arXiv subjects

Prabhat Kumar Mishra

Publications and source records attributed to Prabhat Kumar Mishra.

5 recordsLinked to original sources

arXiv2026

Determining newforms via arithmetic relations among Fourier coefficients

We investigate the distribution of primes satisfying arithmetic inequalities involving the Fourier coefficients of two non-CM newforms at prime powers. More precisely, we establish asymptotic formulas for the number of primes for which the differences, products, and ratios of the Fourier coefficients satisfy prescribed inequalities, together with explicit estimates for the corresponding densities. The proofs combine an effective joint Sato--Tate theorem with a geometric analysis of the associated semi-algebraic regions. As applications, we obtain new multiplicity one criteria, improve a theorem of Matom\"aki on small differences between Fourier coefficients, establish density-one analogues in the spirit of the Atkin--Serre conjecture, and derive a new characterization of twist-equivalence through the distribution of ratios of Fourier coefficients.

math.NT↗

arXiv2026

Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients

We prove an unconditional, effective joint Sato--Tate distribution for the Fourier coefficients of two twist-inequivalent, non-CM newforms $f$ and $f'$. Our result generalises a result of Thorner, which holds for rectangular regions, by extending it to any measurable region of $[-2,2]^2$ whose boundary consists of finitely many continuous curves of finite length. As a consequence, we develop a unified framework to study various arithmetic properties of Fourier coefficients of symmetric power $L$-functions attached to $f$ and $f'$. In particular, for these coefficients (and their polynomial expressions), we obtain effective distribution results, quantitative statements on simultaneous sign behaviour, and bounds for the first sign change.

math.NT↗

arXiv2026

On Lower Bounds for sums of Fourier Coefficients of Twist-Inequivalent Newforms

In this article, we address the lower bounds for the sums $a_f(p)+a_g(p)$ of the $p$-th Fourier coefficients of two twist-inequivalent, non-CM normalized newforms $f$ and $g$. Our main result shows that for such forms with integer Fourier coefficients, the largest prime factor of $a_f(p)+a_g(p)$ satisfies $P(a_f(p)+a_g(p)) > (\log p)^{1/14} (\log \log p)^{3/7-ε}$ for almost all primes $p$ and for any $ε> 0$. Beyond primes, we apply Brun's sieve to show that a similar phenomenon holds for a set of positive integers with natural density one. The main result is further strengthened under the Generalized Riemann Hypothesis, where we establish exponential growth for the absolute value of $a_f(p)+a_g(p)$ in terms of $p$.Additionally, we derive an interesting result related to the multiplicity one theorem, demonstrating that if the sum $a_f(p)+a_g(p)$ is small for a positive-density subset of primes, then $f$ and $g$ must be twist-equivalent by a quadratic character.

math.NT↗

arXiv2024

Certain squarefree levels of reducible modular mod$\,\ell$ Galois representations

Let $k \ge 2$ be an even integer, $ \ell \ge \max\{5, k-1\} $ be a prime, and $N$ be a squarefree positive integer. It is known that if the $\rm{mod}\,\ell$ Galois representation $\overlineρ_f$ associated with a newform $f$ of weight $k$, level $N$, and trivial nebentypus is reducible, then $\overlineρ_f \simeq 1 \oplus \overlineχ_\ell^{k-1}$, up to semisimplification, where $\overlineχ_\ell^{}$ is the $\rm{mod}\,\ell$ cyclotomic character. In this paper, we determine the necessary and sufficient conditions under which the $\rm{mod}\,\ell$ representation $1 \oplus \overlineχ_\ell^{k-1}$ arises from a newform of weight $k$, level $N$ with exactly two prime factors with specified Atkin-Lehner eigenvalues. Specifically, this proves a conjecture of Billerey and Menares when $N$ is a product of two primes under some mild assumption. As an application, we show that for any $\ell\ge 5$ and $k=2$ or $\ell+1$, there exist a large class of distinct primes $p$ and $q$ such that the $\rm{mod}\,\ell$ representation $1 \oplus \overlineχ_\ell^{k-1}$ arises from a newform of weight $k$ and level $pq$ with explicit Atkin-Lehner eigenvalues.

math.NT↗

arXiv2018

Output feedback stable stochastic predictive control with hard control constraints

We present a stochastic predictive controller for discrete time linear time invariant systems under incomplete state information. Our approach is based on a suitable choice of control policies, stability constraints, and employment of a Kalman filter to estimate the states of the system from incomplete and corrupt observations. We demonstrate that this approach yields a computationally tractable problem that should be solved online periodically, and that the resulting closed loop system is mean-square bounded for any positive bound on the control actions. Our results allow one to tackle the largest class of linear time invariant systems known to be amenable to stochastic stabilization under bounded control actions via output feedback stochastic predictive control.

math.OC↗