Goal
Define the Weil polynomial (characteristic polynomial of Frobenius) of an abelian variety A/𝔽_q — P_A(t) = det(t − F | T_ℓ A), monic of degree 2g, ℓ-independent — its reverse the L-polynomial, and state the Honda–Tate theorem (isogeny classes ↔ Weil q-numbers / admissible Weil polynomials).
What already exists
- AINTLIB
HasseWeil: Tate module T_ℓ(E), Frobenius, Weil pairing, the Hasse bound (the g = 1 case). Sibling repos WeilConjectures, Hasse-Weil.
- mathlib: finite fields, char polynomials; little on abelian varieties as such.
What's missing
weilPolynomial A, lPolynomial A, the functional equation t^{2g} P_A(q/t) = q^g P_A(t), the Weil-number condition (|root| = √q), and the Honda–Tate bijection (research-level; Waterhouse's admissibility condition).
Test cases
g = 1: P_E(t) = t² − a_q t + q (recover the Hasse bound). A product E₁ × E₂. An ordinary vs supersingular example.
LMFDB targets
New area, not yet in the Verso blueprint — links go to the LMFDB knowls.
Goal
Define the Weil polynomial (characteristic polynomial of Frobenius) of an abelian variety
A/𝔽_q—P_A(t) = det(t − F | T_ℓ A), monic of degree2g, ℓ-independent — its reverse the L-polynomial, and state the Honda–Tate theorem (isogeny classes ↔ Weilq-numbers / admissible Weil polynomials).What already exists
HasseWeil: Tate moduleT_ℓ(E), Frobenius, Weil pairing, the Hasse bound (theg = 1case). Sibling reposWeilConjectures,Hasse-Weil.What's missing
weilPolynomial A,lPolynomial A, the functional equationt^{2g} P_A(q/t) = q^g P_A(t), the Weil-number condition (|root| = √q), and the Honda–Tate bijection (research-level; Waterhouse's admissibility condition).Test cases
g = 1:P_E(t) = t² − a_q t + q(recover the Hasse bound). A productE₁ × E₂. An ordinary vs supersingular example.LMFDB targets
av.fq.weil_polynomialav.fq.l-polynomialav.fq.honda_tateav.fq.frobenius_anglesNew area, not yet in the Verso blueprint — links go to the LMFDB knowls.