50 citations · 143 across the 21 of their papers we have counts for
27 papers · 1 filter
Rational torsion on simple genus two Jacobians
Jennifer S. Balakrishnan, Filip Najman, Ari Shnidman +1
We exhibit new subgroups of rational torsion points in geometrically simple Jacobians of genus-two curves over . The largest group, which has order 96 and invariants [2,…
A refined Chabauty--Coleman bound for surfaces
Jennifer S. Balakrishnan, Jerson Caro
Caro and Pasten gave an explicit upper bound on the number of rational points on a hyperbolic surface that is embedded in an abelian variety of rank at most one. We show how to use…
Ogg's Torsion conjecture: Fifty years later
Jennifer S. Balakrishnan, Barry Mazur
Andrew Ogg's mathematical viewpoint has inspired an increasingly broad array of results and conjectures. His results and conjectures have earmarked fruitful turning points in our s…
Even values of Ramanujan's tau-function
Jennifer S. Balakrishnan, Ken Ono, Wei-Lun Tsai
In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer is a value of . For odd , Murty, Mur…
Quadratic Chabauty for modular curves: Algorithms and examples
Jennifer S. Balakrishnan, Netan Dogra, Jan Steffen Müller +2
We describe how the quadratic Chabauty method may be applied to explicitly determine the set of rational points on modular curves of genus whose Jacobians have Mordell--Weil…
Variations of Lehmer's Conjecture for Ramanujan's tau-function
Jennifer S. Balakrishnan, William Craig, Ken Ono
We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for we prove that $$τ(n)\not \in \{\pm 1, \pm 3, \pm 5, \…
Variants of Lehmer's speculation for newforms
Jennifer S. Balakrishnan, William Craig, Ken Ono +1
In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of or is a Fourier…
Two recent p-adic approaches towards the (effective) Mordell conjecture
Jennifer S. Balakrishnan, Alex J. Best, Francesca Bianchi +4
We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usual…
Explicit quadratic Chabauty over number fields
Jennifer S. Balakrishnan, Amnon Besser, Francesca Bianchi +1
We generalize the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary nu…
Chabauty-Coleman experiments for genus 3 hyperelliptic curves
Jennifer S. Balakrishnan, Francesca Bianchi, Victoria Cantoral-Farfán +2
We describe a computation of rational points on genus 3 hyperelliptic curves defined over whose Jacobians have Mordell-Weil rank 1. Using the method of Chabauty an…
An effective Chabauty-Kim theorem
Jennifer Balakrishnan, Netan Dogra
The Chabauty--Kim method is a method for finding rational points on curves under certain technical conditions, generalising Chabauty's proof of the Mordell conjecture for curves wi…
Explicit Chabauty-Kim for the Split Cartan Modular Curve of Level 13
Jennifer S. Balakrishnan, Netan Dogra, J. Steffen Müller +2
We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compu…
Explicit Coleman integration for curves
Jennifer S. Balakrishnan, Jan Tuitman
The Coleman integral is a -adic line integral that plays a key role in computing several important invariants in arithmetic geometry. We give an algorithm for explicit Coleman i…
Quadratic Chabauty and rational points II: Generalised height functions on Selmer varieties
Jennifer S. Balakrishnan, Netan Dogra
We give new instances where Chabauty--Kim sets can be proved to be finite, by developing a notion of "generalised height functions" on Selmer varieties. We also explain how to comp…
Shadow lines in the arithmetic of elliptic curves
Jennifer S. Balakrishnan, Mirela Ciperiani, Jaclyn Lang +2
Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a correspond…
Constructing genus 3 hyperelliptic Jacobians with CM
Jennifer S. Balakrishnan, Sorina Ionica, Kristin Lauter +1
Given a sextic CM field , we give an explicit method for finding all genus 3 hyperelliptic curves defined over whose Jacobians are simple and have complex multiplic…
Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks
Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan +3
Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava-Shankar studying the average sizes…
Quadratic Chabauty and rational points I: p-adic heights
Jennifer S. Balakrishnan, Netan Dogra
We give the first explicit examples beyond the Chabauty-Coleman method where Kim's nonabelian Chabauty program determines the set of rational points of a curve defined over $\mathb…
Computing integral points on hyperelliptic curves using quadratic Chabauty
Jennifer S. Balakrishnan, Amnon Besser, J. Steffen Müller
We give a method for the computation of integral points on a hyperelliptic curve of odd degree over the rationals whose genus equals the Mordell-Weil rank of its Jacobian. Our appr…
Quadratic Chabauty: p-adic height pairings and integral points on hyperelliptic curves
Jennifer S. Balakrishnan, Amnon Besser, J. Steffen Müller
We give a formula for the component at p of the p-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for finding p-ad…
Comparing arithmetic intersection formulas for denominators of Igusa class polynomials
Jacqueline Anderson, Jennifer S. Balakrishnan, Kristin Lauter +2
Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for CM(K).G_1 under s…
A p-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties
Jennifer S. Balakrishnan, J. Steffen Müller, William A. Stein
Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordi…
A non-abelian conjecture of Tate-Shafarevich type for hyperbolic curves
Jennifer Balakrishnan, Ishai Dan-Cohen, Minhyong Kim +1
We state a conjectural criterion for identifying global integral points on a hyperbolic curve over in terms of Selmer schemes inside non-abelian cohomology functors wi…
Coleman-Gross height pairings and the -adic sigma function
Jennifer S. Balakrishnan, Amnon Besser
We give a direct proof that the Mazur-Tate and Coleman-Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman-Gross height to the case of divisors…