Prime numbers, Diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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41
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2k views

the topology of arithmetic progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...
41
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3k views

Grothendieck-Teichmuller conjecture

(1) In "Esquisse d'un programme", Grothendieck conjectures Grothendieck-Teichmuller conjecture: the morphism $$ G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T}) $$ is an isomorphism. Here ...
35
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1k views

To what extent does Spec R determine Spec of the Witt vector ring over R?

Let $R$ be a perfect $\mathbb{F}_p$-algebra and write $W(R)$ for the Witt ring [i.e., ring of Witt vectors -- PLC] on $R$. I want to know how much we can deduce about $\text{Spec } W(R)$ from ...
34
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1k views

What does the theta divisor of a number field know about its arithmetic?

This question is about a remark made by van der Geer and Schoof in their beautiful article "Effectivity of Arakelov divisors and the theta divisor of a number field" (from '98) (link). Let me first ...
32
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1k views

Constructing non-torsion rational points (over Q) on elliptic curves of rank > 1.

Consider an elliptic curve E defined over Q. Assume that the rank of E(Q) is >=2. (Assume the Birch-Swinnerton-Dyer conjecture if needed, so that analytic rank = algebraic rank.) How do you construct ...
29
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741 views

(Approximately) bijective proof of $\zeta(2)=\pi^2/6$ ?

Given $A,B\in {\Bbb Z}^2$, write $A \leftrightarrows B$ if the interior of the line segment AB misses ${\Bbb Z}^2$. For $r>0$, define $S_r:=\{ \{A, B\} | A,B\in {\Bbb Z}^2,||A||<r,||B||<r, ...
28
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609 views

“Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
27
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494 views

The exponent of Ш of y^2 = x^3 + px, where p is a Fermat prime

For $d$ a non-zero integer, let $E_d$ be the elliptic curve $$ E_d \colon y^2 = x^3+dx. $$ When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD, $$ \# ...
26
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2k views

A generalisation of the equation n = ab + ac + bc

In a result I am currently studying (completely unrelated to number theory) I had to examine the solvability of the equation $n = ab+ac+bc$ where $n,a,b,c$ are positive integers $0 < a < b < ...
24
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983 views

Peano Arithmetic and the Field of Rationals

In 1949 Julia Robinson showed the undecidability of the first order theory of the field of rationals by demonstrating that the set of natural numbers $\Bbb{N}$ is first order definable in $(\Bbb{Q}, ...
23
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743 views

Optimization problem arising from the study of zeta zeros

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I ...
22
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620 views

Does a proof of Selberg's 3.2 inequality exist?

A well-known inequality of Montgomery and Vaughan (generalizing a result of Hilbert) states that $$ \left |\sum_{r \neq s} \frac{w_{r} \overline{w_{s}} }{\lambda_r - \lambda_s} \right| \leq \pi ...
22
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0answers
841 views

Ramanujan's $\tau(n)$ and continued fractions

In D.H. Lehmer's paper "Ramanujan's function $\tau(n)$, (Duke J. Math v. 10 1943, pp. 483-492), Lehmer states the Ramanujan conjecture $|\tau( p )|< 2p^{11/2}$, so that $p^{-11/2}\tau( p ...
21
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640 views

Monstrous Moonshine for Thompson group $Th$?

I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2, $$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...
20
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1k views

Name of amateur who gave a new proof of the Ramanujan-Nagell theorem?

In an article by George Johnson in the New York Times back in 1999, it says that an amateur mathematician from India once sent Ian Stewart a proof of the Ramanujan-Nagell theorem that the Diophantine ...
19
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554 views

Derivative of Class number of real quadratic fields

Let $\Delta$ be a fundamental quadratic discriminant, set $N = |\Delta|$, and define the Fekete polynomials $$ F_N(X) = \sum_{a=1}^N \Big(\frac{\Delta}a\Big) X^a. $$ Define $$ f_N(X) = ...
18
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545 views

Eichler-Shimura over Totally Real Fields

By Eichler-Shimura over totally real fields I mean the following conjecture. Conjecture. Let $K$ be a totally real field. Let $f$ be a Hilbert eigenform with rational eigenvalues, of parallel weight ...
18
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1k views

Permutations of $(Z/pZ)^*$

Let $p$ be a prime integer, and let $(\mathbb Z/p\mathbb Z)^*$ be the set of non-zero elements of $\mathbb Z/p \mathbb Z$. Denote by $S((\mathbb Z/p \mathbb Z)^*)$ the group of permutations of ...
18
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825 views

When does a modular form satisfy a differential equation with rational coefficients?

Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...
18
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1k views

What are the possible singular fibers of an elliptic fibration over a higher dimensional base?

An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one. It is often required for the ...
18
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1k views

Trigonometry related to Rogers--Ramanujan identities

For integers $n\ge2$ and $k\ge2$, fix the notation $$ [m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad [m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}. $$ Consider the following trigonometric ...
17
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790 views

Orders in number fields

Let $K$ be a degree $n$ extension of ${\mathbb Q}$ with ring of integers $R$. An order in $K$ is a subring with identity of $R$ which is a ${\mathbb Z}$-module of rank $n$. Question: Let $p$ be an ...
17
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329 views

probability of zero subset sum

Almost 17 years ago, I asked the following question on USENET, motivated by a method in numerology (I kid you not). Pick integers $n \ge 2$, $k \ge 1$. Toss $n$ $k$-sided dice. The sides of each die ...
17
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655 views

Could unramified Galois groups satisfy a version of property tau?

This is an experiment: there is a question I want to mention in an article I'm writing, and I am not sure it's a sensible question, so I will ask it here first, in the hopes that if it's insensible ...
16
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714 views

Does this variant of a theorem of Hasse (really due to Gauss) have an “elementary” proof?

BACKGROUND Here are 3 theorems of varying difficulty. Let $M$ be the $Z/2$ subspace of $Z/2[[x]]$ spanned by $f^k$, with the $k>0$ and odd, and $f=x+x^9+x^{25}+x^{49}+\cdots$. For $g$ in $M$, let ...
16
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0answers
546 views

Can one compare integral structures on de Rham and crystalline cohomology?

Suppose $\mathfrak{X}$ is a smooth projective scheme of finite type over $\mathbb{Z}_p$, with generic fibre $X$. Then there are comparison theorems relating de Rham and crystalline cohomology, ...
16
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549 views

Smooth proper schemes over Z with points everywhere locally

This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question. Question. Is there a smooth proper scheme ...
15
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527 views

K-theory and rings of integers

From the works of Borel and Quillen there is a connection between the $K$-theory of the ring of integers $\mathfrak{o}_K$ in a number field $K$ and the arithmetic of the number field. In fact, it is ...
15
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634 views

Most “natural” proof of the existence of Hilbert class fields

Assume that you have proved the two inequalities of class field theory, and that you want to show that the Hilbert class field, i.e., the maximal unramified abelian extension, of a number field $K$ ...
15
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392 views

Cohomological characterization of CM curves

In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable ...
14
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671 views

Polynomial bijection from ZxZ to Z?

It is known that the polynomial $f(n,m)=\frac{1}{2}(n+m)(n+m+1)+m$ defines bijection $\mathbb{N}\times\mathbb{N}\to\mathbb{N}$ (Put pairs of $\mathbb{N}$ into the semi-infinite matrix and count them ...
14
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362 views

Erdos-Kac for squarefree numbers

In its usual form, the Erdos-Kac Theorem states that if $f(n) : \mathbb{N} \rightarrow \mathbb{R}$ is a strongly additive function with $|f(p)| \le 1$ for all primes $p$, then $$\frac{\\#\{n \le x : ...
14
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0answers
380 views

Torsion points of abelian varieties in the perfect closure of a function field

The following is a problem, which was recently brought to my attention by H. Esnault and A. Langer. Let $K$ be the function field of a smooth curve over the algebraic closure $k$ of the finite field ...
14
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663 views

Fake CM elliptic curves

Suppose one has an elliptic curve $E$ over $\mathbb{Q}$ with conductor $N < k^3$ for some (large) positive $k$, with the property that its Fourier coefficients satisfy $$ a_p=0, \; \mbox{ for all ...
14
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390 views

