Antonio Lei
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View article: GENERALISED MAZUR’S GROWTH NUMBER CONJECTURE
GENERALISED MAZUR’S GROWTH NUMBER CONJECTURE Open
Let F be a totally real field. Let $\mathsf {A}$ be a simple modular self-dual abelian variety defined over F . We study the growth of the corank of Selmer groups of $\mathsf {A}$ over $\mathbb {Z}_p$ -extensions of a complex multiplicatio…
View article: Mazur's Growth Number Conjecture in the Rank One Case
Mazur's Growth Number Conjecture in the Rank One Case Open
Let $p\geq 5$ be a prime number. Let $\mathsf{E}/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. Let $K$ be an imaginary quadratic field where $p$ splits, and such that the generalized Heegner hypothesis holds. Under …
View article: Isogeny graphs with level structures arrising from the Verschiebung map
Isogeny graphs with level structures arrising from the Verschiebung map Open
We enhance an isogeny graph of elliptic curves by incorporating level structures defined by bases of the kernels of iterates of the Verschiebung map. We extend several previous results on isogeny graphs with level structures defined by geo…
View article: On $\mathbb{Z}_p$-towers of graph coverings arising from a constant voltage assignment
On $\mathbb{Z}_p$-towers of graph coverings arising from a constant voltage assignment Open
We investigate properties of $\mathbb{Z}_p$-towers of graph coverings that arise from a constant voltage assignment. We prove the existence and uniqueness (up to isomorphisms) of such towers. Furthermore, we study the Iwasawa invariants of…
View article: On the Iwasawa Invariants of Mazur--Tate elements of elliptic curves at additive primes
On the Iwasawa Invariants of Mazur--Tate elements of elliptic curves at additive primes Open
We investigate the $λ$-invariants of Mazur--Tate elements of elliptic curves defined over the field of rational numbers at primes of additive reduction. We explain their growth and how these invariants relate to other better understood inv…
View article: On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers
On the structure of the Bloch--Kato Selmer groups of modular forms over anticyclotomic $\mathbf{Z}_p$-towers Open
Let $p$ be an odd prime number and let $K$ be an imaginary quadratic field in which $p$ is split. Let $f$ be a modular form with good reduction at $p$. We study the variation of the Bloch--Kato Selmer groups and the Bloch--Kato--Shafarevic…
View article: The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\mathbb{Z}_p$-extensions at inert primes
The growth of Tate-Shafarevich groups of $p$-supersingular elliptic curves over anticyclotomic $\mathbb{Z}_p$-extensions at inert primes Open
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption th…
View article: On ordinary isogeny graphs with level structures
On ordinary isogeny graphs with level structures Open
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,…
View article: Artin formalism for $p$-adic $L$-functions of modular forms at non-ordinary primes
Artin formalism for $p$-adic $L$-functions of modular forms at non-ordinary primes Open
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 $L$-fu…
View article: Anticyclotomic Iwasawa theory of abelian varieties of GL2-type at non-ordinary primes
Anticyclotomic Iwasawa theory of abelian varieties of GL2-type at non-ordinary primes Open
Let p≥5 be a prime number, E/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 I…
View article: On a conjecture of Mazur predicting the growth of Mordell--Weil ranks in $\mathbb{Z}_p$-extensions
On a conjecture of Mazur predicting the growth of Mordell--Weil ranks in $\mathbb{Z}_p$-extensions Open
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…
View article: Asymptotic formula for Tate–Shafarevich groups of p-supersingular elliptic curves over anticyclotomic extensions
Asymptotic formula for Tate–Shafarevich groups of p-supersingular elliptic curves over anticyclotomic extensions Open
Let p⩾5 be a prime number and E/Q an elliptic curve with good supersingular reduction at p. Under the generalized Heegner hypothesis, we investigate the p-primary subgroups of the Tate–Shafarevich groups of E over number fields contained i…
View article: Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes II
Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes II Open
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 $K$, Castella and Wan f…
View article: ON THE DISTRIBUTION OF IWASAWA INVARIANTS ASSOCIATED TO MULTIGRAPHS
ON THE DISTRIBUTION OF IWASAWA INVARIANTS ASSOCIATED TO MULTIGRAPHS Open
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…
View article: On towers of Isogeny graphs with full level structure
On towers of Isogeny graphs with full level structure Open
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 f…
View article: A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes
A remark on the characteristic elements of anticyclotomic Selmer groups of elliptic curves with complex multiplication at supersingular primes Open
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 $\mathb…
View article: On the Zeta functions of supersingular isogeny graphs and modular curves
On the Zeta functions of supersingular isogeny graphs and modular curves Open
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 Iha…
View article: On ordinary isogeny graphs with level structure
On ordinary isogeny graphs with level structure Open
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 $…
View article: Growth of $p$-parts of ideal class groups and fine Selmer groups in $\mathbb{Z}_q$-extensions with $p\neq q$
Growth of $p$-parts of ideal class groups and fine Selmer groups in $\mathbb{Z}_q$-extensions with $p\neq q$ Open
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 hypoth…
View article: Non-vanishing modulo $p$ of Hecke $L$-values over imaginary quadratic fields
Non-vanishing modulo $p$ of Hecke $L$-values over imaginary quadratic fields Open
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,…
View article: On the Iwasawa invariants of BDP Selmer groups and BDP P-adic L-fucntions
On the Iwasawa invariants of BDP Selmer groups and BDP P-adic L-fucntions Open
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…
View article: BCM volume 65 issue 4 Cover and Front matter
BCM volume 65 issue 4 Cover and Front matter Open
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View article: On the BDP Iwasawa main conjecture for modular forms
On the BDP Iwasawa main conjecture for modular forms Open
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…
View article: Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes
Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes Open
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 f…
View article: Lambda-invariants of Mazur--Tate elements attached to Ramanujan's tau function and congruences with Eisenstein series
Lambda-invariants of Mazur--Tate elements attached to Ramanujan's tau function and congruences with Eisenstein series Open
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…
View article: Studying Hilbert's 10th problem via explicit elliptic curves
Studying Hilbert's 10th problem via explicit elliptic curves Open
N.García-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. We improve their proporti…
View article: On the distribution of Iwasawa invariants associated to multigraphs
On the distribution of Iwasawa invariants associated to multigraphs Open
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 an…