All Questions
Tagged with reference-request nt.number-theory
388 questions with no upvoted or accepted answers
41
votes
0
answers
2k
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 ...
- 2,029
29
votes
0
answers
3k
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 ...
- 3,022
26
votes
0
answers
567
views
Elliptic analogue of primes of the form $x^2 + 1$
I have a project in mind for an undergraduate to investigate next quarter -- a curiosity really, but I'm surprised I can't find it in the literature. I do not want a detailed analysis here... but ...
- 13.3k
24
votes
0
answers
1k
views
Exotic 4-spheres and the Tate-Shafarevich Group
The title is a talk given by Sir M. Atiyah in a conference with the following abstract:
I will explain a deep analogy between 4-dimensional smooth geometry (Donaldson theory)...
- 1,629
19
votes
0
answers
523
views
univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
- 109k
18
votes
0
answers
718
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
$$\frac{|\{n \le x : \...
- 211
17
votes
0
answers
367
views
Average value of j-invariant at infinity
Let $\xi\in\mathbb{R}$ and consider the average value (with respect to hyperbolic length) of the $j$-invariant ($j(z)=q^{-1}+744+196884q+\ldots$, $q=e^{2\pi iz}$) along a geodesic aimed at $\xi$:
$$
\...
- 609
17
votes
0
answers
891
views
An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$
This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...
- 10.5k
16
votes
0
answers
11k
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Deligne's letter to Jean-Pierre Serre
I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform ...
15
votes
0
answers
673
views
Exposition of Drinfeld's proof of function field Langlands for GL(2)
I know, or think I know, the vague outline of the proof: the Galois-to-automorphic direction is "classical," i.e. follows from converse theorems due to Grothendieck et al., and for the ...
- 311
15
votes
0
answers
591
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 ...
- 605
15
votes
0
answers
476
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 ...
- 201
14
votes
0
answers
358
views
How do we deduce the Jacquet-Langlands correspondence from Fargues' two towers?
In trying to understand the geometric proof of the local-Langlands and Jacquet-Langlands correspondence which uses Fargues's two tower theorem, I am having trouble finding a nice source on this, and I ...
- 3,539
14
votes
0
answers
481
views
If $ab^2$ is a sum of three squares, then so is $a$. How to see it quickly?
Here $a, b$ are positive integers, and the squares are the squares of integers. This follows from Legendre's three squares theorem, but is there a direct way?
- 109k
14
votes
0
answers
644
views
Can there be arbitrarily many cubic fields unramified outside $\{p,\infty\}$?
Observe, trivially, that since quadratic fields correspond to rational integers modulo squares (viz. discriminants), there are (roughly about, but certainly at most) $2^{|S|+1}$ quadratic fields ...
- 13.8k
13
votes
0
answers
328
views
Upper bound on prime powers in interval
I just spent a full day on the brutish and thankless task of proving that the Brun-Titchmarsh bound holds for prime powers (including primes), and not just for primes, in the following senses:
(a) the ...
- 20.2k
13
votes
0
answers
523
views
Euler Subgroups and Automorphic L-functions
Recently, I have read about the Whittaker expansion for $\mathrm{GL}_n$ and was struck by the utility of the mirabolic subgroup, $\mathrm{P}_n\subset \mathrm{GL}_n$ of matrices with bottom row $(0\; 0 ...
- 2,516
11
votes
0
answers
290
views
Color your partitions by parity
Let $a_c(n)$ be the number of ways to partition a positive integer $n$ where each even part comes in $c$ colors. Then, we can supply the generating function
$$\sum_{n\geq0}a_c(n)q^n=\prod_{k\geq1}\...
- 43.2k
11
votes
0
answers
374
views
Example of abelian variety over finite field which doesn't lift
What is an example of an abelian variety over a finite field $\mathbb{F}_p$ which doesn't lift to $\mathbb{Z}_p$? This question seems to hint that they should exist, but no example is given.
Note that ...
- 21.3k
11
votes
0
answers
269
views
Proving a group with two generators is not free that uses the Brahamagupta-Pell equation
Hello I encountered the following while reading a set of notes on free groups. It's not a homework question.
"Does there exist a rational number $\alpha$ with $0 <|\alpha| < 2$ such that the ...
