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MATH-GA 2210.001: Homework Number theory 3

1. Let m, n∈ Z with n ≥ 1 and let L⊂ R2 be the lattice L ={(a, b) ∈ Z, a ≡ mbmod n}. Find a Z-basis of L.

2. Show that {(a, b, c) ∈ Z3, a ≡ b mod 5, b ≡ a+cmod 2} is a lattice in R3. Compute its covolume and give a Z-basis.

3. Let p be a prime such that −6 is a square modulo p. Show that p (resp. 2p) could be written asp=a2+ 6b2 iff p≡ ±1mod 8(resp. p≡ ±3mod 8.) 4. (the lattice E8) Let

D8 ={(x1, . . . x8)∈Z8,X

xi ≡0mod 2}

and lete∈R8 be the vector:

e= (1 2,1

2,1 2,1

2,1 2,1

2,1 2,1

2).

We consider the standard scalar product on R8. (a) Show that D8 is a lattice in R8, of covolume 2.

(b) Deduce that E8 :=Ze+D8 is a lattice in R8 of covolume1.

(c) Verify that for any v, w∈E8, one has v·v ∈2Z and v·w∈Z.

(d) Deduce that E8 is not of the form g(Z8) for g in the orthogonal group g ∈O8(R).

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