arXiv subjects

Barinder S. Banwait

Publications and source records attributed to Barinder S. Banwait.

13 recordsLinked to original sources

arXiv2026

Second derivatives of $p$-adic $L$-functions and the Shafarevich--Tate group of rank-two CM elliptic curves

For an elliptic curve $E/\mathbb{Q}$ of rank two with complex multiplication, Coates, Liang and Sujatha gave a criterion for the vanishing of $Sha(E/\mathbb{Q})[p^\infty]$ at a good ordinary prime $p$ and applied it to five such curves for $p < 30{,}000$. We prove a cyclotomic criterion of the same kind: outside an explicit set of primes, the normalised second Taylor coefficient of the Mazur-Tate-Teitelbaum $p$-adic $L$-function at the central point is a $p$-adic unit if and only if the cyclotomic $p$-adic regulator is a unit and $Sha(E/\mathbb{Q})[p^\infty] = 0$, and, by the theory of Bannai and Kobayashi, if and only if an explicit combination of three critical Hecke $L$-values of weight $2p - 1$ has valuation exactly two. Following the algorithm of Stein and Wuthrich, we compute the regulator for the same five curves at every good ordinary prime below $30{,}000$: it is a unit at all but three of the $8{,}050$ primes outside the excluded set. The one case that the criterion of Coates, Liang and Sujatha left open, $p = 577$ for $y^2 = x^3 + 34x$, is settled by the new criterion. A Lean 4 formalisation of the first equivalence, assuming stated results from the literature, is provided.

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arXiv2026

Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves

We investigate the extent to which the reduced minimal Weierstrass coefficients of an elliptic curve over $\mathbb{Q}$ may be computed from it's Frobenius traces. Decision tree models reveal that the first two reduced minimal Weierstrass coefficients can be recovered with perfect accuracy from the Frobenius traces at the primes $2$ and $3$, and the third by supplementing these two traces with the conductor parity. We subsequently prove explicit formulae for these coefficients using the Frobenius traces and conductor parity. These formulae appear to be new. In particular, we deduce that the first three reduced minimal Weierstrass coefficients of an elliptic curve are determined by its isogeny class.

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arXiv2026

A formal proof of the Ramanujan--Nagell theorem in Lean 4

We present a complete formalization, in the Lean interactive theorem prover with the Mathlib library, of the Ramanujan--Nagell theorem: the only integer solutions to the Diophantine equation $x^2 + 7 = 2^n$ are $(n,x) \in \{(3,\pm1),(4,\pm3),(5,\pm5),(7,\pm11),(15,\pm181)\}$. The formalization includes all dependencies, notably the computation of the ring of integers of the quadratic field $\mathbb{Q}(\sqrt{-7})$, its class number, and unit group. We describe the proof strategy, the architecture of the formalization, and the challenges encountered in bridging the gap between textbook proofs and their machine-checked counterparts, with particular attention to the algebraic number theory infrastructure required.

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arXiv2026

On the Visibility category of the Shafarevich--Tate group

Given an elliptic curve $E$ over $\Q$ and a nontrivial element $\sigma$ of its Shafarevich--Tate group $\Sha(E)$, we introduce the \textbf{Visualization category} $\V(E; \sigma)$ of abelian varieties that ``visualize'' $\sigma$ in the sense of Mazur, and we study minimal objects in this category. In particular, we show that there can be several minimal visualizing abelian varieties of different dimensions, answering a question of Mazur. We revisit two constructions of visualizing abelian varieties: restriction of scalars (as in the work of Agashe and Stein), and a construction due to de Jong (as in the work of Cremona and Mazur). We show that restriction of scalars typically produces minimal visualizations. When $\sigma$ has order $2$ or $3$, we build upon the de Jong construction and make it totally explicit. While the de Jong construction can produce non-minimal objects, an appropriate choice in the construction for order $2$ elements $\sigma$ yields an explicit genus $2$ curve whose Jacobian is a minimal visualization. For order $3$ elements we apply our algorithmic construction to Fisher's database of such elements, and obtain computational evidence that, in the absence of a $3$-isogeny, the de Jong construction yields a minimal visualization.

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arXiv2026

On the Identification of Elliptic Curves That Admit Infinitely Many Twists Satisfying the Birch-Swinnerton-Dyer Conjecture

Recent work of Burungale-Skinner-Tian-Wan established the first infinite families of quadratic twists of non-CM elliptic curves over $\mathbb{Q}$ for which the strong Birch-Swinnerton-Dyer (BSD) conjecture holds. Building on their results, we encode the required hypotheses into an explicit algorithm and apply it to the database of elliptic curves in the $L$-functions and Modular Forms Database (LMFDB), identifying all elliptic curves $E$ of conductor at most $500{,}000$ that admit infinitely many quadratic twists satisfying the strong BSD conjecture. Our computations provide certain numerical evidence for a conjecture of Radziwi{\l}{\l} and Soundararajan predicting Gaussian behavior in the analytic order of the Shafarevich-Tate group, while also observing a systematic positive bias within the BSD-satisfying subfamily.

