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Eyal Kaplan

Publications and source records attributed to Eyal Kaplan.

At least 19 recordsLinked to original sources
arXiv2026

Preserving the Ultrapower Axiom by Forcing over Nonoverlapping Extender Models

We identify a class of simple yet nontrivial forcing notions that preserve the Ultrapower Axiom (UA) over canonical inner models, reaching the level of a strong cardinal. The forcings we consider are discrete product forcings whose index sets are chosen and spaced according to the structure of the underlying nonoverlapping extender sequence. As an application, we show that UA is consistent with the existence of a strong cardinal $κ$ such that $V\neq \text{HOD}_X$ for every $X\in V_κ$, answering a question of Goldberg.

math.LO↗
arXiv2026

The number of measures on very large measurable cardinals

We study the possible number of normal measures on a measurable cardinal in settings where inner model techniques are unavailable. Instead, we exploit consequences of the Ultrapower Axiom to obtain our theorems. We show that the classical Kimchi-Magidor result -that the first $n$ measurable cardinals can be strongly compact- can be combined with an arbitrary prescribed pattern for the number of normal measures they carry. We also prove that the first measurable cardinal above a supercompact cardinal can carry any given number of normal measures; the same conclusion is established for the first measurable limit of supercompact cardinals. As further applications of our techniques, we strengthen an unpublished theorem of Goldberg--Woodin and a theorem of Goldberg, Osinski, and Poveda. Our analysis circumvents both the reliance of Friedman--Magidor on core model methods and the limitations of the Prikry-type forcing iterations of Gitik--Kaplan.

math.LO↗
arXiv2025

A note on the Ketonen order and Lipschitz reducibility between ultrafilters

In his study of the Ultrapower Axiom (UA), Goldberg revealed a connection between UA and the determinacy of certain games that witness Lipschitz reducibility between ultrafilters. In particular, he analyzed the relationship between the Ketonen and Lipschitz orders - two natural extensions of the Mitchell order from normal measures to arbitrary $σ$-complete ultrafilters - and proved that the Lipschitz order extends the Ketonen order. He further observed that under UA the two orders coincide. Goldberg asked if it's consistent that the orders differ from each other. We show that the answer is positive. In fact, even the Weak Ultrapower Axiom does not imply that the Ketonen and Lipschitz orders coincide.

math.LO↗
arXiv2025

The number of normal measures, revisited

We present a new version of the Friedman-Magidor theorem: for every measurable cardinal $κ$ and $τ\leqκ^{++}$, there exists a forcing extension $V\subseteq V[G]$ such that any normal measure $U\in V$ on $κ$ has exactly $τ$ distinct lifts in $V[G]$, and every normal measure on $κ$ in $V[G]$ arises as such a lift. This version differs from the original Friedman-Magidor theorem in several notable ways. First, the new technique does not involve forcing over canonical inner models or rely on any fine-structural tools or assumptions, allowing it to be applied in the realm of large cardinals beyond the current reach of the inner model program. Second, in the case where $τ\leq κ^+$, all lifts of a normal measure $U\in V$ on $κ$ to $V[G]$ have the same ultrapower. Finally, the technique generalizes to a version of the Friedman-Magidor theorem for extenders. An additional advantage is that the forcing used is notably simple, relying only on nonstationary support product forcing.

math.LO↗
arXiv2025

Simple supercuspidal L-packets of split special orthogonal groups over dyadic fields

We consider the split special orthogonal group $\mathrm{SO}_{N}$ defined over a $p$-adic field. We determine the structure of any $L$-packet of $\mathrm{SO}_{N}$ containing a simple supercuspidal representation (in the sense of Gross--Reeder). We also determine its endoscopic lift to a general linear group. Combined with the explicit local Langlands correspondence for simple supercuspidal representations of general linear groups, this leads us to get an explicit description of the $L$-parameter as a representation of the Weil group of $F$. Our result is new when $p=2$ and our method provides a new proof even when $p\neq2$.

math.NT↗
arXiv2024

A Kunen-Like Model with a Critical Failure of the Continuum Hypothesis

We construct a model of the form $L[A,U]$ that exhibits the simplest structural behavior of $σ$-complete ultrafilters in a model of set theory with a single measurable cardinal $κ$ , yet satisfies $2^κ= κ^{++}$. This result establishes a limitation on the extent to which structural properties of ultrafilters can determine the cardinal arithmetic at large cardinals, and answers a question posed by Goldberg concerning the failure of the Continuum Hypothesis at a measurable cardinal in a model of the Ultrapower Axiom. The construction introduces several methods in extensions of embeddings theory and fine-structure-based forcing, designed to control the behavior of non-normal ultrafilters in generic extensions.

math.LO↗
arXiv2024

Doubling constructions: Global functoriality for non-generic cuspidal representations

We study the generalized doubling method for pairs of representations of $G\times GL_k$ where $G$ is a symplectic group, split special orthogonal group or split general spin group. We analyze the poles of the local integrals, and prove that the global completed $L$-function with a cuspidal representation of $GL_k$ twisted by a highly ramified Hecke character is entire. We obtain a new proof of the weak functorial transfer of cuspidal automorphic representations of $G$ to the natural general linear group, which is independent of the trace formula and its prerequisites, by combining our results with the Converse Theorem.

