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3 votes
0 answers
680 views

Birch/Swinnerton-Dyer "Notes on Elliptic Curves II"

I would like to know if any of you know if there is a more general treatment to what Birch and Swinnerton-Dyer did in "Notes on Elliptic Curves II" (http://www.ams.org/mathscinet-getitem?mr=179168). ...
BlackAdder's user avatar
2 votes
1 answer
543 views

main conjecture of Iwasawa theory implies Herbrand-Ribet

I need a reference/proof for "main conjecture of Iwasawa theory => refinement of Herbrand-Ribet ($v_p(B_{p-i}) = v_p(|A_i|)$, where $A_i$ denotes the $i$-th eigenspace of the Galois group acting on ...
user avatar
3 votes
0 answers
293 views

multiplicity of automorphic representation of unitary similitude group

Let $G$ be a unitary similitude group over $\mathbb{Q}$ (as in the book of Harris-Taylor), $\pi$ an irreducible automorphic representation of $G(\mathbb{A})$. I'm looking for some results on its ...
Devine's user avatar
  • 66
8 votes
0 answers
335 views

Irreducibility of Galois representations attached to unitary groups

If $G$ is a unitary group in $n$ variables over $\mathbb Q$, attached to an hermitian form for an imaginary quadratic extension $E/\mathbb Q$ and if we suppose that the hermitian form is definite over ...
Joël's user avatar
  • 26k
2 votes
1 answer
240 views

Congruence for the Apery Numbers

Is it true that $$A_n\equiv (-1)^n\;\;(\mathrm{mod}\;3)\;\;?$$ Here $A_n$ is the Apery number: $$A_n=\sum\limits_{k=0}^n\binom{n}{k}^2\binom{n+k}{k}^2.$$ What is known about congruence properties ...
Zurab Silagadze's user avatar
1 vote
0 answers
285 views

Davenport's proof that almost all integers are the sum of 4 cubes

Where can I find a pdf that describes Davenport's proof that almost all integers are the sum of $4$ cubes?
Mayank Pandey's user avatar
1 vote
0 answers
70 views

A non-surjective coboundary map induced by a central extension

Let $k$ be a number field and $$ 1\to A \to B \to C \to 1$$ be a central extension of finite groups over $\mathcal{O}_k$ (the ring of integers of $k$), with $B$ non-commutative. Consider the induced ...
Coquelicot's user avatar
1 vote
0 answers
331 views

Where can I find the article of A. Borel: "Values of zeta-functions at integers, cohomology and polylogarithms"? [closed]

Where on the internet can I find this article? I know that it is in this book: Current trends in mathematics and physics, Narosa, New Delhi, 1995.
user avatar
7 votes
2 answers
522 views

How large (small) can be the measure of a set where a polynomial takes small values ?

A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question how large ( and small) can be the measure of a set where a polynomial takes small values ? This, and other ...
Vagabond's user avatar
  • 1,795
2 votes
0 answers
159 views

The dimension of the space of automorphic forms with multiplier system

Let $\Gamma$ be a discrete subgroup of $SL_{2}(\mathbb{Z})$ and $\vartheta$ a multiplier system of weight $k$ for $\Gamma$, by which we mean a function $\vartheta:\Gamma \rightarrow \mathbb{C}$ ...
M.Souf's user avatar
  • 433
1 vote
0 answers
122 views

Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...
user avatar
11 votes
3 answers
461 views

citation for first statement of the Re(s) = 6 conjecture on zeros of Ramanujan L function

Hi, for the bibliography of a paper I'm writing I seek a citation for the first statement of the conjecture that the nontrivial zeros of $F(s) = \sum_n\tau(n)n^{-s}$ all lie on the line Re(s) = 6. (...
Barry Brent's user avatar
2 votes
0 answers
179 views

Questions about transformation or integral transformation

I have asked several mathematicians about the following questions,but all of them think they are good questions,but can not give a complete answer.Now I have to come here to ask mathematicians all ...
XL _At_Here_There's user avatar
2 votes
1 answer
164 views

A problem on the finiteness of solutions to a Diophantine equations

Given two positive integers $a,b$, and an odd prime $p$, I want to know whether the number of solutions to the following equation is finite: $X^2=a+bp^{Y}$ where $X,Y$ are variables and are integers....
Tao Feng's user avatar
0 votes
1 answer
202 views

Reference request: on sums of the form $ax^m + by^n = h$

I know that equations of the form $$\displaystyle ax^d + by^d = h$$ with $a,b,h \in \mathbb{Z}$ have been thoroughly investigated as a special (and interesting) case of the Thue-Mahler equation, for ...
Stanley Yao Xiao's user avatar
1 vote
0 answers
165 views

Square-free sieve over number fields

I am trying to work on extending various methods to study square-free values of polynomials (or more generally, $k$-free values) over general rings of integers, and a literature review has yielded ...
Stanley Yao Xiao's user avatar
3 votes
1 answer
528 views

