Questions tagged [periods]

A period is a number that can be expressed as an integral of an algebraic function over an algebraic domain.

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Evaluation of mock modular forms at elliptic points

The holomorphic function $$F(\tau)=-\frac{1}{\vartheta_4(\tau)}\sum_{n\in\mathbb Z}\frac{(-1)^nq^{\frac{n^2}{2}-\frac 18}}{1-q^{n-\frac12}}=2q^{\frac38}(1+3q^{\frac12}+7q+14q^{\frac32}+\dots),$$ is a ...
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7 votes
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Can we make a hierarchy of complex numbers by repeatedly iterating the construction that produces periods?

Given a set $S$, which is a subset of the complex numbers, we can form the smallest field which contains $S$, which we will denote by $S_F$ by taking the intersection of all complex fields which ...
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8 votes
2 answers
446 views

What is the rank of the period lattice of modular forms?

Let $f$ be a weight $2$ cusp form for the group $\Gamma_0(N)$. I was experimenting with integrals of the form $$ \int_r^s f(z) \, dz$$ where $r, s \in \mathbf{P}^1(\mathbf{Q})$ and the integral above ...
4 votes
0 answers
279 views

Automorphisms of the ring of periods

The set of periods $\mathcal{P}$ introduced by Kontsevich and Zagier forms a ring, see for example https://en.m.wikipedia.org/wiki/Period_(algebraic_geometry). Moreover J. Wan introduced in 2011 in ...
4 votes
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757 views

Showing that $\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$

Prove, without evaluating the integrals that: $$\int_0^\pi\frac{x\ln(1-\sin x)}{\sin x}dx=3\int_0^\frac{\pi}{2}\frac{x\ln(1-\sin x)}{\sin x}dx$$ Originally I posted this here on MSE, however it's ...
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3 votes
1 answer
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How to compute period polynomial of a meromorphic cuspform explicitly?

I am looking for an algorithm to compute the period polynomial $$P(z,f) := \int_C f(\tau) (z-\tau)^{k-2} d \tau$$ for a cusp form $f(\tau)$ of weight-k, where $C$ is a path connecting $\tau =0$ and $\...
3 votes
1 answer
363 views

Why does $\mathbb C_p$ not contain the periods?

I am reading the following article of Berger, p8 and I don't understand the idea: $C_p:=\widehat{\overline{\mathbb Q_p}}$ does not contain the periods The text seem to reason as follows (under some ...
0 votes
0 answers
75 views

The loss of double periodicity (ellipticity)

Consider a meromorphic function $f(\mathfrak{a}_1, \mathfrak{a}_2)$, such that $$ \begin{align} f(\mathfrak{a}_1, \mathfrak{a}_2) = f (\mathfrak{a}_1 + 1, \mathfrak{a}_2) = f(\mathfrak{a}_1 + \tau, \...
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3 votes
1 answer
208 views

Can one define a degree of a period?

In the context of Feynman integrals, certain values of the zeta function appear at certain loop orders. Given that the tree-level amplitudes are rational, and assuming the zeta values to be ...
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3 votes
0 answers
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Galois theory of periods of algebraic varieties PhD project

I finished my MMath at Exeter in July of this year and I would like to undertake a PhD at the interface of algebraic geometry, number theory and Galois theory. More specifically, I recently came ...
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34 votes
1 answer
929 views

Quaternionic and octonionic analogues of the Basel problem

I asked this question in MSE around 3 months ago but I have received no answer yet, so following the suggestion in the comments I decided to post it here. It is a well-known fact that $$\sum_{0\neq n\...
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3 votes
0 answers
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Comparing an integral to zero, by slicing and stacking

Let $f \colon [0,1] \rightarrow {\mathbb R}$ be a nice function -- real-analytic, or maybe definable in some o-minimal structure, let's say. Let $0 < \alpha_1 < \cdots < \alpha_n < 1$ be ...
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4 votes
2 answers
419 views

Kinds of differentials and algebraic groups

This Wikipedia article mentions that the analogues of differentials of the first/second/third kind for algebraic groups are abelian varieties/algebraic tori/linear algebraic groups. I guess ...
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6 votes
1 answer
588 views

$l$-adic periods?

For an algebraic variety $X$ over $\mathbb{Q}$ the comparison isomorphism between Betti and de Rham cohomologies provides the theory of periods with a motivic context whose reformulation as motivic ...
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10 votes
1 answer
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periods of higher weight modular forms

For now let $f\in S_2(\Gamma_0(N))$ and define $I(\alpha,\beta)=\int_\alpha^\beta f(z)dz$. If $M\in\Gamma_0(N)$ it is immediate to prove that $I(\alpha,M(\alpha))$ is independent of $\alpha$: this ...
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8 votes
1 answer
399 views

How to compute cohomology using differentials of the second kind on a Fermat curve?