math.NT↗
arXiv2024

On fresh sets in iterations of Prikry type forcing notions

We examine the existence (and mostly non-existence) of fresh sets in commonly used iterations of Prikry type forcing notions. Results of [4] are generalized. As an application, a question of a referee of [9] is answered. In addition stationary sets preservation is addressed.

math.LO↗
arXiv2023

On Easton support iteration of Prikry type forcing notions

We consider here Easton support iterations of Prikry type forcing notions. New ways of constructing normal ultrafilters in extensions are presented. It turns out that, in contrast with other supports, seemingly unrelated measures or extenders can be involved here.

math.LO↗
arXiv2022

The Magidor Iteration and Restrictions of Ultrapowers to the Ground Model

We study the Magidor iteration of Prikry forcings below a measurable limit of measurables $ κ$. We first characterize all the normal measures $ κ$ carries in the generic extension, building on and extending the main result of \cite{ben2014forcing}. Then, for every such normal measure, we prove that the restriction of its ultrapower, from the generic extension to the ground model, is an iterated ultrapower of $ V $ by normal measures. This is done without core model theoretic assumptions; $ \mbox{GCH}_{\leq κ} $ in the ground model suffices.

math.LO↗
arXiv2021

The generalized doubling method: $(k,c)$ models

One of the key ingredients in the recent construction of the generalized doubling method is a new class of models, called $(k,c)$ models, for local components of generalized Speh representations. We construct a family of $(k,c)$ representations, in a purely local setting, and discuss their realizations using inductive formulas. Our main result is a uniqueness theorem which is essential for the proof that the generalized doubling integral is Eulerian.

math.NT↗
arXiv2021

Non-stationary support iterations of Prikry Forcings and Restrictions of Ultrapower Embeddings to the Ground Model

We study the nonstationary-support iteration of Prikry forcings below a measurable cardinal κ, characterizing all the normal measures it carries in the generic extension. We then analyze the restriction of ultrapower embeddings, taken with such a normal measure in the generic extension, to the ground model. We prove that every such restriction is an iterated ultrapwer of the ground model, and provide a sufficient condition for its definability there. This is done without core-model theoretic arguments: the assumption that GCH holds in the ground model up to κsuffices.

math.LO↗
arXiv2021

Doubling Constructions: the complete L-function for coverings of the symplectic group

We develop the local theory of the generalized doubling method for the $m$-fold central extension $Sp_{2n}^{(m)}$ of Matsumoto of the symplectic group. We define local $γ$-, $L$- and $ε$-factors for pairs of genuine representations of $Sp_{2n}^{(m)}\times\widetilde{GL}_k$ and prove their fundamental properties, in the sense of Shahidi. Here $\widetilde{GL}_k$ is the central extension of $GL_k$ arising in the context of the Langlands--Shahidi method for covering groups of $Sp_{2n}\times GL_k$. We then construct the complete $L$-function for cuspidal representations and prove its global functional equation. Possible applications include classification results and a Shimura type lift of representations from covering groups to general linear groups (a global lift is sketched here for $m=2$).

math.NT↗
arXiv2021

Multiplicity one theorems for the generalized doubling method

In this work we prove the local multiplicity at most one theorem underlying the definition and theory of local $γ$-, $ε$- and $L$-factors, defined by virtue of the generalized doubling method, over any local field of characteristic 0. We also present two applications: one to the existence of local factors for genuine representations of covering groups, the other to the global unfolding argument of the doubling integral.

math.NT↗
arXiv2020

Doubling Constructions and Tensor Product $L$-Functions: coverings of the symplectic group

In this work we develop an integral representation for the partial $L$-function of a pair $π\timesτ$ of genuine irreducible cuspidal automorphic representations, $π$ of the $m$-fold covering of Matsumoto of the symplectic group $Sp_{2n}$, and $τ$ of a certain covering group of $GL_k$, with arbitrary $m$, $n$ and $k$. Our construction is based on the recent extension by Cai, Friedberg, Ginzburg and the author, of the classical doubling method of Piatetski-Shapiro and Rallis, from rank-$1$ twists to arbitrary rank twists. We prove a basic global identity for the integral and compute the local integrals with unramified data. Possible applications include an analytic definition of local factors for representations of covering groups, and a Shimura type lift of representations from covering groups to general linear groups.

math.NT↗
arXiv2019

On the Langlands parameter of a simple supercuspidal representation: even orthogonal groups

Let $π$ be a simple supercuspidal representation of the split even special orthogonal group. We compute the Rankin-Selberg $γ$-factors for rank 1-twists of $π$ by quadratic tamely ramified characters of $F^*$. We then use our results to determine the Langlands parameter of $π$ up to its restriction to the wild inertia subgroup, subject to an analogue of a work of Blondel, Henniart, and Stevens for $SO_{2l}$. In the particular case of the field $\mathbb{Q}_2$, we are able to describe the parameter completely.

math.RT↗