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46 research outputs found

Constructing Galois representations with prescribed Iwasawa $\lambda$ -invariant

Author: Ray Anwesh
Publication venue
Publication date: 11/01/2024
Field of study

Let

p\geq 5

be a prime number. We consider the Iwasawa

\lambda

-invariants associated to modular Bloch-Kato Selmer groups, considered over the cyclotomic

\mathbb{Z}_p

-extension of

\mathbb{Q}

. Let

g

be a

p

-ordinary cuspidal newform of weight

2

and trivial nebentype. We assume that the

\mu

-invariant of

g

vanishes, and that the image of the residual representation associated to

g

is suitably large. We show that for any number greater

n

greater than or equal to the

\lambda

-invariant of

g

, there are infinitely many newforms

f

that are

p

-congruent to

g

, with

\lambda

-invariant equal to

n

. We also prove quantitative results regarding the levels of such modular forms with prescribed

\lambda

-invariant.Comment: Version 2: accepted for publication in the Bulletin of the London Math Societ

arXiv.org e-Print Archive

The $T$ -adic Galois representation is surjective for a positive density of Drinfeld modules

Author: Ray Anwesh
Publication venue
Publication date: 27/12/2023
Field of study

Let

\mathbb{F}_q

be the finite field with

q\geq 5

elements,

A:=\mathbb{F}_q[T]

and

F:=\mathbb{F}_q(T)

. Assume that

q

is odd and take

|\cdot|

to be the absolute value at

\infty

that is normalized by

|T|=q

. Given a pair

w=(g_1, g_2)\in A^2

with

g_2\neq 0

, consider the associated Drinfeld module

\phi^w: A\rightarrow A\{\tau\}

of rank

2

defined by

\phi_T^w=T+g_1\tau+g_2\tau^2

. Fix integers

c_1, c_2\geq 1

and define

|w|:=max\{|g_1|^{\frac{1}{c_1}}, |g_2|^{\frac{1}{c_2}}\}

. I show that when ordered by height, there is a positive density of pairs

w=(g_1, g_2)

, such that the

T

-adic Galois representation attached to

\phi^w

is surjective

arXiv.org e-Print Archive

Lower bounds for the number of number fields with Galois group $GL_2(\mathbb{F}_\ell)$

Author: Ray Anwesh
Publication venue
Publication date: 14/05/2023
Field of study

Let

\ell\geq 5

be a prime number and

\mathbb{F}_\ell

denote the finite field with

\ell

elements. We show that the number of Galois extensions of the rationals with Galois group isomorphic to

GL_2(\mathbb{F}_\ell)

and absolute discriminant bounded above by

X

is asymptotically at least

\frac{X^{\frac{\ell}{12(\ell-1)\# GL_2(\mathbb{F}_\ell)}}}{\log X}

. We also obtain a similar result for the number of surjective homomorphisms

\rho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{F}_\ell)

ordered by the prime to

\ell

part of the Artin conductor of

\rho

.Comment: Version 3: Added two paragraphs of exposition in the introduction following the statement of the main resul

arXiv.org e-Print Archive

Galois representations ramified at one prime and with suitably large image

Author: Ray Anwesh
Publication venue
Publication date: 03/06/2023
Field of study

Let

p\geq 7

be a prime and

n>1

be a natural number. We show that there exist infinitely many Galois representations

\varrho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_{n}(\mathbb{Z}_p)

which are unramified outside

\{p, \infty\}

with large image. More precisely, the Galois representations constructed have image containing the kernel of the mod-

p^t

reduction map

SL_n(\mathbb{Z}_p)\rightarrow SL_n(\mathbb{Z}/p^t\mathbb{Z})

, where

t:=8(n^2-n)\left(3+\lfloor log_p(2^n+1)\rfloor\right)+8

. The results are proven via a purely Galois theoretic lifting construction. When

p\equiv 1\mod{4}

, our results are conditional since in this case, we assume a very weak version of Vandiver's conjecture.Comment: Version 2: minor revisions following referee report, accepted for publication in Trans. Amer. Math. So

arXiv.org e-Print Archive

Statistics for p-ranks of Artin-Schreier covers

Author: Ray Anwesh
Publication venue
Publication date: 14/10/2022
Field of study

Given a prime

p

and

q

a power of

p

, we study the statistics of

p

-ranks of Artin--Schreier covers of given genus defined over

\mathbb{F}_q

, in the large

q

-limit. We refer to this problem as the geometric problem. We also study an arithmetic variation of this problem, and consider Artin--Schreier covers defined over

\mathbb{F}_p

, letting

p

go to infinity. Distribution of

p

-ranks has been previously studied for Artin--Schreier covers over a fixed finite field as the genus is allowed to go to infinity. The method requires that we count isomorphism classes of covers that are unramified at

\infty

arXiv.org e-Print Archive

Remarks on Greenberg's conjecture for Galois representations associated to elliptic curves

