arXiv subjects

Nahid Walji

Publications and source records attributed to Nahid Walji.

12 recordsLinked to original sources

arXiv2021

On the distribution of traces of Frobenius for families of elliptic curves and the Lang-Trotter conjecture on average

We obtain distribution results for traces of Frobenius for various families of elliptic curves with respect to the Lang-Trotter conjecture, extremal primes, and the central limit theorem. This includes some generalisations and bounds related to the work of Sha-Shparlinski on the average Lang-Trotter conjecture for single-parametric families of elliptic curves and the work of various authors on the trace of Frobenius for primes in congruence classes. Some results are also obtained for modular forms.

math.NT↗

arXiv2021

On the conjectural decomposition of symmetric powers of automorphic representations for GL(3) and GL(4)

Given a cuspidal automorphic representation $π$ for GL(3) over a number field and a positive integer $k$, assume that the symmetric $m$th power lifts of $π$ are isobaric automorphic for $m \leq k$, cuspidal for $m \leq k-1$, and that certain associated Rankin-Selberg products are isobaric automorphic. Then the number of cuspidal isobaric summands in the $k$th symmetric power lift is bounded above by 3 when $k \geq 7$, and bounded above by 2 when $k \geq 19$ with $k \equiv 1 \pmod 3$. We then investigate the analogous problem for GL(4).

math.NT↗

arXiv2020

A conjectural refinement of strong multiplicity one for GL(n)

Given a pair of distinct unitary cuspidal automorphic representations for GL(n) over a number field, let S denote the set of finite places at which the automorphic representations are unramified and their associated Hecke eigenvalues differ. In this note, we demonstrate how conjectures on the automorphy and possible cuspidality of adjoint lifts and Rankin-Selberg products imply lower bounds on the size of S. We also obtain further results for GL(3).

math.NT↗

arXiv2020

On the occurrence of Hecke eigenvalues in sectors

Let $π$ be a non-self-dual unitary cuspidal automorphic representation of non-solvable polyhedral type for GL(2) over a number field. We show that $π$ has a positive upper Dirichlet density of Hecke eigenvalues in any sector whose angle is at least 2.63 radians.

math.NT↗

arXiv2019

On the occurrence of large positive Hecke eigenvalues for GL(2)

Let $π$ be a cuspidal automorphic representation for GL(2)/$\mathbb{Q}$ that is self-dual. In this Note we show that there exists a positive upper Dirichlet density of primes at which the associated Hecke eigenvalues of $π$ are larger than a specified positive constant.

math.NT↗

arXiv2017

Distinguishing finite group characters and refined local-global phenomena

Serre obtained a sharp bound on how often two irreducible degree $n$ complex characters of a finite group can agree, which tells us how many local factors determine an Artin $L$-function. We consider the more delicate question of finding a sharp bound when these objects are primitive, and answer these questions for $n=2,3$. This provides some insight on refined strong multiplicity one phenomena for automorphic representations of GL$(n)$. For general $n$, we also answer the character question for the families PSL$(2,q)$ and SL$(2,q)$.

math.GR↗

arXiv2015

On the distribution of Hecke eigenvalues for cuspidal automorphic representations for GL(2)

Given a self-dual cuspidal automorphic representation for GL(2) over a number field, we establish the existence of an infinite number of Hecke eigenvalues that are greater than an explicit positive constant, and an infinite number of Hecke eigenvalues that are less than an explicit negative constant. This provides an answer to a question of Serre. We also consider analogous problems for cuspidal automorphic representations that are not self-dual.

math.NT↗

arXiv2015

Matching densities for Galois representations

Given a pair of n-dimensional complex Galois representations over Q, we define their matching density to be the density, if it exists, of the set of places at which the traces of Frobenius of the two Galois representations are equal. We will show that the set of matching densities of such pairs of irreducible Galois representations (for all n) is dense in the interval [0, 1]. Under the strong Artin conjecture, this also implies the corresponding statement for cuspidal automorphic representations.

math.NT↗

arXiv2014

On the Occurrence of Hecke Eigenvalues and a Lacunarity Question of Serre

Let πbe a unitary cuspidal automorphic representation for GL(n) over a number field. We establish upper bounds on the number of Hecke eigenvalues of πequal to a fixed complex number. For GL(2), we also determine upper bounds on the number of Hecke eigenvalues with absolute value equal to a fixed number γ; in the case γ=0, this answers a question of Serre. These bounds are then improved upon by restricting to non-dihedral representations. Finally, we obtain analogous bounds for a family of cuspidal automorphic representations for GL(3).

math.NT↗

arXiv2013

Further refinement of strong multiplicity one for GL(2)

We obtain a sharp refinement of the strong multiplicity one theorem for the case of unitary non-dihedral cuspidal automorphic representations for GL(2). Given two unitary cuspidal automorphic representations for GL(2) that are not twist-equivalent, we also find sharp lower bounds for the number of places where the Hecke eigenvalues are not equal, for both the general and non-dihedral cases. We then construct examples to demonstrate that these results are sharp.

math.NT↗

arXiv2013

On the size of Satake parameters for unitary cuspidal automorphic representations for GL(4)

Let Π be a cuspidal automorphic representation for GL(4) over a number field F. We obtain unconditional lower bounds on the number of places at which the Satake parameters are not "too large". In the case of self-dual Π with non-trivial central character, our results imply that the set of places at which Π is tempered has an explicit positive lower Dirichlet density. Our methods extend those of Ramakrishnan by careful analysis of the hypothetical possibilities for the structure of the Langlands conjugacy classes, as well as their behaviour under functorial lifts. We then discuss the analogous problem in GL(3).

math.NT↗

arXiv2013

Supersingular distribution on average for congruence classes of primes

We demonstrate the existence of a congruence class bias in the distribution of supersingular primes on average for elliptic curves over $\Q$. For example, we show that on average there are twice as many supersingular primes congruent to 2 mod 3 as there are congruent to 1 mod 3. Our result is obtained using the averaging approach of Fouvry-Murty along with ideas of David-Pappalardi.

math.NT↗