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Eknath Ghate

Publications and source records attributed to Eknath Ghate.

At least 19 recordsLinked to original sources
arXiv2026

Which numbers are $u$-invariants of perfectoid fields?

Let $p\neq 2$ be a prime. We determine the set of numbers that occur as the $u$-invariant of perfectoid subfields of $\mathbb{C}_p$. Given any number that is the $u$-invariant of a field of characteristic $p$, we give an explicit construction of a perfectoid field with the same $u$-invariant. In particular, if a number is the $u$-invariant of a field of characteristic $p$, it is also the $u$-invariant of a field of characteristic zero.

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arXiv2026

Reductions Of Crystalline Representations Of Fractional Slope $<p-1$

Let $p$ be an odd prime and let $V_{k,a_p}$ be the two-dimensional crystalline representation of the Galois group of ${\mathbb Q}_p$ of weight $k \geq 2$ and parameter $a_p \in \bar{\mathbb{Q}}_p$. We study the semi-simplification $\bar{V}_{k,a_p}$ of the mod $p$ reduction of $V_{k,a_p}$ when the slope (valuation of $a_p$) is a positive fraction $< p-1$ using the mod $p$ local Langlands correspondence. We describe the $\textit{exact shape}$ of $\bar{V}_{k,a_p}$ for all such slopes and all (sufficiently large, depending on the slope) weights $k$, as long as certain Jordan-H\"older factors of dimension $p-1$ do not intervene in the computation (when $k$ is odd), though we also provide some criteria which further determine the shape of $\bar{V}_{k,a_p}$ in some of these exceptional cases. To keep this paper a reasonable length, we assume that for certain bad congruence classes of $k$ mod $p$, the slope is less than the representative - taken in the range $[1,p-1]$ - of the congruence class of $k-2$ mod $(p-1)$, which is generically the case if the slope is small. Finally, a folklore conjecture predicts that the reduction $\bar{V}_{k,a_p}$ is $\textit{irreducible}$ for fractional slopes if $k$ is even. We deduce this conjecture for all fractional slopes $< p-2$ and all (sufficiently large, even) weights $k$ under the aforementioned slope assumption.

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arXiv2026

Reductions of $\mathrm{GL}_2(\mathbb Q_{p^f})$-Banach spaces of slopes in $(0,1)$

Let $p$ be an odd prime and $f \geq 1$. We consider a $p$-adic locally algebraic $\text{GL}_2(\mathbb Q_{p^f})$-representation attached to a tuple of $f$ weights $k=(k_i)$ for $0 \leq i \leq f-1$ and a $p$-adic integer $a_p$ with valuation in $(0,1)$. We give conditions under which the irreducible quotients of the subquotients in a filtration on the reduction mod $p$ of the natural integral structure on this space are supercuspidal. We also check that for small $k$ and $f$ the integral structure is a lattice so that the mod $p$ reduction is nonzero.

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arXiv2026

Reduction mod $p$ of semi-stable representations of some super-Breuil weights

We determine the mod $p$ reductions of the semi-stable representations $V_{k, \mathcal{L}}$ of weight $k \in [p + 5, 2p]\cup[2p + 6, 3p + 1]$ and $v_p(\mathcal{L}) < 1-k/2$ for primes $p \geq 5$. In particular, this shows that the techniques introduced in [CG24] involving the $p$-adic and mod $p$ local Langlands correspondences can be used to compute the reduction of $V_{k, \mathcal{L}}$ outside the range $k \in [3, p + 1]$. Moreover, this shows that the bound on $v_p(\mathcal{L})$ given by Bergdall-Levin-Liu [BLL23] can be improved, at least for weights $k \in [2p + 6, 3p + 1]$.

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arXiv2026

On the Brauer class of Modular Endomorphism Algebras

We investigate the Brauer class of the endomorphism algebra of the motive attached to a non-CM form. The ramification of the algebra is shown in many cases to be controlled by the normalized slopes of the form.