Author: Ray Anwesh
Publication venue
Publication date: 12/08/2023
Field of study

Let

E_{/\mathbb{Q}}

be an elliptic curve and

p

be an odd prime number at which

E

has good ordinary reduction. Let

Sel_{p^\infty}(\mathbb{Q}_\infty, E)

denote the

p

-primary Selmer group of

E

considered over the cyclotomic

\mathbb{Z}_p

-extension of

\mathbb{Q}

. The (algebraic) \emph{

\mu

-invariant} of

Sel_{p^\infty}(\mathbb{Q}_\infty, E)

is denoted

\mu_p(E)

. Denote by

\bar{\rho}_{E, p}:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_2(\mathbb{Z}/p\mathbb{Z})

the Galois representation on the

p

-torsion subgroup of

E(\bar{\mathbb{Q}})

. Greenberg conjectured that if

\bar{\rho}_{E, p}

is reducible, then there is a rational isogeny

E\rightarrow E'

whose degree is a power of

p

, and such that

\mu_p(E')=0

. In this article, we study this conjecture by showing that it is satisfied provided some purely Galois theoretic conditions hold that are expressed in terms of the representation

\bar{\rho}_{E,p}

. In establishing our results, we leverage a theorem of Coates and Sujatha on the algebraic structure of the fine Selmer group. Furthermore, in the case when

\bar{\rho}_{E, p}

is irreducible, we show that our hypotheses imply that

\mu_p(E)=0

provided the classical Iwasawa

\mu

-invariant vanishes for the splitting field

\mathbb{Q}(E[p]):=\bar{\mathbb{Q}}^{ker\bar{\rho}_{E,p}}

.Comment: 23 pages, initial date of submission 28 May 202

arXiv.org e-Print Archive

On large Iwasawa $\lambda$ -invariants of imaginary quadratic function fields

Author: Ray Anwesh
Publication venue
Publication date: 05/03/2023
Field of study

Let

\ell

be a prime number and

q

be a power of

\ell

. Given an odd prime number

p

and an imaginary quadratic extension

F

of the rational function field

\mathbb{F}_q(T)

, let

\lambda_p(F)

denote the Iwasawa

\lambda

-invariant of the constant

\mathbb{Z}_p

-extension of

F

. We show that for any number

r>0

and all large enough values of

q\not\equiv 1\mod{p}

, there is a positive proportion of imaginary quadratic fields

F/\mathbb{F}_q(T)

with the property that

\lambda_p(F)\geq r

. The main result is proved as a consequence of recent unconditional theorems of Ellenberg-Venkatesh-Westerland on the distribution of class groups of imaginary quadratic function fields.Comment: 8 pages; accepted for publication in the Ramanujan Journa

arXiv.org e-Print Archive

Constructing Galois representations with large Iwasawa $\lambda$ -Invariant

Author: Ray Anwesh
Publication venue
Publication date: 11/01/2023
Field of study

Let

p\geq 5

be a prime. We construct modular Galois representations for which the

\mathbb{Z}_p

-corank of the

p

-primary Selmer group (i.e.,

\lambda

-invariant) over the cyclotomic

\mathbb{Z}_p

-extension is large. More precisely, for any natural number

n

, one constructs a modular Galois representation such that the associated

\lambda

-invariant is

\geq n

. The method is based on the study of congruences between modular forms, and leverages results of Greenberg and Vatsal. Given a modular form

f_1

satisfying suitable conditions, one constructs a congruent modular form

f_2

for which the

\lambda

-invariant of the Selmer group is large. A key ingredient in acheiving this is the Galois theoretic lifting result of Fakruddin-Khare-Patrikis, which extends previous work of Ramakrishna. The results are subject to certain additional hypotheses, and are illustrated by explicit examples.Comment: Version 3: Final version, accepted for publication in Annales Math Quebe

arXiv.org e-Print Archive

Statistics for Iwasawa invariants of elliptic curves, $\rm{III}$

Author: Ray Anwesh
Publication venue
Publication date: 13/04/2024
Field of study

Given a prime

p\geq 5

, a conjecture of Greenberg predicts that the

\mu

-invariant of the

p

-primary Selmer group should vanish for most elliptic curves with good ordinary reduction at

p

. In support of this conjecture, I show that the

5

-primary Iwasawa

\mu

- and

\lambda

-invariants simultaneously vanish for an explicit positive density of elliptic curves

E_{/\mathbb{Q}}

. The elliptic curves in question have good ordinary reduction at

5

, and are ordered by their height. The results are proven by leveraging work of Bhargava and Shankar on the distribution of

5

-Selmer groups of elliptic curves defined over

\mathbb{Q}

arXiv.org e-Print Archive

Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$ -extension

Author: Ray Anwesh
Publication venue
Publication date: 21/11/2024
Field of study

Let

n

be a cubefree natural number and

p\geq 5

be a prime number. Assume that

n

is not expressible as a sum of the form

x^3+y^3

, where

x,y\in \mathbb{Q}

. In this note, we study the solutions (or lack thereof) to the equation

n=x^3+y^3

, where

x

and

y

belong to the cyclotomic

\mathbb{Z}_p

-extension of

\mathbb{Q}

. As an application, consider the case when

n

is not a sum of rational cubes. Then, we prove that

n

cannot be a sum of two cubes in certain large families of prime cyclic extensions of

\mathbb{Q}

.Accepted for publication in the Tohoku Math

arXiv.org e-Print Archive