ZETA FUNCTIONS. NO.10

YOSHIFUMI TSUCHIMOTO

Problems

According to Proposition 5.2, a congruent zeta function of an elliptic curve E is given as follows.

\frac{1 − aT + qT^{2}}{(1 − T)(1 − qT)}

where a is an integer. We note that by using a Taylor expansion

∑_{r=1}^{∞} \frac{#E(𝔽_{q^{r}})}{r} = log(\frac{1 − aT + qT^{2}}{(1 − T)(1 − qT)}) = (q − a + 1)T + O(T^{2})

we have

a = q + 1 − #E(𝔽_{q}).

Problem 10.1. Compute the congruent zeta function of an elliptic curve

{[X : Y : Z]; Y Z^{2} = X(X − Z)(X + Z)}

over 𝔽_{p} for a prime of your choice.

Problem 10.2. Find a formula for a congruent zeta function of elliptic curves

{[X : Y : Z]; Y Z^{2} = X(X − Z)(X − λZ)}

over 𝔽_{q} for λ ∈ 𝔽_{q}.

Problem 10.3. Compute the congruent zeta function of an elliptic surface of your choice.

Problem 10.4. Compute the congruent zeta function of a homology plane of your choice. Compare the result with the congruent zeta function of a plane.

Problem 10.5. Describe what happens to the conguent zeta function when we blow up an scheme.

YOSHIFUMI TSUCHIMOTO

Problem 10.6. Let R be a commutative ring which is fninitely generated over ℤ. Let φ : R → R be a ring homomorphism. Let us define “the semi direct product” R ⋊_{φ} ℕ as

R ×_{φ} ℕ = ⟨R, τ; τx = φ(x)τ (∀x ∈ R)⟩

Compute the zeta function of a category (A-modules). Compare it with the category of the dynamical system (Spec R, Spec φ).

Problem 10.7. Is it possible to describe zeta functions of dynamical systems as zeta functions of categories?