Tomorrow's Mathematics Gallery

If there were a mathematics driver's license written exam

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Mark ○ if the following statements are mathematically correct, and × if they are incorrect.

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1 (5 points). The connected component containing the identity element of an algebraic group G is a normal subgroup of finite index in G.

2 (5 points). If an elliptic curve over the rational number field has good reduction at a prime p, then any elliptic curve that is rational-isogenous to it also has good reduction at p.

3 (5 points). Group cohomology is a group.

4 (5 points). Isogenous elliptic curves have the same regulator.

5 (2 points × 3 problems). (1) The Kodaira symbol is a symbol that classifies the isomorphism classes as schemes of the special fiber over F_p-bar of the Neron minimal model of elliptic curves. (2) In type I_0*, the component group ε~(F_p)/ε~_0 of the Neron model has an element of order 4. (3) Elliptic curves with the same Kodaira symbol have the same Tamagawa number.

6 (5 points) Let p be a good prime for an elliptic curve E/Q. H^1(Q_p, E) is 0 because torsors over p-adic fields on which E acts necessarily have rational points.

Solutions:

1. ○ See "The Tamagawa number is the number of connected components of the special fiber of the Neron model" in the "Concrete Example 1" section.

2. ○ Neron-Ogg-Shafarevich criterion.

3. △ (When the group being acted upon is non-abelian, there is no group structure. For example, when Z/2Z acts trivially on S_3, H^1(Z/2Z, S_3) is not closed under the product of f_1: 1 → (12) and f_2: 1 → (13), so it does not form a group.)

4. × Correct except for I_0. This is a typical trick question from a driving school.

5. (1) × (2) × It doesn't have one. This fact has elementary applications including Lemma 2.6 of Klagsbrun's paper. That is, when applying a ramified twist D at v: good reduction, there exists no element of order 4 or higher in E_1^D(Q_v)! At this point, the Kodaira symbol at v is I_0*, and from the answer to this problem, there is no element of order 4. Then from the exact sequence 0 → E_1^D(Q_v) → E_0^D(Q_v) → F_v → 0, E_0 also has no order 4, leading to the conclusion. This shows that while the torsion of the Selmer group is quartic, solvability can essentially be determined by Hilbert symbols alone! (3) × The Kodaira symbol classifies geometry over F_p-bar, while the Tamagawa number is an arithmetic quantity over F_p, and they generally differ except for I_0, II, II*.

6. × Indeed, a genus 1 curve over F_p necessarily has points, and if p is good, this lifts to Q_p by Hensel's lemma. However, a good prime for an elliptic curve is not necessarily a good prime for the torsors it acts on. In fact, for n ≥ 1, H^1(Q_p, E)[n] is the dual of E(Q_p)/nE(Q_p) (Tate local duality) and is non-trivial.

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PPAP?

Question from Pikotaro: Calculate the Hilbert symbol (p,pa)_p , where p is a prime number and a is a unit number in Zp.

(Answer) When p≡1mod4, (a/p) When p≡3mod4, -(a/p)

(See §3.1 "Hilbert symbols" in "The relationship between the Furstenberg topology and the profinite completion of Z" in the "Mathematics PDF" section.

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