- 111
11
votes
0
answers
324
views
Why is the CM-type preserved after base changing from char 0 to char p?
There is a transition in the theory of complex multiplication which seems to be glossed over in all expositions I can find. I would like to explicitly find a theorem that allows me to do this.
...
- 3,539
11
votes
0
answers
576
views
What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
10
votes
0
answers
287
views
Coefficients of polynomials vs trigonometric product
Let's consider the family of sequences of coefficients in the expansion
$$\prod_{i=0}^{n-1}(1+x^{3^i}+x^{3^{i+1}})=\sum_{k\geq0}a_n(k)\, x^k.$$
Remark. Evidently, the RHS is a finite sum.
Here is a ...
- 43.2k
10
votes
0
answers
351
views
How are the hypergeometric motives of WZ-Pairs connected?
If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
- 2,836
10
votes
0
answers
598
views
Does the interior of Pascal's triangle contain three consecutive integers?
This question defeated Math SE, so I am posting it here.
Consider the interior of Pascal's triangle: the triangle without numbers of the form $\binom{n}{0},\binom{n}{1},\binom{n}{n-1},\binom{n}{n}$.
...
- 3,527
10
votes
0
answers
286
views
Published reference on the automorphism group of modular curves $X_1(N)$?
I wish to cite that the automorphism groups of $X_1(N)$ have already been completely calculated, and what they are, but I am having difficulty finding this calculation in the literature.
I have ...
- 3,539
10
votes
0
answers
373
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Local Langlands Correspondence for unramified principal series representations
Let $G$ be a connected, reductive group over a $p$-adic field $k$. Assume $G$ is quasi-split with Borel subgroup $B = TU$. Consider those irreducible admissible representations $\pi$ of $G(k)$ which ...
- 6,180
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
9
votes
1
answer
650
views
Sum of three squares as class numbers and Waldspurger's formula
It is known that the number of ways to express $n \in \mathbb{Z}_{\geq 0}$ as a sum of three squares (let's denote it as $r_3(n)$) can be expressed as Hurwitz-Kronecker class number (certain weighted ...
- 2,215
9
votes
0
answers
462
views
Who realized the finite fields $\mathbb F_{p^n}$ first? Gauss or Galois?
Let $p$ be a prime, and let $n$ be a positive integer. The finite field $\mathbb F_{p^n}$ is often called a Galois field and denoted by $\mathrm{GF}(p^n)$ by researchers on coding theory.
On the other ...
- 15.6k
9
votes
0
answers
358
views
Being even or odd in the product expansion $\prod(1+x^k+x^{k+1})$
Consider the generating function of "partitions with distinct parts"
$$\sum_nQ(n)x^n=\prod_k(1+x^k).$$
It's known that
$$\left[\prod_k(1+x^k)\right] \mod 2=\prod_m(1-x^m)=\sum_{j\in\mathbb{Z}...
- 43.2k
9
votes
0
answers
251
views
Exponential sums over integers with a fixed number of prime divisors
Are there bounds in the literature on sums of the form
$$\sum_{\omega(n)= k} e(\alpha n) \;\;\;\;\;\text{or}\;\;\;\;\; \sum_{\Omega(n)=k} e(\alpha n)$$
for $\alpha$ on minor arcs (i.e., not very close ...
- 20.2k
9
votes
0
answers
886
views
How many ways are there to teach class field theory?
I will soon have to teach class field theory (I do not know whether it will be local or global yet:)) to postgraduate students. I wonder, which approaches to this subject(s) exist now.
I definitely ...
- 16.9k
9
votes
0
answers
271
views
Cancellation in a sum of Möbius evaluated along a quadratic form
Let $Q(x,y)$ be an indefinite binary quadratic form. Suppose $0 < B < \sqrt{A} $ are such that $B \gg \sqrt{A}$.
Is it true one can save an arbitrary power of log from the trivial bound in
$$...
- 2,283
9
votes
0
answers
409
views
The proof of Kazhdan's density theorem (And does it hold over positive characteristic?)
When proving identities about traces of functions on representations of $p$-adic groups, Kazhdan's density theorem indicates one only has to check equalities of traces on tempered representations. ...
