University of Bristol

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211 papers · 1 filter

math.NT2025

Bounded Exponential Sums with Multiplicative Coefficients

Pierre-Alexandre Bazin, Ihor Pylaiev, Fred Tyrrell

We investigate when the exponential sum $S_{f} (x, α) := \sum_{n \leq x} f (n) e (n α)$ is bounded, for a multiplicative function $f$ and $α \in R$ . We show that under natural…

math.NT2024

On Artin's conjecture on average and short character sums

Oleksiy Klurman, Igor E. Shparlinski, Joni Teräväinen

Let $N_{a} (x)$ denote the number of primes up to $x$ for which the integer $a$ is a primitive root. We show that $N_{a} (x)$ satisfies the asymptotic predicted by Artin's conjecture for…

math.NT2024

Conductor exponents for families of hyperelliptic curves

Martin Azon, Mar Curcó-Iranzo, Maleeha Khawaja +2

We compute the conductor exponents at odd places using the machinery of cluster pictures of curves for three infinite families of hyperelliptic curves. These are families of Frey h…

math.NT2024

Computing Euler factors of genus 2 curves at odd primes of almost good reduction

Céline Maistret, Andrew V. Sutherland

We present an efficient algorithm to compute the Euler factor of a genus 2 curve C/Q at an odd prime p that is of bad reduction for C but of good reduction for the Jacobian of C (a…

math.NT2024

Arithmetic Ramsey theory over the primes

Jonathan Chapman, Sam Chow

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_{1} h (x_{1}) + \dots + a_{s} h (x_{s}) = b$ , where the $a_{i}$ and $b$ …

math.NT2023

Murmurations of modular forms in the weight aspect

Jonathan Bober, Andrew R. Booker, Min Lee +1

We prove the existence of "murmurations" in the family of holomorphic modular forms of level $1$ and weight $k \to \infty$ , that is, correlations between their root numbers and Hecke…

math.NT2023★ 7 cited

Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders

Sarah Arpin, James Clements, Pierrick Dartois +3

Orientations of supersingular elliptic curves encode the information of an endomorphism of the curve. Computing the full endomorphism ring is a known hard problem, so one might con…

math.NT2023★ 1 cited

The sum-product problem for integers with few prime factors

Brandon Hanson, Misha Rudnev, Ilya Shkredov +1

It was asked by E. Szemerédi if, for a finite set $A \subset Z$ , one can improve estimates for $max {∣ A + A ∣, ∣ A \cdot A ∣}$ , under the constraint that all integers involved ha…

math.NT2022★ 3 cited

Polynomial progressions in topological fields

Ben Krause, Mariusz Mirek, Sarah Peluse +1

Let $P_{1}, \dots, P_{m} \in K [y]$ be polynomials with distinct degrees, no constant terms and coefficients in a general locally compact topological field $K$ . We give a quantitative…

math.NT2022★ 1 cited

Semi-integral Brauer-Manin obstruction and quadric orbifold pairs

Vladimir Mitankin, Masahiro Nakahara, Sam Streeter

We study local-global principles for two notions of semi-integral points, termed Campana points and Darmon points. In particular, we develop a semi-integral version of the Brauer-M…

math.NT2022

Polynomials represented by norm forms via the beta sieve

Alec Shute

A central question in Arithmetic geometry is to determine for which polynomials $f \in Z [t]$ and which number fields $K$ the Hasse principle holds for the affine equation…

math.NT2022★ 1 cited

Beyond the Erdős discrepancy problem in function fields

Oleksiy Klurman, Alexander P. Mangerel, Joni Teräväinen

We characterize the limiting behavior of partial sums of multiplicative functions $f : F_{q} [t] \to S^{1}$ . In contrast to the number field setting, the characterization depends…

math.NT2021★ 5 cited

The 2-parity conjecture for elliptic curves with isomorphic 2-torsion

Holly Green, Celine Maistret

The Birch and Swinnerton--Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its $L$ -function. In this article we consider a weaker version o…

math.NT2021★ 5 cited

L-series of harmonic Maass forms and a summation formula for harmonic lifts

Nikolaos Diamantis, Min Lee, Wissam Raji +1

We introduce an L-series associated with harmonic Maass forms and prove their functional equations. We establish converse theorems for these L-series and, as an application, we for…

math.NT2021★ 1 cited

Fine-scale distribution of roots of quadratic congruences

Jens Marklof, Matthew Welsh

We establish limit laws for the distribution in small intervals of the roots of the quadratic congruence $μ^{2} \equiv D mod m$ , with $D > 0$ square-free and $D\not\equiv 1 \bmod 4…

