We are expected to solve a conjecture of the title. Reference is Jean-Pierre Serre — Propriétés conjecturales des groupes de Galois motiviques et des représentations l-adiques. Precisely;
Conjecture A:
k:an algebraic number field
v:a non archimedean place of k
E:a motive with good reduction at v
For E, we have a corresponding unramified l-adic representation. The image of the decomposition group of that representation is topologically generated by a Frobenius element.
The conjecture is that the characteristic polynomial of that Frobenius is ℚ-coefficients and if E is pure of weight i, its roots in ℂ has absolute value Nv^i/2. Here Nv is the number of elements of the residue field of kv.
We also have Weil conjecture;
F:a finite field of q elements
X:a smooth projective variety over F
The characteristic polynomial of frobenius of its l-adic cohomology of degree i has absolute value q^i/2.
In addition we have two standard conjectures on algebraic cycles;
Lefschetz standard conjecture asserts that an abstract analogue of the Λ-operator of Hodge theory is induced by an algebraic cycle on two products of smooth projective varieties over an algebraic closed field.
Hodge standard conjecture asserts that there is an abstract version of the Hodge index theorem for ℚ-vector space of classes of algebraic cycles.
I have the following three questions.
1.Conjecture A and Weil conjecture both concern the roots of weight of a characteristic polynomial. How are these two related?
2.Weil conjecture is proved under the assumptions of Lefschetz and Hodge standard conjectures. Is the conjecture A also proved using them?
3.Is the conjecture A proved for abelian varieties over any number field?
Thank you for your answers.
Edit:motives are defined by numerical equivalence of algebraic cycles, good reduction of motives are defined by the corresponding l-adic representation is unramified
