The zeta function of an elliptic curve over a finite field #
Let W be an elliptic curve over a finite field F with q elements, π its q-power
Frobenius endomorphism and a_q = q + 1 - #W(F) its Frobenius trace. The zeta function of W is
the formal power series
Z(W/F, T) = exp (∑_{n ≥ 1} Nₙ Tⁿ / n) ∈ ℚ⟦T⟧,
where Nₙ = #W(𝔽_{qⁿ}) counts the points over an extension of degree n, the point at infinity
included. Here Nₙ is taken to be deg (1 - π ^ n), which is the number of points of W fixed
by π ^ n over a separable closure, and the number of points over any extension of F of degree
n (coeff_finrank_logOf_zetaFunction). The definition therefore depends on no chosen field with
q ^ n elements.
The zeta function is rational (Silverman V.2.4):
Z(W/F, T) = (1 - a_q T + q T²) / ((1 - T) (1 - q T)).
The traces tₙ = q ^ n + 1 - Nₙ of the powers of Frobenius satisfy t₀ = 2, t₁ = a_q and
tₙ₊₂ = a_q tₙ₊₁ - q tₙ (WeierstrassCurve.frobeniusPowTrace_add_two), and an exponential of
power sums ∑ (1 + qⁿ - tₙ) Tⁿ / n with such a sequence t is this rational function
(PowerSeries.subst_exp_mul_one_sub_X_mul_one_sub_C_mul_X). No roots of T² - a_q T + q are
introduced.
The closed form is an element zetaRatFunc of the rational function field ℚ(T), whose Laurent
expansion is Z(W/F, T). The functional equation Z(W/F, 1 / (q T)) = Z(W/F, T) is an identity
in ℚ(T), for the substitution T ↦ 1 / (q T) is the linear fractional transformation of ℚ(T)
with coefficient matrix !![0, 1; q, 0] but is not an operation on power series.
The Hasse bound a_q² ≤ 4q then gives the Riemann hypothesis for W: every complex zero of the
numerator 1 - a_q T + q T² has absolute value q^{-1/2}.
Main definitions #
WeierstrassCurve.zetaFunction: the zeta functionZ(W/F, T) ∈ ℚ⟦T⟧.WeierstrassCurve.zetaRatFunc: the rational function(1 - a_q T + q T²) / ((1 - T) (1 - q T))inℚ(T).
Main results #
WeierstrassCurve.logOf_zetaFunction: the logarithm ofZ(W/F, T)is∑ deg (1 - π ^ n) Tⁿ / n.WeierstrassCurve.coeff_finrank_logOf_zetaFunction: for a finite extensionE/Fof degreen, thenth coefficient of the logarithm is#W(E) / n.WeierstrassCurve.zetaFunction_mul_one_sub_X_mul_one_sub_C_mul_XandWeierstrassCurve.zetaFunction_eq_mul_inv: rationality,Z(W/F, T) = (1 - a_q T + q T²) / ((1 - T) (1 - q T)).WeierstrassCurve.coe_zetaRatFunc: the Laurent expansion ofzetaRatFuncisZ(W/F, T).WeierstrassCurve.mobiusAutOf_zetaRatFunc:zetaRatFuncis invariant underT ↦ 1 / (q T); for an elliptic curve this is the functional equationZ(W/F, 1 / (q T)) = Z(W/F, T)inℚ(T).WeierstrassCurve.norm_eq_inv_sqrt_card_of_one_sub_frobeniusTrace_mul_add_card_mul_sq_eq_zero: the Riemann hypothesis, the zeros of1 - a_q T + q T²have absolute valueq^{-1/2}.
References #
The zeta function as a rational function: the element
(1 - a_q T + q T²) / ((1 - T) (1 - q T)) of ℚ(T), where q is the number of elements of F
and a_q the Frobenius trace of W.
For an elliptic curve its Laurent expansion is the zeta function Z(W/F, T) (coe_zetaRatFunc).
Like the Frobenius trace, the formula is defined at every Weierstrass model, but only at an
elliptic one is it a zeta function.
Instances For
The defining equation of zetaRatFunc.
The rational function zetaRatFunc is invariant under the substitution T ↦ 1 / (q T),
where q is the number of elements of F. This substitution is the linear fractional
transformation with coefficient matrix !![0, 1; q, 0].
The identity is algebraic and holds at every Weierstrass model. For an elliptic curve, where
zetaRatFunc is the zeta function (coe_zetaRatFunc), it is the functional equation of the zeta
function (Silverman V.2.4): Z(W/F, 1 / (q T)) = Z(W/F, T) in ℚ(T).
The zeta function of an elliptic curve W over a finite field F with q elements:
Z(W/F, T) = exp (∑_{n ≥ 1} Nₙ Tⁿ / n), where Nₙ = deg (1 - π ^ n) for the q-power Frobenius
endomorphism π.
Nₙ is the number of points of W, the point at infinity included, over an extension of F of
degree n (coeff_finrank_logOf_zetaFunction). The zeta function is the rational function
(1 - a_q T + q T²) / ((1 - T) (1 - q T)) (zetaFunction_eq_mul_inv).
Instances For
The defining equation of zetaFunction.
The zeta function has constant coefficient 1.
The logarithm of the zeta function is ∑_{n ≥ 1} deg (1 - π ^ n) Tⁿ / n.
The zeta function counts points over finite extensions: for a finite extension E/F of
degree n, the nth coefficient of the logarithm of Z(W/F, T) is #W(E) / n, where the count
includes the point at infinity.
Rationality of the zeta function (Silverman V.2.4), without inverses:
Z(W/F, T) (1 - T) (1 - q T) = 1 - a_q T + q T², where q is the number of elements of F and
a_q the Frobenius trace of W.
Rationality of the zeta function (Silverman V.2.4):
Z(W/F, T) = (1 - a_q T + q T²) / ((1 - T) (1 - q T)), where q is the number of elements of F
and a_q the Frobenius trace of W.
The zeta function is the Laurent expansion of zetaRatFunc: the rational function
(1 - a_q T + q T²) / ((1 - T) (1 - q T)) in ℚ(T) expands to Z(W/F, T) in ℚ⸨T⸩. Since the
expansion map ℚ(T) → ℚ⸨T⸩ is injective, zetaRatFunc is the only rational function with this
expansion.
The Riemann hypothesis for an elliptic curve over a finite field (Silverman V.2.4): every
complex zero z of the numerator 1 - a_q T + q T² of the zeta function has |z| = q^{-1/2},
where q is the number of elements of F and a_q the Frobenius trace of W.