Karolyi's theorem for finite groups and its extensions

Suppose that $\mathbb A = (A, +)$ is a (possibly non-commutative) group, and denote by $p(\mathbb A)$ the minimum of $|S|$ as $S$ ranges in the set of non-trivial subgroups of $\mathbb A$, with the ...
Salvo Tringali's user avatar
6 votes
0 answers
252 views

How big is the Fourier transform of the log of a polynomial over the p-adic numbers

Let $f(z_1,\dots,z_n)$ be a polynomial with $p$-adic coefficients, and let $g(z):=log\lvert f(z) \rvert$. If $\xi$ is a complex character of $\mathbb{Z}_p^n$ there exists a number $v=v(\xi)$ such that ...
Tzanko Matev's user avatar
2 votes
0 answers
114 views

Distribution of residue classes of totients of (univariate) polynomials

Let $\phi$ denote Euler's totient function and $f$ a non-constant univariate polynomial with integer coefficients such that $f(u) \in \mathbf N^+$ for all $u \in \mathbf N^+$ (assume $f$ is ...
Salvo Tringali's user avatar
8 votes
0 answers
595 views

A property of supersingular $j$-invariants (reference request)

Edit 2: For those who understandably don't want to read such a long post, I think Voloch's suggestion reduces the problem to asking whether $j$-invariants of supersingular curves are 3rd powers in $\...
DCT's user avatar
  • 1,537
7 votes
0 answers
695 views

modularity of elliptic curves with cm

I'd like to ask for references on the status of modularity results for elliptic curves with CM which are not necessarily defined over $\mathbb Q$. In the case of an elliptic curve with CM defined over ...
user5831's user avatar
  • 2,029
3 votes
0 answers
133 views

Reference for a lemma on the asymptotic upper density of special sets with large gaps and intervals

Update. Based on Anthony Quas' comment below, the proof can be made sensibly shorter and the lemma can be slightly generalized by weakening the old assumption (iii). In a joint paper that I am ...
Salvo Tringali's user avatar
2 votes
0 answers
127 views

Nonnegative integers represented by $\prod_{i=1}^m \sum_{j=1}^n a_{i,j} x_j $, where the $a_{i,j}$ are positive integers

Fix $m, n \in \mathbf N^+$ with $m+n \ge 3$, and let $A = (a_{i,j})_{1 \le i \le m, 1 \le j \le n}$ be an $m$-by-$n$ matrix of positive integers. What is known about the asymptotic behavior of the ...
Salvo Tringali's user avatar
1 vote
1 answer
435 views

Elliptic subfields of a function field

Let $C$ be a curve and $K(C)$ be its function field of genus 2, where $K$ = $\mathbb{C}$. The number of essential elliptic subfields of $K(C)$ is 0 or 2 or $\infty$. Edit: I am looking for a proof. ...
Srilakshmi's user avatar
0 votes
1 answer
510 views

Erdős-Straus with 4 terms

The Erdős-Straus conjecture states that any fraction of the form $\frac{4}{n}$ can be decomposed as an Egyptian fraction with just 3 terms. In related research, I've recently come across conditions on ...
Aeryk's user avatar
  • 2,235
2 votes
1 answer
569 views

Generalization of singular moduli

$j$-invariants of CM curves $E$ over (say) the complex numbers are known as singular moduli. As the theory of complex multiplication explains, singular moduli are algebraic integers of great ...
Tommaso Centeleghe's user avatar
3 votes
3 answers
285 views

Limit connected with a periodic function

I am posting the following question from Math.Stackexchange: Let $f$ be a $1$-periodic function, i.e., $f(x+1)=f(x)$, defined on the interval $(0, 1)$ by the formula $$ f(x)=2x-1. $$ For a real ...
kap44's user avatar
  • 217
0 votes
0 answers
121 views

Bound of Chebyshev function and zeros of zeta function

It is an elementary argument (such as in Multiplicative Number Theory, section 18) that, if the Chebyshev's function $f(x) = \sum_{n \le x} \Lambda(x) = x + O(x^\alpha)$ for some $\alpha < 1$, then ...
Linh Nguyen's user avatar
0 votes
1 answer
939 views

Asymptotic equivalence for functions with zeros

I am considering the relative asymptotic behavior of a pair of real functions on the positive real axis, say $f$ and $g$. There is no $x_0$ such that $f$ and $g$ are non-zero for all $x>x_0$. ...
Kevin Smith's user avatar
  • 2,480
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 : \...
Zev's user avatar
  • 211
2 votes
0 answers
227 views

Kloosterman-like sum with inverse to different moduli

In some recent work, the following strange-looking exponential sum arose: $$ \sum_k \sum_r \sum_s e\bigg( \frac{r \bar s^{(r)} \bar k^{(r^2+s^2)}}{r^2+s^2} \bigg). $$ Here $e(x) = e^{2\pi i x}$ as ...
Greg Martin's user avatar
  • 12.8k
3 votes
3 answers
670 views

