form arising from Ramanujan's tau function. We show that $Δ$ is congruent to an Eisenstein series $E_{k,χ, ψ}$ modulo $p$ for explicit choices of $k$ and Dirichlet characters $χ$ and $ψ$. We then prove formulae describing the Iwasawa invariants of the Mazur--Tate elements attached to $Δ$, confirming numerical data gathered by the authors in a previous work.

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arXiv2024

Artin formalism for $p$-adic $L$-functions of modular forms at non-ordinary primes

Let $p$ be an odd prime number. Let $f$ be a normalized Hecke eigen-cuspform that is non-ordinary at $p$. Let $K$ be an imaginary quadratic field in which $p$ splits. We study the Artin formalism for the two-variable signed $p$-adic $L$-functions attached to $f$ over $K$. In particular, we give evidence of a prediction made by Castella--Ciperiani--Skinner--Sprung.

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arXiv2023

Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes

Let $p\ge 5$ be a prime number, $E/\mathbb{Q}$ an elliptic curve with good supersingular reduction at $p$ and $K$ an imaginary quadratic field such that the root number of $E$ over $K$ is $+1$. When $p$ is split in $K$, Darmon and Iovita formulated the plus and minus Iwasawa main conjectures for $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$, and proved one-sided inclusion: an upper bound for plus and minus Selmer groups in terms of the associated $p$-adic $L$-functions. We generalize their results to two new settings: 1. Under the assumption that $p$ is split in $K$ but without assuming $a_p(E)=0$, we study Sprung-type Iwasawa main conjectures for abelian varieties of $\mathrm{GL}_2$-type, and prove an analogous inclusion. 2. We formulate, relying on the recent work of the first named author with Kobayashi and Ota, plus and minus Iwasawa main conjectures for elliptic curves when $p$ is inert in $K$, and prove an analogous inclusion.

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arXiv2023

On the Zeta functions of supersingular isogeny graphs and modular curves

Let $p$ and $q$ be distinct prime numbers, with $q\equiv 1\pmod{12}$. Let $N$ be a positive integer that is coprime to $pq$. We prove a formula relating the Hasse--Weil zeta function of the modular curve $X_0(qN)_{\mathbb{F}_q}$ to the Ihara zeta function of the $p$-isogeny graphs of supersingular elliptic curves defined over $\overline{\mathbb{F}_q}$ equipped with a $Γ_0(N)$-level structure. When $N=1$, this recovers a result of Sugiyama.

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arXiv2023

On the Iwasawa invariants of BDP Selmer groups and BDP P-adic L-fucntions

Let $p$ be an odd prime. Let $f_1$ and $f_2$ be weight-two Hecke eigen-cuspforms with isomorphic residual Galois representations at $p$. Greenberg--Vatsal and Emerton--Pollack--Weston showed that if $p$ is a good ordinary prime for the two forms, the Iwasawa invariants of their $p$-primary Selmer groups and $p$-adic $L$-functions over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ are closely related. The goal of this article is to generalize these results to the anticyclotomic setting. More precisely, let $K$ be an imaginary quadratic field where $p$ splits. Suppose that the generalized Heegner hypothesis holds with respect to both $(f_1,K)$ and $(f_2,K)$. We study relations between the Iwasawa invariants of the BDP Selmer groups and the BDP $p$-adic $L$-functions of $f_1$ and $f_2$.

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arXiv2023

On the distribution of Iwasawa invariants associated to multigraphs

Let $\ell$ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell$-towers of multigraphs. In this context, growth patterns are realized by certain analogues of Iwasawa invariants, which depend on the prime $\ell$ and the abelian $\ell$-tower of multigraphs. We formulate and study statistical questions about the behaviour of the Iwasawa $μ$ and $λ$ invariants.

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arXiv2023

Algebraic structure and characteristic ideals of fine Mordell--Weil groups and plus/minus Mordell--Weil groups

Given an elliptic curve defined over a number field $F$, we study the algebraic structure and prove a control theorem for Wuthrich's fine Mordell--Weil groups over a $\mathbb{Z}_p$-extension of $F$, generalizing results of Lee on the usual Mordell--Weil groups. In the case where $F=\mathbb{Q}$, we show that the characteristic ideal of the Pontryagin dual of the fine Mordell--Weil group over the cyclotomic $\mathbb{Z}_p$-extension coincides with Greenberg's prediction for the characteristic ideal of the dual fine Selmer group. If furthermore $E$ has good supersingular reduction at $p$ with $a_p(E)=0$, we generalize Wuthrich's fine Mordell--Weil groups to define "plus and minus Mordell--Weil groups". We show that the greatest common divisor of the characteristic ideals of the Pontryagin duals of these groups coincides with Kurihara--Pollack's prediction for the greatest common divisor of the plus and minus $p$-adic $L$-functions.