Division fields of abelian varieties over function fields

Let $k$ be a finitely generated field (for example a finite field or a number field) and $K/k$ a finitely generated regular extension with $trdeg(K/k)=1$. Let $A/K$ be a principally polarized abelian ...
14
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624 views

Computation of low weight Siegel modular forms

We have these huge tables of elliptic curves, which were generated by computing modular forms of weight 2 and level Gamma_0(N) as N increased. For abelian surfaces over $\mathbf{Q}$ we have very ...
13
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561 views

Rational iterations on $\mathbb{P}^1$ defined over $\mathbb{Q}$ and possessing a totally $2$-adic point of a high finite order

A (final) remark (9/29). Now that the question is open for bounty anyway, and hence cannot be deleted even though everything boils down to the line added on 9/28, I may as well record -- for anyone ...
13
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314 views

Integer-valued power towers

$$\text{Let }f_n(a)=\underbrace{2^{2^{.^{.^{.^{2^a}}}}}}_{\text{$n$ 2s}}.$$ Obviously, $f_n(a)$ is an integer for every positive integer $n$ and non-negative integer $a$. Are there any positive ...
13
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0answers
351 views

What can be the dimension of a pointless smooth proper Z-scheme?

What is the smallest dimension $d$ such that there is a smooth proper morphism $X \to \operatorname{Spec} \mathbb Z$ of relative dimension $d$, with $X$ nonempty, without a section? Of course, ...
13
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407 views

Bloch-Kato conjecture and Wiles' numerical criterion

I already asked this question some days ago on http://math.stackexchange.com/questions/158747/bloch-kato-conjecture-and-wiles-numerical-criterion but didn't receive any response. In the introduction ...
13
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504 views

What's about N. M. Katz' “over-world” of exp. sums?

Having just read in N. M. Katz' beautiful old survey on exponential sums a d differential eq.s, I wonder what became out of his question (on p. 297 - 300) on a "general conceptual framework in which ...
13
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774 views

Defining $\mathbb{Z}$ in $\mathbb{Q}$

It was proved by Poonen that $\mathbb{Z}$ is definable in $\mathbb{Q}$ using $\forall \exists$ formula. Koenigsmann has shown that $\mathbb{Z}$ is in fact definable by universal formula. What is the ...
13
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348 views

Any references on zeta-function like sums of inverse determinants over lattices of matrices?

I'm sorry for the title, it was little difficult to phrase.. Let us consider a matrix lattice $L\subset M_n(\mathbb{C})$. By this I mean a discrete additive group in $M_n(\mathbb{C})$. Let us ...
13
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710 views

Deciding whether a given power series is modular or not

The degree 3 modular equation for the Jacobi modular invariant $$ \lambda(q)=\biggl(\frac{\sum_{n\in\mathbb Z}q^{(n+1/2)^2}}{\sum_{n\in\mathbb Z}q^{n^2}}\biggr)^4 $$ is given by $$ ...
13
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0answers
540 views

Galois theory: Generalization of Abel's Theorem?

Let $L$ stand for the field obtained by adjoining to ${\Bbb Q}$ all roots of all polynomials of the form $x^n+ax+b$, $a,b\in {\Bbb Q}$. What polynomials $p$ don't split over $L$? In particular, ...
13
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713 views

Special values of Artin L-functions

This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so. Background: Stark's conjecture interprets ...
13
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1k views

sums of digits of powers of integers

It is known (Senge and Straus, 1971, see also C.L.Stewart, 1980) that for every natural $a$, not a power of 10, and every natural $s$, there are only finitely many $k$ such that the sum of decimal ...
12
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342 views

Should the number of small solutions in Roth's theorem be bounded uniformly, assuming the target is an algebraic integer?

Consider, on the one hand, algebraic integers $\alpha$ and their rational approximants to within a varying exponent $\kappa > 2$; and on the other hand, smooth projective geometrically irreducible ...
12
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499 views

How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
12
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230 views

For how many primes does an elliptic curve over a totally imaginary field have supersingular reduction?

An elliptic curve over a finite field, $k$, of characteristic p is called supersingular if it has no $p$-torsion over $k^{\mathrm{alg}}$, or equivalently, if $\mathrm{End}(E)$ is an order in a ...