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Rylan Gajek-Leonard

Publications and source records attributed to Rylan Gajek-Leonard.

8 recordsLinked to original sources
arXiv2026

A canonical construction of signed $p$-adic $L$-functions for non-ordinary modular forms of weight $\leq p+1$

Fix an odd prime $p$ and let $f$ be a $p$-non-ordinary cuspidal eigen-newform of weight $2\leq k\leq p+1$. We construct a pair of bounded $p$-adic $L$-functions associated to $f$ by decomposing the unbounded $p$-adic $L$-functions in terms of an explicit logarithm-type matrix whose definition does not require $p$-adic Hodge theory. Using this decomposition, we compute asymptotic formulas for the Iwasawa invariants of Mazur--Tate elements attached to non-ordinary forms of weight $\leq p+1$. As a corollary, we obtain a relation between the signed Iwasawa invariants of $p$-non-ordinary and $p$-congruent cuspforms of weights 2 and $p+1$, generalizing previous results in the $a_p=0$ case.

math.NT↗
arXiv2026

Mazur-Tate elements of non-ordinary modular forms with Serre weight larger than two

Fix an odd prime $p$ and let $f$ be a non-ordinary eigen-cuspform of weight $k$ and level coprime to $p$. Assuming $p>k-1$, we compute asymptotic formulas for the Iwasawa invariants of the Mazur-Tate elements attached to $f$ in terms of the corresponding invariants of the signed $p$-adic $L$-functions. By combining this with a version of mod $p$ multiplicity one, we also obtain descriptions of the $λ$-invariants of Mazur-Tate elements attached to certain higher weight modular forms with Serre weight $<p+1$, generalizing results of Pollack and Weston in the Serre weight 2 case.

math.NT↗
arXiv2025

Iwasawa invariants of modular forms with $a_p=0$

Fix a prime $p$ and a cuspidal newform $f$ of level coprime to $p$ with $a_p=0$. Attached to $f$ are signed $p$-adic $L$-functions $L_p^\pm(f)$ and Mazur-Tate elements $θ_n(f)$, both of which encode arithmetic data about $f$ along the cyclotomic $\mathbf{Z}_p$-extension of $\mathbf{Q}$. We compute the Iwasawa invariants of Mazur-Tate elements in terms of the corresponding invariants of the signed $p$-adic $L$-functions. As corollaries, we determine the $p$-adic valuation of critical values of the $L$-function of $f$, and describe a relation between the Iwasawa invariants of congruent modular forms of weights 2 and $p+1$. Our results provide an asymptotic method for computing the signed Iwasawa invariants attached to newforms of any weight $k\geq 2$ with $a_p=0$.

math.NT↗
arXiv2025

Stretching Newton polygons using pure polynomials

The $p$-adic Newton polygon is a visual tool that encodes information about the roots and factorization of a polynomial relative to a prime $p$. In this article, we investigate how the Newton polygon changes under polynomial composition. If $f$ and $g$ are polynomials with rational (or $p$-adic) coefficients and the Newton polygon of $g$ is pure (has only one segment), we show under some mild conditions that the Newton polygon of $f\circ g$ is the same as that of $f$, but stretched horizontally by $\operatorname{deg}(g)$. When $f=g$, this implies that all iterates of certain pure polynomials are irreducible, recovering a classical result of Robert Odoni on the irreducibility of iterated Eisenstein polynomials.

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arXiv2025

Entanglement of elliptic curves upon base extension

Fix distinct primes $p$ and $q$ and let $E$ be an elliptic curve defined over a number field $K$. The $(p,q)$-entanglement type of $E$ over $K$ is the isomorphism class of the group $\operatorname{Gal}(K(E[p])\cap K(E[q])/K)$. The size of this group measures the extent to which the image of the mod $pq$ Galois representation attached to $E$ fails to be a direct product of the mod $p$ and mod $q$ images. In this article, we study how the $(p,q)$-entanglement group varies over different base fields. We prove that for each prime $\ell$ dividing the greatest common divisor of the size of the mod $p$ and $q$ images, there are infinitely many fields $L/K$ such that the entanglement over $L$ is cyclic of order $\ell$. We also classify all possible $(2,q)$-entanglement types that can occur as the base field $L$ varies.

math.NT↗
arXiv2025

A bound on the $μ$-invariants of supersingular elliptic curves

Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be a prime of good supersingular reduction. Attached to $E$ are pairs of Iwasawa invariants $μ_p^\pm$ and $λ_p^\pm$ which encode arithmetic properties of $E$ along the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. A well-known conjecture of B. Perrin-Riou and R. Pollack asserts that $μ_p^\pm=0$. We provide support for this conjecture by proving that for any $\ell\geq 0$, we have $μ_p^\pm\leq 1$ for all but finitely many primes $p$ with $λ_p^\pm=\ell$. Assuming a recent conjecture of D. Kundu and A. Ray, our result implies that $μ_p^\pm\leq 1$ holds on a density 1 set of good supersingular primes for $E$.

math.NT↗
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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