Search ResultsShowing 1-14 of 14
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Deformations of reducible Galois representations with large Selmer $p$-rank
Comments: Version 2: 43 pages; minor corrections incorporatedCategories: math.NTLet $p\geq 5$ 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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arXiv:2311.03740 (Published 2023-11-07)
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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arXiv:2211.12114 (Published 2022-11-22)
Zig-zag holds on inertia for large weights
Categories: math.NTThe zig-zag conjecture essentially says that the reduction of a two-dimensional crystalline $p$-adic Galois representation of exceptional weight $k$ and half-integral slope $\frac{1}{2} \leq v \leq \frac{p-1}{2}$ is given by an alternating sequence of irreducible and reducible representations depending on the relative sizes of two parameters. The conjecture has so far only been proved for some small slopes. We prove that the conjecture holds on the inertia subgroup for $\frac{1}{2} \leq v \leq \frac{p-3}{2}$ and for $k$ sufficiently close to $2v+2$.
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arXiv:2210.07281 (Published 2022-10-13)
Non-admissible irreducible representations of $p$-adic $\mathrm{GL}_{2}$ in characteristic $p$
Comments: 8 pagesLet $p>3$ 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 authors for $F$ unramified over $\mathbb{Q}_{p}$. The construction is uniform and uses the theory of diagrams of Breuil and Pa\v{s}k$\mathrm{\bar{u}}$nas.
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arXiv:2003.00781 (Published 2020-03-02)
On non-admissible irreducible modulo $p$ representations of $GL_{2}(\mathbb{Q}_{p^{2}})$
Comments: 6 pagesWe use a Diamond diagram attached to a $2$-dimensional reducible split mod $p$ Galois representation of $\mathrm{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p^{2}})$ to construct a non-admissible smooth irreducible mod $p$ representation of $GL_{2}(\mathbb{Q}_{p^{2}})$ following the approach of Daniel Le.
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arXiv:1908.10095 (Published 2019-08-27)
$p$-adic Asai $L$-functions attached to Bianchi cusp forms
Comments: 21 pages, conference proceedingsCategories: math.NTKeywords: bianchi cusp form, adic asai, abstract kummer congruences, twisted asai l-values, rationality resultTags: conference paperWe 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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arXiv:1906.10364 (Published 2019-06-25)
Reductions of Galois representations and the theta operator
Comments: 22 pagesCategories: math.NTLet $p\ge 5$ be a prime, and let $f$ be an 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 some mild assumptions, we prove that there exists an 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 cuspforms with slopes in the interval $[0,1)$ were determined by Deligne, Buzzard, Gee and for slopes in $[1,2)$ by Bhattacharya, 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, recall that Wan has given upper bounds on the radii of Coleman families. The methods of this paper allow us to obtain lower bounds on the radii of certain Coleman families.
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arXiv:1903.08996 (Published 2019-03-20)
A zig-zag conjecture and local constancy for Galois representations
Comments: arXiv admin note: text overlap with arXiv:1901.01728Categories: math.NTWe make a zig-zag conjecture describing the reductions of irreducible crystalline two-dimensional representations of $G_{{\mathbb{Q}}_p}$ of half-integral slopes and exceptional weights. Such weights are two more than twice the slope mod $(p-1)$. We show that zig-zag holds for half-integral slopes at most $\frac{3}{2}$. We then explore the connection between zig-zag and local constancy results in the weight. First we show that known cases of zig-zag force local constancy to fail for small weights. Conversely, we explain how local constancy forces zig-zag to fail for some small weights and half-integral slopes at least $2$. However, we expect zig-zag to be qualitatively true in general. We end with some compatibility results between zig-zag and other results.
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arXiv:1901.01728 (Published 2019-01-07)
Reductions of Galois representations of Slope $\frac{3}{2}$
Comments: 78 pagesCategories: math.NTWe prove a zig-zag conjecture describing the reductions of irreducible crystalline two-dimensional representations of $G_{{\mathbb{Q}}_p}$ of slope $\frac{3}{2}$ and exceptional weights. This along with previous works completes the description of the reduction for all slopes less than $2$. The proof involves computing the reductions of the Banach spaces attached by the $p$-adic LLC to these representations, followed by an application of the mod $p$ LLC to recover the reductions of these representations.
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arXiv:1901.00628 (Published 2019-01-03)
$p$-adic Rankin product $L$-functions
We describe Panchishkin's construction of the $p$-adic Rankin product $L$-function.
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arXiv:1612.04548 (Published 2016-12-14)
Sums of Fractions and Finiteness of Monodromy
We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.
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arXiv:1504.03838 (Published 2015-04-15)
Reduction of Galois Representations of slope 1
We compute the reductions of irreducible crystalline two-dimensional representations of $G_{\mathbb{Q}_p}$ of slope 1, for primes $p > 3$. We give a complete answer for all weights $k \geq 2$, except for weights $k \equiv 4 \mod (p-1)$ where we provide partial results. The proof uses the mod $p$ Local Langlands Correspondence.
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arXiv:1504.00148 (Published 2015-04-01)
Reductions of Galois representations for slopes in $(1,2)$
We describe the semi-simplification of the mod $p$ reduction of certain crystalline two dimensional local Galois representations of slopes in the interval $(1,2)$ and all weights. The proof uses the mod $p$ Local Langlands Correspondence for $GL_2(Q_p)$. We also give a complete description of the submodules generated by the second highest monomial in the mod $p$ symmetric power representations of $GL_2(F_p)$.
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arXiv:1109.6676 (Published 2011-09-29)
On uniform large Galois images for modular abelian varieties
DOI: 10.1112/blms/bds041Categories: math.NTKeywords: uniform large galois images, modular abelian varieties, large galois image properties, exceptional image case, question regarding uniform versionsTags: journal articleWe formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.