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Preston Wake

Publications and source records attributed to Preston Wake.

At least 19 recordsLinked to original sources
arXiv2026

Counting level-raising congruences using modular representation theory

We introduce a new method for studying mod-$\ell$ congruences between eigenforms through the modular representation theory of $\mathrm{PGL}_2(\mathbb{F}_p)$. When $p\equiv \pm 1 \pmod{\ell}$, we use this theory to construct and describe extra structures on spaces of modular forms with $\Gamma_0(p^2)$-level at $p$ and a fixed mod-$\ell$ Galois representation. The structural results we obtain can be viewed as a refinement of classical level-raising theorems since they not only allow us to prove the existence of congruences, but also to count the number of such congruences. Our methods work equally well in the residually irreducible and residually reducible cases, allowing us to prove several new instances of congruences between Eisenstein series and cuspforms (as well as independently rederiving classical results of Mazur and more recent results of Lang--Wake). Notably, our approach proves these results without computing constant terms of Eisenstein series, without Galois deformation theory and $R=\mathbb{T}$ theorems, and without the Jacquet--Langlands correspondence.

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arXiv2025

Modular analogs of character formulas and minimal lifts of modular forms

If $f$ is a mod-$3$ eigenform of weight 2 and level $\Gamma_0(\ell^2)$ for a prime $\ell$ such that $\ell \equiv -1 \pmod{3}$, and $\ell$ is a vexing prime for $f$, we show that there is no obstruction to finding a minimal lift of $f$, but that there is an obstruction to finding a nonminimal lift. The key new ingredient that we prove is a modular analog of the standard character formula for a cuspidal representation of $\mathrm{GL}_2(\mathbb{F}_\ell)$, an enhancement that allows us to easily compute the group cohomology of a $3$-adic lattice in such a representation. In fact, we provide a general framework for proving such modular analogs for a broader class of representations using results of Brou\'e and Puig in modular representation theory. We show that this class includes certain Deligne--Lusztig representations and representations coming from higher-depth supercuspidal representations of $\mathrm{GL}_2$.

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arXiv2025

The Eisenstein ideal at prime-square level has constant rank

Let $N$ and $p$ be prime numbers with $p \geq 5$ such that $p || (N + 1)$. In a previous paper, we showed that there is a cuspform $f$ of weight 2 and level $\Gamma_0(N^2)$ whose $\ell$-th Fourier coefficient is congruent to $\ell + 1$ modulo a prime above $p$ for all primes $\ell$. In this paper, we prove that this form $f$ is unique up to Galois conjugacy, and the extension of $\mathbb{Z}_p$ generated by the coefficients of $f$ is exactly $\mathbb{Z}_p[\zeta_p + \zeta_p^{-1}]$. We also prove similar results when a higher power of $p$ divides $N + 1$.

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arXiv2024

Iwasawa invariants in residually reducible Hida families

We study the variation of $\mu$-invariants of modular forms in a cuspidal Hida family in the case that the family intersects an Eisenstein family. We allow for intersections that occur because of "trivial zeros" (that is, because $p$ divides an Euler factor) as in Mazur's Eisenstein ideal paper, and pay special attention to the case of the 5-adic family passing through the elliptic curve $X_0(11)$.

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arXiv2022

Explicit non-Gorenstein R=T via rank bounds I: Deformation theory

Ribet has proven remarkable results about non-optimal levels of residually reducible Galois representations. We focus on a non-optimal level $N$ that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. We prove a Galois-theoretic criterion for the deformation ring to be as small as possible -- that is, for there to be a unique newform of level $N$ with reducible residual representation. When this criterion is satisfied, we deduce an $R=\mathbb{T}$ theorem.

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arXiv2022

Explicit non-Gorenstein R=T via rank bounds II: Computational aspects

This is the second in a pair of papers about residually reducible Galois deformation rings with non-optimal level. In the first paper, we proved a Galois-theoretic criterion for the deformation ring to be as small as possible. This paper focuses on the computations needed to verify this criterion. We adapt a technique developed by Sharifi to compute number fields with twisted-Heisenberg Galois group and prescribed ramification, and compute the splitting behavior of primes in these extensions.

