Local Factors in the BSD Conjecture: A Unified Statistical View
Abstract
For elliptic curves E/ℚ in short Weierstrass form E=E(a4,a6): y2=x3+a4x+a6, we derive a multivariable Euler product generating function which encodes the Tamagawa product Tam(E)=∏p cp(E). Using this generating function, we compute limiting distributions, exact covariances, and moment and tail bounds for four important statistics on the Tamagawa number; for example we show that more than half of all curves in this height ordering have trivial Tamagawa product, about 42.2% have exactly one prime with nontrivial local Tamagawa number, and only about 6.8% have exactly two such primes. The product is obtained by specializing an Euler product indexed by local reduction data that we derive from Tate's algorithm. The results in this paper were autoformalized in Lean by AxiomProver.
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BibTeX
@misc{angdinata2026local,
title = {{Local Factors in the BSD Conjecture: A Unified Statistical View}},
author = {David Kurniadi Angdinata and Kenny Lau and Ken Ono and Ashvin Swaminathan and Sameera Vemulapalli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10243},
eprint = {2610.10243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10243},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Local Factors in the BSD Conjecture: A Unified Statistical View
AU - David Kurniadi Angdinata
AU - Kenny Lau
AU - Ken Ono
AU - Ashvin Swaminathan
AU - Sameera Vemulapalli
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10243
UR - https://arxiv.org/abs/2610.10243
AB - For elliptic curves E/ℚ in short Weierstrass form E=E(a_4,a_6): y^2=x^3+a_4x+a_6, we derive a multivariable Euler product generating function which encodes the Tamagawa product Tam(E)=∏_p c_p(E). Using this generating function, we compute limiting distributions, exact covariances, and moment and tail bounds for four important statistics on the Tamagawa number; for example we show that more than half of all curves in this height ordering have trivial Tamagawa product, about 42.2% have exactly one prime with nontrivial local Tamagawa number, and only about 6.8% have exactly two such primes. The product is obtained by specializing an Euler product indexed by local reduction data that we derive from Tate's algorithm. The results in this paper were autoformalized in Lean by AxiomProver.
ER -