Preston Wake

activity

20122026

most citedThe Eisenstein ideal with squarefree level

19 citations · 80 across the 16 of their papers we have counts for

collaborators

18 papers

math.NT2026

Eisenstein congruences in tame families and the class group of a metabelian extension

Catherine Hsu, Alice Pozzi, Preston Wake +1

Ribet's 1976 proof of the converse to Herbrand's theorem created a new paradigm in algebraic number theory by illustrating that unramified abelian Galois extensions of number field…

math.NT2025

The Eisenstein ideal at prime-square level has constant rank

Jaclyn Lang, Preston Wake

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 $Γ_{0} (N^{2})$ whose $ℓ$ …

math.NT2022★ 2 cited

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

Catherine Hsu, Preston Wake, Carl Wang-Erickson

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 th…

math.NT2022

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

Catherine Hsu, Preston Wake, Carl Wang-Erickson

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…

math.NT2022★ 3 cited

Another look at rational torsion of modular Jacobians

Kenneth A. Ribet, Preston Wake

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 conjectu…

math.NT2021

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

Jaclyn Lang, Preston Wake

We show that for primes $N, p \geq 5$ with $N \equiv - 1 mod p$ , the class number of $Q (N^{1/ p})$ is divisible by $p$ . Our methods are via congruences between Eisenstein…

math.NT2020★ 5 cited

Generalized Bockstein maps and Massey products

Yeuk Hay Joshua Lam, Yuan Liu, Romyar Sharifi +2

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 p…

math.NT2020

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

Preston Wake

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 $extra reducibility$ : the Eisenst…

math.NT2018★ 19 cited

The Eisenstein ideal with squarefree level

Preston Wake, Carl Wang-Erickson

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…

math.NT2017★ 6 cited

Deformation conditions for pseudorepresentations

Preston Wake, Carl Wang-Erickson

Given a property of representations satisfying a basic stability condition, Ramakrishna developed a variant of Mazur's Galois deformation theory for representations with that prope…

math.NT2017★ 14 cited

The rank of Mazur's Eisenstein ideal

Preston Wake, Carl Wang-Erickson

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 co…

math.NT2015

Primitive elements in $p$ -divisible groups

Robert Kottwitz, Preston Wake

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. Pri…

math.NT2015★ 10 cited

Ordinary pseudorepresentations and modular forms

Preston Wake, Carl Wang-Erickson

In this short note, we observe that the techniques of our recent work "Pseudo-modularity and Iwasawa theory" can be used to provide a new proof of some of the residually reducible…

math.NT2015★ 3 cited

Full level structures revisited: pairs of roots of unity

Preston Wake

We introduce a notion of full level structure for the group scheme $μ_{p} \times μ_{p}$ , and show that scheme of full level structures is flat over $Z$ .

math.NT2015★ 17 cited

Pseudo-modularity and Iwasawa theory

Preston Wake, Carl Wang-Erickson

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,…

math.NT2014★ 1 cited

Eisenstein Hecke algebras and conjectures in Iwasawa theory

Preston Wake

We formulate a weak Gorenstein property for the Eisenstein component of the p-adic Hecke algebra associated to modular forms. We show that this weak Gorenstein property holds if an…

math.NT2013

The Lambda-adic Eichler-Shimura isomorphism and p-adic etale cohomology

Preston Wake

We give a new proof of Ohta's Lambda-adic Eichler-Shimura isomorphism using p-adic Hodge theory and the results of Bloch-Kato and Hyodo on p-adic etale cohomology.

math.NT2012

Hecke algebras associated to $Λ$ -adic modular forms

Preston Wake

We show that if an Eisenstein component of the $p$ -adic Hecke algebra associated to modular forms is Gorenstein, then it is necessary that the plus-part of a certain ideal class gr…

Eisenstein congruences in tame families and the class group of a metabelian extension

The Eisenstein ideal at prime-square level has constant rank

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

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

Another look at rational torsion of modular Jacobians

A modular construction of unramified p-extensions of Q(N1/p)

Generalized Bockstein maps and Massey products

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

The Eisenstein ideal with squarefree level

Deformation conditions for pseudorepresentations

The rank of Mazur's Eisenstein ideal

Primitive elements in p-divisible groups

Ordinary pseudorepresentations and modular forms

Full level structures revisited: pairs of roots of unity

Pseudo-modularity and Iwasawa theory

Eisenstein Hecke algebras and conjectures in Iwasawa theory

The Lambda-adic Eichler-Shimura isomorphism and p-adic etale cohomology

Hecke algebras associated to Λ-adic modular forms

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

Primitive elements in $p$ -divisible groups

Hecke algebras associated to $Λ$ -adic modular forms