Differentials of the second kind Gross and Rohrlich in the paper On the periods of abelian integrals and a formula of Chowla and Selberg state the claim below without citation (pg. 198), giving an ...
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3 votes
1 answer
239 views

Higher Genus Surfaces That "Look Like" Genus-1 Surfaces

It is my understanding that a genus-$g$ Riemann surface has $2g$ independent cycles that satisfy the usual intersection rules: $$a_i \cap a_j = 0$$ $$b_i \cap b_j = 0$$ $$a_i \cap b_j = \delta_{ij}$$ ...
5 votes
1 answer
370 views

Simple integral representation for a beta function with more than two variables

The beta function $B(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)}$ can be written for $\Re x, \Re y > 0$ symmetrically as $$ B(x,y) = \int_{-\frac12}^{\frac12}(\frac12-t)^{x-1}(\frac12+t)^{y-1}\,...
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9 votes
1 answer
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Feynman diagrams and periods of motives

A recent article in the online science magazine Quanta, Strange Numbers Found in Particle Collisions, discusses experimental evidence of a connection between Feynman integrals and periods of motives. ...
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2 votes
1 answer
465 views

What is explicitly, $ \mathcal{P} \mathrm{er} ( X ) / \mathbb{Q} $?

In the following link$^{[1]}$, page $2$, we find the following question : Let $X$ be a smooth $ \mathbb{Q} $ - variety and let $\mathcal{P} \mathrm{er} (X)$ be the subfield of $\mathbb{C}$ ...
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2 votes
0 answers
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Open Period Integrals of Elliptically Fibered K3 surfaces

Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, \,\...
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1 vote
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Can there be a nonzero period integral of this form?

I have been trying to compute the following integral: $$\displaystyle \int_{\gamma} \frac{g \Omega_{\mathbb{P}^3}}{f^2}$$ where: $\gamma$ is the torus cycle $\{ |z_1| =|z_2| =|z_3| =|z_4| =1 \}$, $...
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12 votes
3 answers
721 views

Is there a closed form of $\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$?

For naturals $n\ge m$, define $$I(n,m):=\int_0^\frac12\dfrac{\text{arcsinh}^nx}{x^m}dx$$ with $\text{arcsinh}\ x=\ln(x+\sqrt{1+x^2} )$, so $\text{arcsinh} \frac12=\ln \frac{\sqrt{5}+1}2 $. Is it ...
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17 votes
1 answer
493 views

Is special value of Epstein zeta function in 3 variables a period?

Kontsevich-Zagier's article "Periods" contains the following question Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period? ($\sum'$ means we do not sum ...
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32 votes
1 answer
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$\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?

QUESTION Numerical calculation with gp (first to the default 38-digit precision, then tripled) supports the conjecture that $$ \int_0^\infty x \, [J_0(x)]^5 \, dx = \frac{\Gamma(1/15) \, \Gamma(2/15) ...
16 votes
1 answer
541 views

Computer software for periods

Kontsevich and Zagier define a period as an integral of a rational function (over $\mathbb{Q}$) defined on a $\mathbb{Q}$-semialgebraic set. They conjecture that if two periods are equal, then the ...
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14 votes
1 answer
767 views

Are there integral representations of the Mertens constant?

It is well-known that the Euler constant $$\gamma=\lim\limits_{n\to \infty}\biggl( \sum\limits_{k\le n}\dfrac{1}{k}-\ln{n}\biggr ) $$ has a bunch of integral representations, e.g. $$\gamma=-\int\...
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75 votes
4 answers
4k views

What are reasons to believe that e is not a period?

In their 2001 paper defining periods, Kontsevich and Zagier (pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are....
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-1 votes
1 answer
413 views

What is the probability that a randomly chosen number from set of c.e.number is period(number)?

What is the probability that a randomly chosen number from the set of c.e.numbers is period(number)? What is the probability that a randomly chosen number from the set of computable numbers is ...
16 votes
2 answers
1k views

What is the relationship between these two notions of "period"?

The motivation for this question is to understand a recent theorem of Francis Brown which implies that all periods of mixed Tate motives over $\mathbb{Z}$ lie in $\mathcal{Z}[\frac{1}{2\pi i}]$, where ...
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35 votes
1 answer
3k views

Special values of L-functions as periods

If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles. For example, when $M=\...
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73 votes
2 answers
5k views

Is it known that the ring of periods is not a field?

I have just learned here that we know numbers that are not periods; is it known meanwhile that the ring of periods is not a field? I know that it is conjectured that $1/\pi$ is not a period, but the ...