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arXiv2025

The semi-stable Local Langlands Correspondence

We start with background that goes into an Iwahori-theoretic reformulation of the mod $p$ Local Langlands Correspondence (\S 2). We then explain some classical $p$-adic functional analytic results (\S 3) that go into defining the $p$-adic Banach space (\S 4) attached to a two-dimensional semi-stable representation $V_{k,{\mathcal L}}$ of the Galois group of ${\mathbb Q}_p$ of weight $k$ and ${\mathcal L}$-invariant ${\mathcal L}$ under the $p$-adic Local Langlands correspondence. We then sketch how to compute the reduction of a lattice in this Banach space, which along with the Iwahori mod $p$ LLC, allows one to completely determine the mod $p$ reduction of $V_{k,{\mathcal L}}$ for all weights $3 \leq k \leq p+1$ and all ${\mathcal L}$ for $p \geq 5$ (\S 5). These notes are a summary of our joint work with Anand Chitrao [CG24]. Emphasis is placed on motivation and background rather than completeness.

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arXiv2025

Clebsch-Gordan and the theta filtration for modular representations of $\mathrm{GL}_2({\mathbb F}_q)$

Let $p$ be a prime. We solve two problems in the mod $p$ representation theory of $\mathrm{GL}_2(\mathbb{F}_{q})$ where $q=p^f$. We first prove a Clebsch-Gordan decomposition theorem for the tensor product of two mod $p$ representations of $\mathrm{GL}_2(\mathbb{F}_{q})$. As an application, we use this to guess the structure of quotients of symmetric power representations of $\mathrm{GL}_2(\mathbb{F}_{q})$ by submodules in the theta filtration. We then give a direct proof of this structure showing that such quotients are built out of principal series representations.

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arXiv2025

Restriction problem for mod $p$ representations of $\text{GL}_2$ over a finite field

Let $\mathbb{F}_q$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod $p$ representations of $G_q = \text{GL}_2({\mathbb{F}_q})$ to $G_p = \text{GL}_2(\mathbb{F}_p)$. We first study the restrictions of principal series which are obtained by induction from a Borel subgroup $B_q$. We then analyze the restrictions of inductions from an anisotropic torus $T_q$ which are related to cuspidal representations. Complete decompositions are given in both cases according to the parity of $f$. The proofs depend on writing down explicit orbit decompositions of $G_p \backslash G_q / H$ where $H = B_q$ or $T_q$ using the fact that $G_q / H$ is an explicit orbit in a certain projective line, along with Mackey theory.

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arXiv2025

Deformations of reducible Galois representations with large Selmer $p$-rank

Let $p\geq 5$ be a prime number. In this paper, we construct Galois representations associated with modular forms for which the dimension of the $p$-torsion in the Bloch-Kato Selmer group can be made arbitrarily large. Our result extends similar results known for small primes, such as Matsuno's work on Tate-Shafarevich groups of elliptic curves. Extending the technique of Hamblen and Ramakrishna, we lift residually reducible Galois representations to modular representations for which the associated Greenberg Selmer groups are minimally generated by a large number of elements over the Iwasawa algebra. We deduce that there is an isogenous lattice for which the Bloch-Kato Selmer group has large $p$-rank.

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arXiv2023

Reductions of semi-stable representations using the Iwahori mod $p$ Local Langlands Correspondence

We determine the mod $p$ reductions of all two-dimensional semi-stable representations $V_{k,\mathcal{L}}$ of the Galois group of $\mathbb{Q}_p$ of weights $3 \leq k \leq p+1$ and $\mathcal{L}$-invariants $\mathcal{L}$ for primes $p \geq 5$. In particular, we describe the constants appearing in the unramified characters completely. The proof involves computing the reduction of Breuil's $\mathrm{GL}_2(\mathbb{Q}_p)$-Banach space $\tilde{B}(k,\mathcal{L})$, by studying certain logarithmic functions using background material developed by Colmez, and then applying an Iwahori theoretic version of the mod $p$ Local Langlands Correspondence.

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arXiv2023

Modular representations of $\mathrm{GL}_2({\mathbb F}_q)$ using calculus

We show that certain modular induced representations of $\mathrm{GL}_2({\mathbb F}_q)$ can be written as cokernels of operators acting on symmetric power representations of $\mathrm{GL}_2({\mathbb F}_q)$. When the induction is from the Borel subgroup, respectively the anisotropic torus, the operators involve multiplication by newly defined twisted Dickson polynomials, respectively, twisted Serre operators. Our isomorphisms are explicitly defined using differential operators. As a corollary, we improve some periodicity results for quotients in the theta filtration.