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arXiv2022

Another look at rational torsion of modular Jacobians

We study the rational torsion subgroup of the modular Jacobian $J_0(N)$ for $N$ a square-free integer. We give a new proof of a result of Ohta on a generalization of Ogg's conjecture: for a prime number $p \nmid 6N$, the $p$-primary part of the rational torsion subgroup equals that of the cuspidal subgroup. Whereas previous proofs of this result used explicit computations of the cardinalities of these groups, we instead use their structure as modules for the Hecke algebra.

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arXiv2021

A modular construction of unramified $p$-extensions of $\mathbb{Q}(N^{1/p})$

We show that for primes $N, p \geq 5$ with $N \equiv -1 \bmod p$, the class number of $\mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N \equiv -1 \bmod p$, there is always a cusp form of weight $2$ and level $\Gamma_0(N^2)$ whose $\ell$-th Fourier coefficient is congruent to $\ell + 1$ modulo a prime above $p$, for all primes $\ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree $p$ extension of $\mathbb{Q}(N^{1/p})$.

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arXiv2020

Generalized Bockstein maps and Massey products

Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of the rational numbers and to give a new proof of the vanishing of triple Massey products in Galois cohomology.

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arXiv2020

The Eisenstein ideal for weight k and a Bloch-Kato conjecture for tame families

We study the Eisenstein ideal for modular forms of even weight $k>2$ and prime level $N$. We pay special attention to the phenomenon of $\mathit{extra \ reducibility}$: the Eisenstein ideal is strictly larger than the ideal cutting out reducible Galois representations. We prove a modularity theorem for these extra reducible representations. As consequences, we relate the derivative of a Mazur-Tate $L$-function to the rank of the Hecke algebra, generalizing a theorem of Merel, and give a new proof of a special case of an equivariant main conjecture of Kato. In the second half of the paper, we recall Kato's formulation of this main conjecture in the case of a family of motives given by twists by characters of conductor $N$ and $p$-power order and its relation to other formulations of the equivariant main conjecture.

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arXiv2019

The rank of Mazur's Eisenstein ideal

We use pseudodeformation theory to study Mazur's Eisenstein ideal. Given prime numbers $N$ and $p>3$, we study the Eisenstein part of the $p$-adic Hecke algebra for $Γ_0(N)$. We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, answering a question of Mazur and generalizing a result of Calegari-Emerton. We also also give new proofs of Merel's result on this rank and of Mazur's results on the structure of the Hecke algebra.

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arXiv2019

Deformation conditions for pseudorepresentations

Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur's Galois deformation theory for representations with that property. We introduce an axiomatic definition of pseudorepresentations with such a property. Among other things, we show that pseudorepresentations with a property enjoy a good deformation theory, generalizing Ramakrishna's theory to pseudorepresentations.

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arXiv2018

The Eisenstein ideal with squarefree level

We use pseudodeformation theory to study the analogue of Mazur's Eisenstein ideal with certain squarefree levels. Given a prime number $p>3$ and a squarefree number $N$ satisfying certain conditions, we study the Eisenstein part of the $p$-adic Hecke algebra for $\Gamma_0(N)$, and show that it is a local complete intersection and isomorphic to a pseudodeformation ring. We also show that in certain cases, the Eisenstein ideal is not principal and that the cuspidal quotient of the Hecke algebra is not Gorenstein. As a corollary, we prove that "multiplicity one" fails for the modular Jacobian in these cases. In a particular case, this proves a conjecture of Ribet.

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arXiv2017

Pseudo-modularity and Iwasawa theory

We prove, assuming Greenberg's conjecture, that the ordinary eigencurve is Gorenstein at an intersection point between the Eisenstein family and the cuspidal locus. As a corollary, we obtain new results on Sharifi's conjecture. This result is achieved by constructing a universal ordinary pseudodeformation ring and proving an $R = \mathbb T$ result.

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arXiv2017

Primitive elements in $p$-divisible groups

We introduce the notion of primitive elements in arbitrary truncated $p$-divisible groups. By design, the scheme of primitive elements is finite and locally free over the base. Primitive elements generalize the "points of exact order $N$," developed by Drinfeld and Katz-Mazur for elliptic curves.

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