math.NT2020★ 20 cited

Accurate estimation of sums over zeros of the Riemann zeta-function

Richard P. Brent, David J. Platt, Timothy S. Trudgian

We consider sums of the form $\sum ϕ (γ)$ , where $ϕ$ is a given function, and $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We…

math.NT2020★ 2 cited

A harmonic sum over nontrivial zeros of the Riemann zeta-function

Richard P. Brent, David J. Platt, Timothy S. Trudgian

We consider the sum $\sum 1/ γ$ , where $γ$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in an interval $(0, T]$ , and consider the behaviour of the sum a…

math.NT2020★ 4 cited

Correlations of multiplicative functions in function fields

Oleksiy Klurman, Alexander P. Mangerel, Joni Teräväinen

We develop an approach to study character sums, weighted by a multiplicative function $f : F_{q} [t] \to S^{1}$ , of the form \begin{equation} \sum_{G\in \mathcal{M}_N}f(G)χ(G)ξ(G)…

math.NT2020★ 2 cited

Exact $p$ -adic computation in Magma

Christopher Doris

We describe a new arithmetic system for the Magma computer algebra system for working with $p$ -adic numbers exactly, in the sense that numbers are represented lazily to infinite $p…

math.NT2020

Parameterizing roots of polynomial congruences

Matthew Welsh

We use the arithmetic of ideals in orders to parameterize the roots $μ (mod m)$ of the polynomial congruence $F (μ) \equiv 0 (mod m)$ , $F (X) \in Z [X]$ monic, irreducible and…

math.NT2020★ 16 cited

On a question of Mordell

Andrew R. Booker, Andrew V. Sutherland

We make several improvements to methods for finding integer solutions to $x^{3} + y^{3} + z^{3} = k$ for small values of $k$ . We implemented these improvements on Charity Engine's global compu…

math.NT2020★ 10 cited

On the moments of the moments of $ζ (1/2 + i t)$

E. C. Bailey, J. P. Keating

Taking $t$ at random, uniformly from $[0, T]$ , we consider the $k$ th moment, with respect to $t$ , of the random variable corresponding to the $2 β$ th moment of $ζ (1/2 + i x)$ over the i…

math.NT2020★ 1 cited

The Riemann hypothesis is true up to $3 \cdot 1 0^{12}$

Dave Platt, Tim Trudgian

We verify numerically, in a rigorous way using interval arithmetic, that the Riemann hypothesis is true up to height $3 \cdot 1 0^{12}$ . That is, all zeroes $β + iγ$ of the Riemann zet…

math.NT2020

Computing the Galois group of a polynomial over a $p$ -adic field

Christopher Doris

We present a family of algorithms for computing the Galois group of a polynomial defined over a $p$ -adic field. Apart from the "naive" algorithm, these are the first general algori…

Bounded Exponential Sums with Multiplicative Coefficients

On Artin's conjecture on average and short character sums

Conductor exponents for families of hyperelliptic curves

Computing Euler factors of genus 2 curves at odd primes of almost good reduction

Arithmetic Ramsey theory over the primes

Murmurations of modular forms in the weight aspect

Finding Orientations of Supersingular Elliptic Curves and Quaternion Orders

The sum-product problem for integers with few prime factors

Polynomial progressions in topological fields

Semi-integral Brauer-Manin obstruction and quadric orbifold pairs

Polynomials represented by norm forms via the beta sieve

Beyond the Erdős discrepancy problem in function fields

The 2-parity conjecture for elliptic curves with isomorphic 2-torsion

L-series of harmonic Maass forms and a summation formula for harmonic lifts

Fine-scale distribution of roots of quadratic congruences

Accurate estimation of sums over zeros of the Riemann zeta-function

A harmonic sum over nontrivial zeros of the Riemann zeta-function

Correlations of multiplicative functions in function fields

Exact p-adic computation in Magma

Parameterizing roots of polynomial congruences

On a question of Mordell

On the moments of the moments of ζ(1/2+it)

The Riemann hypothesis is true up to 3⋅1012

Computing the Galois group of a polynomial over a p-adic field

Exact $p$ -adic computation in Magma

On the moments of the moments of $ζ (1/2 + i t)$

The Riemann hypothesis is true up to $3 \cdot 1 0^{12}$

Computing the Galois group of a polynomial over a $p$ -adic field