• Aucun résultat trouvé

Show that p (resp


Academic year: 2022

Partager "Show that p (resp"


Texte intégral


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


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).



Documents relatifs

By Baire’s theorem, an infinite dimensional Fréchet space cannot have a countable algebraic

M2R Université Grenoble Alpes 2019-2020 Introduction to analytic geometry (course by Jean-Pierre Demailly).. Sheet number 7, 21/11/2019 The goal of this sheet is to detail the proof

(15) (*) A topological space is quasi-compact if one can extract from any open covering a finite open covering.. Show that a Noetherian topological space

When it is asked to describe an algorithm, it has to be clearly and carefully done: input, output, initialization, loops, conditions, tests, etc.. Exercise 1 [ Solving

Montrer que l’on d´ efinit ainsi deux cartes compatibles de S n , qui munissent la sph` ere d’une structure de vari´ et´ e diff´ erentiable.. Tore de

In 1980, artist, curator and writer John Watt produced, curated and contributed to Television by Artists, a series of video art that was aired on Rogers Cable TV between M ay

We also give in Section 4 an alternative proof of Theorem 1.1 that uses the complex analytic version of the Abhyankar-Jung Theorem and the Artin Approximation Theorem..

Calculate the probability such that there is exactly one goal during the first half condi- tionally upon the fact that “les girondins” scored two goals during the football match..

n a n x n ayant un rayon de convergence R > 0 d´ efinit sur l’intervalle ] − R, R[ (ou sur le disque |x| < R si l’on travaille avec une variable complexe) une fonction

Dans toute la feuille, K désigne un corps commutatif..

[j -invariant] Let k be an algebraically closed field of characteristic different from 2

Also use the formulas for det and Trace from the


n} having exactly k fixed points.. (hint: argue


Can −1 be a quadratic residue in some congruence

.} be the set of positive integers which have the last digit 1 when written in base 10.. Determine all Dirichlet characters modulo 8 and


Show that a (connected) topological group is generated (as a group) by any neighborhood of its


.} be the set of positive integers which have the last digit 1 when written in


Among these ideals, which are