Proof of infinitude of primes whose reversal in base 10 is also prime [closed]

Is there any proof of infinitude of A007500 primes? If you want to generate them here is trivial and naive python program. ...
Pratik Deoghare's user avatar
2 votes
2 answers
491 views

Summation of certain series

Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then $\displaystyle\sum_{n=1}^{\infty}\frac{f(n)}{n}=-\frac{1}{q}\...
Subhajit Jana's user avatar
5 votes
0 answers
79 views

Some questions about the Lévy monoid of certain densities

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$. Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...
Salvo Tringali's user avatar
1 vote
1 answer
1k views

Lacunary sequence

Is there a standard definition for a lacunary sequence? Suppose $0 < a_1 < a_2 < \cdots.$ I've read two papers using the term recently. One requires $$ \liminf_n\frac{a_{n+1}}{a_n}>1 $$ ...
Charles's user avatar
  • 9,114
4 votes
1 answer
928 views

Primes and Ackermann's function

If $A(m,n)$ is Ackermann's function, and $c$ is a fixed integer, are there any heuristics/conjectures/obvious things that can be said about primes of the form $A(m,n)+c$, $m \geq 4$,at all? EDIT: I ...
Timothy Foo's user avatar
  • 1,075
6 votes
2 answers
976 views

References for modular polynomials

I am teaching a graduate "classical" course on modular forms. I try to achieve the most elementary level for presenting modular polynomials. Serge Lang's "Elliptic functions" cover the topic quite ...
Wadim Zudilin's user avatar
3 votes
0 answers
168 views

Invariant Theory over finite adeles

Classical invariant theory, among the other things, classifies polynomial functions over a vector space $V$ endowed with a quadratic form $Q$ which are invariant under the action of $SO(V,Q)$. I am ...
Giulio's user avatar
  • 2,384
3 votes
1 answer
151 views

Definability of orderings on a formally real number field

For vector basis $b_1,..,b_n$ on a finite extension $F$ of $\mathbb{Q}$, where $-1$ is not a sum of squares, each linear order on $F$ is determined by an order on the basis. This uses information ...
Colin McLarty's user avatar
4 votes
0 answers
413 views

Maximal order of Hooley's Delta function?

There is a large literature on Hooley's $$ \Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1 $$ giving its normal and average order. What is known of its maximal order? Clearly $\Delta(n)\le d(n)$ ...
Charles's user avatar
  • 9,114
1 vote
1 answer
446 views

Existence of Solutions to an Equation Involving the Sum-of-Divisors Function [Reference Request]

Let $\sigma(x) = \sigma_1(x)$ denote the sum of all the positive divisors of $x$. If $n \in \mathbb{N}$ is odd and $\gcd(n, \sigma(n)) = 1$, then do there exist any solutions to the following ...
Jose Arnaldo Bebita's user avatar
3 votes
0 answers
163 views

Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding $$ \int_0^T L\left(\tfrac{1}{2} + it, f \...
davidlowryduda's user avatar
2 votes
1 answer
596 views

Principal term of the Dirichlet Divisor problem, from the work of A.F. Lavrik?

Ivic writes, at the beginning of chapter 13 of his The Riemann Zeta Function, about a method of expressing the principal terms of the Dirichlet Divisor Problem as polynomials of $log\\ n $ with ...
Nathan McKenzie's user avatar
2 votes
0 answers
99 views

Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?

Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...
Salvo Tringali's user avatar
3 votes
0 answers
102 views

Localized at $p$ integral representations of finite elementary $p$-groups

Let $C_p$ be a cyclic group of prime order $p$. Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times). I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$. However, ...
Mikhail Borovoi's user avatar
4 votes
0 answers
117 views

Best constant for Maier's theorem?

Maier proved that, for fixed $\lambda>1,$ $$ \limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\pi(x)}{\log^{\lambda-1}x}>1 $$ and in particular $$ \limsup_{x\to\infty}\frac{\pi(x+\log^\lambda x)-\...
Charles's user avatar
  • 9,114
2 votes
0 answers
207 views

n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form. Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to $Q(x_1,\ldots,...
Eric Rowell's user avatar
  • 1,639
3 votes
0 answers
158 views

Conics over number fields

I am looking for a reference for the following fact. Let $k$ be a number field and let $S$ be a finite set of places of $k$ of even cardinality. Then there exists a unique conic $C$ over $k$ such ...
Daniel Loughran's user avatar
1 vote
1 answer
1k views

Good Minkowski Theory and Commutative Algebra Books

I am not so familiar with the theory of measures which Andre Weil uses to develope the Class Field Theory. However, I am interested in learning algebraic number theory and I recently found that the ...
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 ...
Lloyd Yu-West's user avatar