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**49**

votes

**4**answers

6k views

### How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...

**45**

votes

**5**answers

2k views

### If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?

Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous and amazingly tricky proof says that if we ...

**31**

votes

**2**answers

2k views

### Formal group laws and L-series

Let E be an elliptic curve, let $L(s) = \sum a_n n^{-s}$
denote its L-function, and set
$$ f(x) = \sum a_n \frac{x^n}{n}. $$
Then Honda has observed that
$$ F(X,Y) = f^{-1}(f(X) + f(Y)) $$
defines ...

**18**

votes

**1**answer

1k views

### Number of distinct values taken by x^x^…^x with parentheses inserted in all possible ways

For what positive x's the number of distinct values taken by x^x^...^x with parentheses inserted in all possible ways is not represented by the sequence A000081? Is it exactly the set of positive ...

**14**

votes

**3**answers

2k views

### sum of squares in ring of integers

Lagrange proved that every (positive) rational integer is a sum of 4 squares.
Are there general results like this for ring of integers of a number field? Is this class field theory?
Explicity, ...

**40**

votes

**13**answers

4k views

### Erratum for Cassels-Froehlich

Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).
IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has ...

**13**

votes

**1**answer

476 views

### components of E[p], E universal in char p.

I have just realised that a group scheme I've known and loved for years is probably a bit wackier than I'd realised.
In this question, in Charles Rezk's answer, I erroneously claim that his ...

**13**

votes

**6**answers

1k views

### Text for Algebraic Number Theory

I have the privilege of teaching an algebraic number theory course next fall, a rare treat for an algebraic topologist, and have been pondering the choice of text. The students will know some ...

**5**

votes

**2**answers

634 views

### Explicit examples of algebraic Hecke characters with infinite image?

Jerry Shurman has a lovely set of notes explaining the classical definition of Hecke characters, the idelic definition of Hecke characters, their relationship, and the classification of algebraic ...

**4**

votes

**1**answer

180 views

### Reduction mod $p$ of units in a ring of integers

Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$ and let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_k$. I'm looking for conditions on $k$ and $\mathfrak{p}$ which ...

**3**

votes

**1**answer

178 views

### The existence of elliptic curves with prescribed supersingular primes

For a given infinite set of primes, not too big, eg, satisfying Lang-Trotter conjecture, can we always find an E.C. with supersingular reduction (at least) at these primes? How about E.C. without CM?
...

**7**

votes

**1**answer

302 views

### When can number rings be spanned (as $\mathbb{Z}$-modules) by units?

Let $\mathcal{O}$ be the ring of integers in an algebraic number field. Define $R \subset \mathcal{O}$ to be the set of all $\mathbb{Z}$-linear combinations of units. Since the product of two units ...

**5**

votes

**3**answers

732 views

### Crystalline Characters

Let $K$, $L$ be finite extensions of the $p$-adic numbers. Suppose $\chi:G_K\rightarrow L^{\times}$ is crystalline. It is my understanding that if either $K$ or $L=\mathbb{Q}_p$, then $\chi$ must be a ...

**6**

votes

**2**answers

826 views

### Eigenvalues of nonnegative integer matrices

Edit
I realized that the key piece of information that I need is question 1, and so I'd like to rephrase this post:
What are the possible eigenvalues of nonnegative integer matrices?
Any answer ...

**4**

votes

**2**answers

174 views

### Even unimodular lattices with root system $32 A_1$

I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question.
For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ ...

**1**

vote

**0**answers

107 views

### Reference request for a basic result on relative differents & discriminants

I am looking for a better reference for the results in this extremely short and elementary paper:
Tôyama, Hiraku,
`A note on the different of the composed field',
Kōdai Math. Sem. Rep. 7 (1955), ...