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arXiv2022

Zig-zag for Galois Representations

The zig-zag conjecture says that the reductions of two-dimensional crystalline representations of the Galois group of ${\mathbb {Q}}_p$ of large exceptional weights and half-integral slopes up to $\frac{p-1}{2}$ vary through an alternating sequence of irreducible and reducible mod $p$ representations. We prove this conjecture in smoothly varying families of such representations for $p \geq 5$. The proof uses a limiting argument due to Chitrao-Ghate-Yasuda to reduce to the case of semi-stable representations of weights at most $p+1$, and then appeals to the work of Breuil-M\'ezard, Guerberoff-Park and Chitrao-Ghate.

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arXiv2022

Non-admissible irreducible representations of $p$-adic $\mathrm{GL}_{n}$ in characteristic $p$

Let $p>3$ and $F$ be a non-archimedean local field with residue field a proper finite extension of $\mathbb{F}_p$. We construct smooth absolutely irreducible non-admissible representations of $\mathrm{GL}_2(F)$ defined over the residue field of $F$ extending the earlier results of the authors for $F$ unramified over $\mathbb{Q}_{p}$. This construction uses the theory of diagrams of Breuil and Paskunas. By parabolic induction, we obtain smooth absolutely irreducible non-admissible representations of $\mathrm{GL}_n(F)$ for $n>2$.

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arXiv2021

Semi-stable representations as limits of crystalline representations

We construct an explicit sequence $V_{k_n,a_n}$ of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semi-stable representation $V_{k,{\mathcal{L}}}$ of $\mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$. The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid analytic setting and may be of independent interest. Also, we recover a formula of Stevens expressing the ${\mathcal{L}}$-invariant as a logarithmic derivative. Our result can be used to compute the reduction of $V_{k,{\mathcal{L}}}$ in terms of the reductions of the $V_{k_n,a_n}$. For instance, using the zig-zag conjecture we recover (resp. extend) the work of Breuil-M\'ezard and Guerberoff-Park computing the reductions of the $V_{k,{\mathcal{L}}}$ for weights at most $p-1$ (resp. $p+1$), at least on the inertia subgroup. In the cases where zig-zag is known, we are further able to obtain some new information about the reductions for small odd weights. Finally, we explain some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight.

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arXiv2019

$p$-adic Asai $L$-functions attached to Bianchi cusp forms

We establish a rationality result for the twisted Asai L-values attached to a Bianchi cusp form and construct distributions interpolating these L-values. Using the method of abstract Kummer congruences, we then outline the main steps needed to show that these distributions come from a measure.

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arXiv2019

Reductions of Galois representations and the theta operator

Let $p\ge 5$ be a prime, and let $f$ be a cuspidal eigenform of weight at least $2$ and level coprime to $p$ of finite slope $\alpha$. Let $\bar{\rho}_f$ denote the mod $p$ Galois representation associated with $f$ and $\omega$ the mod $p$ cyclotomic character. Under an assumption on the weight of $f$, we prove that there exists a cuspidal eigenform $g$ of weight at least $2$ and level coprime to $p$ of slope $\alpha+1$ such that $$\bar{\rho}_f \otimes \omega \simeq \bar{\rho}_g,$$ up to semisimplification. The proof uses Hida-Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions of the local Galois representations associated to cusp forms with slopes in the interval $[0,1)$ were determined by Deligne, Buzzard and Gee and for slopes in $[1,2)$ by Bhattacharya, Ganguli, Ghate, Rai and Rozensztajn. We show that these reductions, in spite of their somewhat complicated behavior, are compatible with the displayed equation above. Moreover, the displayed equation above allows us to predict the shape of the reductions of a class of Galois representations attached to eigenforms of slope larger than $2$. Finally, the methods of this paper allow us to obtain upper bounds on the radii of certain Coleman families.

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arXiv2019

The Monomial Lattice in Modular Symmetric Power Representations

Let $p$ be a prime. We study the structure of and the inclusion relations among the terms in the monomial lattice in the modular symmetric power representations of $\mathrm{GL}_2(\mathbb{F}_p)$. We also determine the structure of certain related quotients of the symmetric power representations which arise when studying the reductions of local Galois representations of slope at most $p$. In particular, we show that these quotients are periodic and depend only on the congruence class modulo $p(p-1)$. Many of our results are stated in terms of the sizes of various sums of digits in base $p$-expansions and in terms of the vanishing or non-vanishing of certain binomial coefficients modulo $p$.

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