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49
votes
4answers
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
5answers
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
2answers
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
1answer
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
3answers
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
13answers
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
1answer
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
6answers
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
2answers
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
1answer
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
1answer
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
1answer
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
3answers
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
2answers
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
2answers
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
0answers
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), ...