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arXiv2023

A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes

Let $p\ge5$ be a prime number. Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication by an imaginary quadratic field $K$ such that $p$ is inert in $K$ and that $E$ has good reduction at $p$. Let $K_\infty$ be the anticyclotomic $\mathbb{Z}_p$-extension of $K$. Agboola--Howard defined Kobayashi-type signed Selmer groups of $E$ over $K_\infty$ and showed that exactly one of them is cotorsion over the corresponding Iwasawa algebra. In this short note, we discuss a link between the characteristic ideals of the cotorsion signed Selmer group and the fine Selmer group building on a recent breakthrough of Burungale--Kobayashi--Ota on the structure of local points.

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arXiv2023

Iwasawa theory of twists of elliptic modular forms over imaginary quadratic fields at inert primes

Our primary goal in this article is to study the Iwasawa theory for semi-ordinary families of automorphic forms on $\mathrm{GL}_2\times\mathrm{Res}_{K/\mathbb{Q}}\mathrm{GL}_1$, where $K$ is an imaginary quadratic field where the prime $p$ is inert. We prove divisibility results towards Iwasawa main conjectures in this context, utilizing the optimized signed factorization procedure for Perrin-Riou functionals and Beilinson--Flach elements for a family of Rankin--Selberg products of $p$-ordinary forms with a fixed $p$-non-ordinary modular form. The optimality enables an effective control on the $μ$-invariants of Selmer groups and $p$-adic $L$-functions as the modular forms vary in families, which is crucial for our patching argument to establish one divisibility in an Iwasawa main conjecture in three variables.

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arXiv2023

On the BDP Iwasawa main conjecture for modular forms

Let $K$ be an imaginary quadratic field where $p$ splits, $p\geq5$ a prime number and $f$ an eigen-newform of even weight and level $N>3$ that is coprime to $p$. Under the Heegner hypothesis, Kobayashi--Ota showed that one inclusion of the Iwasawa main conjecture of $f$ involving the Bertolini--Darmon--Prasanna $p$-adic $L$-function holds after tensoring by $\mathbb{Q}_p$. Under certain hypotheses, we improve upon Kobayahsi--Ota's result and show that the same inclusion holds integrally. Our result implies the vanishing of the Iwasawa $μ$-invariants of several anticyclotomic Selmer groups.

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arXiv2023

Growth of $p$-parts of ideal class groups and fine Selmer groups in $\mathbb{Z}_q$-extensions with $p\neq q$

Fix two distinct odd primes $p$ and $q$. We study "$p\ne q$" Iwasawa theory in two different settings. Let $K$ be an imaginary quadratic field of class number 1 such that both $p$ and $q$ split in $K$. We show that under appropriate hypotheses, the $p$-part of the ideal class groups is bounded over finite subextensions of an anticyclotomic $\mathbb{Z}_q$-extension of $K$. Let $F$ be a number field and let $A_{/F}$ be an abelian variety with $A[p]\subseteq A(F)$. We give sufficient conditions for the $p$-part of the fine Selmer groups of $A$ over finite subextensions of a $\mathbb{Z}_q$-extension of $F$ to stabilize.

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arXiv2023

Non-vanishing modulo $p$ of Hecke $L$-values over imaginary quadratic fields

Let $p$ and $q$ be two distinct odd primes. Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split. Let $Ψ$ be a Hecke character over $K$ of infinity type $(k,j)$ with $0\le-j< k$. Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters $κ$ over $K$ which factor through the $\mathbb{Z}_q^2$-extension of $K$, the $p$-adic valuation of the algebraic part of the $L$-value $L(\overline{κΨ},k+j)$ is a constant independent of $κ$. In addition, when $j=0$ and certain technical hypothesis holds, this constant is zero.

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arXiv2023

The non-$\ell$-part of the number of spanning trees in abelian $\ell$-towers of multigraphs

Let $\ell$ and $p$ be two distinct primes. We study the $p$-adic valuation of the number of spanning trees in an abelian $\ell$-tower of connected multigraphs. This is analogous to the classical theorem of Washington--Sinnott on the growth of the $p$-part of the class group in a cyclotomic $\mathbb{Z}_\ell$-extension of abelian extensions of $\mathbb{Q}$. Furthermore, we show that under certain hypotheses, the number of primes dividing the number of spanning trees is unbounded in such a tower.

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arXiv2022

Iwasawa theory of automorphic representations of $\mathrm{GL}_{2n}$ at non-ordinary primes

Let $Π$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $Π$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $Π$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $Π$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\mathbb{Q}$ attached to the $p$-adic Galois representation of $Π$ and formulate Iwasawa main conjectures in the spirit of Kobayashi's plus and minus main conjectures for $p$-supersingular elliptic curves.

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