- 181
9
votes
0
answers
230
views
Clozel's unpublished manuscript
I'm looking for Clozel's unpublished manuscript
L. Clozel, Modular properties of automorphic representations I: Applications of
the Selberg trace formula (1993)
cited in Urban's Eigenvarieties ...
- 3,799
9
votes
0
answers
910
views
Grothendieck's motivation of crystalline cohomology
Here Illusie mentions Grothendieck's observation that using Gauss-Manin connection one can give a non-canonical isomorphism between de Rham cohomology of smooth schemes over $W(k)$ with isomorphic ...
- 7,377
9
votes
0
answers
275
views
Complete list of exceptions and efficient algorithm for Waring's problem
2 weeks ago, Samir Siksek https://arxiv.org/abs/1505.00647 proved the more than 150-years-old conjecture that every positive integer other than 15, 22, 23, 50, 114, 167, 175, 186, 212, 231, 238, 239, ...
- 91
9
votes
0
answers
414
views
In which orders can the numbers of prime factors of consecutive integers be?
Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given a ...
- 2,719
9
votes
0
answers
3k
views
"Must read "papers on analytic number theory
Question: What would be some must-read
papers for an aspiring analytic number
theorist? In other words, what are the papers that any analytic number theorist would have read? (Background: Someone ...
9
votes
0
answers
462
views
$C^\infty$ function $f:{\bf C}\mapsto {\bf C}$ such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$
Suppose that $f:{\bf C}\mapsto {\bf C}$ is a $C^\infty$ function such that $f(z)\in\overline{{\bf Q}(z)}$ for all $z\in {\bf C}$, ie $f(z)$ is algebraic over the field ${\bf Q}(z)$ generated by $z$ ...
- 3,076
9
votes
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
9
votes
0
answers
605
views
Hilbert symbol and Weil index, beyond the quadratic case?
Let $F$ be a local nonarchimedean field. Let $n$ be a positive integer for which the group $\mu_n(F)$ of $n^{th}$ roots of unity in $F$ has order $n$. Let $\epsilon: \mu_n(F) \rightarrow C^\times$ ...
- 13.3k
8
votes
0
answers
208
views
Elliptic curves of rank 1 over number fields
I am interested what is known about the following statement:
For every number field $K$, there exists an elliptic curve $E$ defined over $K$ with algebraic rank equal to $1$.
Is this statement known ...
- 502
8
votes
0
answers
367
views
References for Yoichi Miyaoka's work around Fermat's Last Theorem
Apparently, Yoichi Miyaoka made a serious attempt to prove FLT in 1988. See the following question.
What were the main ideas and gaps in Yoichi Miyaoka's attempted proof (1988) of Fermat's Last ...
8
votes
0
answers
481
views
Formal degree of discrete series representations
Let $G$ be a locally compact unimodular group. A continuous irreducible unitary representation $\pi$ of $G$ is said to be a discrete series if its matrix coefficients (defined by $\xi^\pi_{v,w} : g \...
- 5,424
8
votes
0
answers
240
views
Question on calculating character sums
I am wondering if there are any references that would help me with the following problem:
Let $p>2$ be a prime number, $n \in \mathbb{Z}^+$ odd such that $(n,p)=1$, $\chi$ an imaginary quadratic ...
- 707
8
votes
0
answers
346
views
A generalization of Feit–Thompson conjecture, for square-free integers
I asked the following question with my account that I have for these sites Mathematics Stack Exchange and MathOverflow. The bounty that I offered in MSE expired without answers. The post that I refer ...
8
votes
0
answers
169
views
Why is the set of lifts of a p-divisible group canonically the same as the set of lines that span $M(G)/FM(G)$?
Let $M$ be the Dieudonne module of a p-divisible group $G_0$ over $k$, and let a lift of $G_0$ to $A$ be a p-divisible group $G$ over $A$ such that $G \otimes_A k \simeq G_0$. Let $\omega_G$ be the ...
- 3,539
8
votes
0
answers
161
views
Looking for an elliptic curve E st ${\large Ш}(\mathbb Q,E)$ cont. an element of order $p^2$ and certain other properties
I am looking for an elliptic curve $E$ with Weierstraß coefficients in $\mathbb{Q} $ so that for some prime $p$ the following conditions are satisfied:
(1) ${\large Ш}_{p^{\infty}}(\mathbb{Q},E)$ ...
- 1,805