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arXiv2024

Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curves

We train machine learning models to predict the order of the Shafarevich-Tate group of an elliptic curve over $\mathbb{Q}$. Building on earlier work of He, Lee, and Oliver, we show that a feed-forward neural network classifier trained on subsets of the invariants arising in the Birch--Swinnerton-Dyer conjectural formula yields higher accuracies ($> 0.9$) than any model previously studied. In addition, we develop a regression model that may be used to predict orders of this group not seen during training and apply this to the elliptic curve of rank 29 recently discovered by Elkies and Klagsbrun. Finally we conduct some exploratory data analyses and visualizations on our dataset. We use the elliptic curve dataset from the L-functions and modular forms database (LMFDB).

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arXiv2024

Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville

We study the problem of determining the groups that can arise as the torsion subgroup of an elliptic curve over a fixed quadratic field, building on work of Kamienny-Najman, Krumm, and Trbović. By employing techniques to study rational points on curves developed by Bruin and Stoll, we determine the possible torsion subgroups of elliptic curves over quadratic fields $\mathbb{Q}(\sqrt{d})$ for all squarefree $d$ with $|d| < 800$, improving on the previously known range of $-5 < d < 26$. We use our computations to study the validity of a conjecture of Granville concerning how many twists of a given hyperelliptic curve admit a nontrivial rational point.

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arXiv2023

Computing nonsurjective primes associated to Galois representations of genus $2$ curves

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action on $\ell$-torsion points of $A$ is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell's classification of maximal subgroups of $\mathrm{PSp_4}(\mathbb{F}_\ell)$, sampling of the characteristic polynomials of Frobenius, and the Khare--Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus~$2$ curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such curve.

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arXiv2023

Towards strong uniformity for isogenies of prime degree

Let $E$ be an elliptic curve over a number field $k$ of degree $d$ that admits a $k$-rational isogeny of prime degree $p$. We study the question of finding a uniform bound on $p$ that depends only on $d$, and obtain, under a certain condition on the signature of the isogeny, such a uniform bound by explicitly constructing nonzero integers that $p$ must divide. As a corollary we find a uniform bound on torsion points defined over unramified extensions of the base field, generalising Merel's Uniform Boundedness result for torsion.

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arXiv2022

Modularity over $\mathbb{C}$ implies modularity over $\mathbb{Q}$

We give an account of Mazur's proof that, for an elliptic curve over $\mathbb{Q}$, if it admits a nonconstant mapping from $X(N)$ defined over the complex numbers $\mathbb{C}$, for some $N$, then it also admits a nonconstant mapping from $X_0(M)$ for some (possibly different) $M$ defined over the rational numbers $\mathbb{Q}$. We also briefly discuss two open questions of Khare concerning uniformisations of elliptic curves by noncongruence modular curves. This expository note is based on the author's expository talk in March 2022 during the 2nd Trimester Program on "Modularity and the Generalised Fermat Equation" held online and organised by Bhaskaracharya Pratishthana in Pune, India.

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arXiv2022

Cyclic isogenies of elliptic curves over fixed quadratic fields

Building on Mazur's 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over $\mathbb{Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised. In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields $\mathbb{Q}(\sqrt{d})$ with $|d| < 10^4$ we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over $19$ quadratic fields, including $\mathbb{Q}(\sqrt{213})$ and $\mathbb{Q}(\sqrt{-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves $X_0(125)$ and $X_0(169)$, which may be of independent interest.

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arXiv2022

Explicit isogenies of prime degree over number fields

We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of Box-Gajovi\'c-Goodman we determine the first instances of isogenies of prime degree for cubic number fields, as well as for several quadratic fields not previously known. While the correctness of the general algorithm relies on the Generalised Riemann Hypothesis, the algorithm is unconditional for the restricted class of semistable elliptic curves.

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arXiv2022

Explicit isogenies of prime degree over quadratic fields

Let $K$ be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute the primes $p$ for which there exists an elliptic curve over $K$ admitting a $K$-rational $p$-isogeny. This builds on work of David, Larson-Vaintrob, and Momose. Combining this algorithm with work of Bruin-Najman, Özman-Siksek, and most recently Box, we determine the above set of primes for the three quadratic fields $\mathbb{Q}(\sqrt{-10})$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\sqrt{7})$, providing the first such examples after Mazur's 1978 determination for $K = \mathbb{Q}$. The termination of the algorithm relies on the Generalised Riemann Hypothesis.

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