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David Kurniadi Angdinata

Publications and source records attributed to David Kurniadi Angdinata.

7 recordsLinked to original sources
arXiv2026

Tamagawa numbers and torsion of elliptic curves over function fields

We study the divisibilities of Tamagawa numbers $ c(E) $ of elliptic curves $ E $ over global function fields $ K $ in terms of their torsion subgroups $ E(K)_{\operatorname{tors}} $. In particular, for a non-isotrivial elliptic curve $ E / k(t) $, where $ k $ is a finite field of characteristic greater than $ 3 $, we prove that $ |E(k(t))_{\operatorname{tors}}|^2 $ divides $ c(E) $, except possibly in four exceptional torsion families. More specifically, we give a complete characterisation of divisibilities for $ c(E) $ in each torsion family, and provide explicit examples to prove that they are best possible. Over a general global function field, we also prove that a rational point of prime order $ 5 \le N \le 101 $ on $ E / K $ forces $ N^2 $ to divide $ c(E) $, which motivates our result. Finally, we formulate a conjecture on the leading coefficient of the $ L $-function of $ E / K $, motivated by the integrality of Birch--Swinnerton-Dyer quotients.

math.NT↗
arXiv2026

On the paucity of lattice triangles

A rational triangle $T$ (one whose angles are rational multiples of $π$) unfolds to a translation surface ${X_T}$. The lattice triangle problem asks to classify those $T$ for which ${X_T}$ is a Veech (lattice) surface, which means that the $\operatorname{SL}_2(\mathbb R)$-orbit of ${X_T}$ is closed in its stratum (so its projection to moduli space is a Teichmüller curve). The most mysterious regime is the "hard obtuse window" (largest angle in $(π/2,2π/3]$), where it is conjectured that no lattice triangles exist. Using an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, we prove a quantitative theorem that rules out all but a proportion $n^{-1+o(1)}$ of the triangles in this window with denominator $n$. The main technical result in our proof was autoformalized by AxiomProver in Lean (using mathlib).

math.DS↗
arXiv2026

ABC implies that Ramanujan's tau function misses almost all primes

Lehmer conjectured that Ramanujan's tau-function never vanishes. In a related direction, a folklore conjecture asserts that infinitely many primes arise as absolute values of Ramanujan's tau-function. Recently, Xiong showed that these prime values form a subset of the primes with density at most $2/11$. Assuming the $abc$ Conjecture, we prove the stronger upper bound \[ S(X):=\#\{\ell\le X:\ \ell\ \text{prime and } |τ(n)|=\ell \text{ for some } n\ge 1\} = O(X^{13/22}), \] which implies that Ramanujan's tau-function misses a density 1 subset of the primes. We give a heuristic suggesting that $S(X)$ should nevertheless be infinite, with predicted order of magnitude \[ S(X)\asymp \frac{C X^{\frac{1}{11}}}{(\log X)^2}. \] The main engine in this note was formalized and produced automatically in Lean/Mathlib by AxiomProver from a natural-language statement of the problem.

math.NT↗
arXiv2026

Computing L-functions of $ λ$-adic representations of global function fields

The L-function $ L(ρ_λ, s) $ of an almost everywhere unramified $ λ$-adic representation $ ρ_λ$ of a global function field $ \mathbb{F}_q(C) $ is known to be a rational function in $ q^{-s} $ satisfying a functional equation up to some complex sign $ ε(ρ_λ) $. This paper presents a systematic framework to compute the coefficients of $ L(ρ_λ, s) $ and its sign $ ε(ρ_λ) $ with some explicit examples.

math.NT↗
arXiv2024

L-values of elliptic curves twisted by cubic characters

Given a rational elliptic curve $ E $ of analytic rank zero, its L-function can be twisted by an even primitive Dirichlet character $ χ$ of order $ q $, and in many cases its associated central algebraic L-value $ \mathcal{L}(E, χ) $ is known to be integral. This paper derives some arithmetic consequences from a congruence between $ \mathcal{L}(E, 1) $ and $ \mathcal{L}(E, χ) $ arising from this integrality, with an emphasis on cubic characters $ χ$. These include $ q $-adic valuations of the denominator of $ \mathcal{L}(E, 1) $, determination of $ \mathcal{L}(E, χ) $ in terms of Birch--Swinnerton-Dyer invariants, and asymptotic densities of $ \mathcal{L}(E, χ) $ modulo $ q $ by varying $ χ$.

math.NT↗
arXiv2023

An Elementary Formal Proof of the Group Law on Weierstrass Elliptic Curves in any Characteristic

Elliptic curves are fundamental objects in number theory and algebraic geometry, whose points over a field form an abelian group under a geometric addition law. Any elliptic curve over a field admits a Weierstrass model, but prior formal proofs that the addition law is associative in this model involve either advanced algebraic geometry or tedious computation, especially in characteristic two. We formalise in the Lean theorem prover, the type of nonsingular points of a Weierstrass curve over a field of any characteristic and a purely algebraic proof that it forms an abelian group.

cs.LO↗