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