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Francis Brown

Publications and source records attributed to Francis Brown.

At least 37 records · Page 2Linked to original sources
arXiv2019

A multi-variable version of the completed Riemann zeta function and other $L$-functions

We define a generalisation of the completed Riemann zeta function in several complex variables. It satisfies a functional equation, shuffle product identities, and has simple poles along finitely many hyperplanes, with a recursive structure on its residues. The special case of two variables can be written as a partial Mellin transform of a real analytic Eisenstein series, which enables us to relate its values at pairs of positive even points to periods of (simple extensions of symmetric powers of the cohomology of) the CM elliptic curve corresponding to the Gaussian integers. In general, the totally even values of these functions are related to new quantities which we call multiple quadratic sums. More generally, we cautiously define multiple-variable versions of motivic $L$-functions and ask whether there is a relation between their special values and periods of general mixed motives. We show that all periods of mixed Tate motives over the integers, and all periods of motivic fundamental groups (or relative completions) of modular groups, are indeed special values of the multiple motivic $L$-values defined here.

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arXiv2019

A class of non-holomorphic modular forms II : equivariant iterated Eisenstein integrals

We introduce a new family of real analytic modular forms on the upper half plane. They are arguably the simplest class of `mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q, \overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first non-trivial functions in this class are real analytic Eisenstein series.

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arXiv2019

From the Deligne-Ihara conjecture to Multiple Modular Values

This is the write-up of a talk given in honour of Prof. Ihara's 80th Birthday conference in Kyoto in 2018. After briefly reviewing the work of Ihara on the projective line minus 3 points, I outline the main ideas in the proof of the Deligne-Ihara conjecture and provide an update on recent progress in this area and raise some new questions. The second part of the talk outlines the main features of the corresponding theory in genus one, i.e., on the moduli stack of elliptic curves. It contains many concrete and elementary examples to illustrate the main features of the theory and to make it more widely accessible. In particular, the case of non-abelian cocycles associated to a pair of modular forms of level one is discussed in some detail. It also contains some new conjectures which have not previously appeared in print.

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arXiv2017

Algebraic de Rham theory for weakly holomorphic modular forms of level one

We establish an Eichler-Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads to formulae for the periods and quasi-periods of modular forms.

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arXiv2017

A class of non-holomorphic modular forms III: real analytic cusp forms for $\mathrm{SL}_2(\mathbb{Z})$

We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit `weak harmonic lifts' for every eigenform of integral weight $k$ and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to $n^{1-k}(a'_n + ρa_n)$ for $n\neq 0$, where $ρ$ is the normalised permanent of the period matrix of the corresponding motive, and $a_n, a'_n$ are the Fourier coefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform, respectively. More generally, this framework provides a conceptual explanation for the algebraicity of the coefficients of mock modular forms in the CM case.

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arXiv2017

A class of non-holomorphic modular forms I

This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms, and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms), as well as the modular graph functions arising in genus one string perturbation theory. In an appendix, we use weakly holomorphic modular forms to write down modular primitives of cusp forms. Their coefficients involve the full period matrix (periods and quasi-periods) of cusp forms.

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arXiv2017

Anatomy of an associator

We study some Lie algebras defined by solutions to the double shuffle equations with poles and construct families of explicit solutions to these equations in all weights and depths. These provide universal coordinates in which to write down `zeta elements': the images of generators of the Lie algebra of the motivic Galois group of mixed Tate motives over the integers. We expect that a similar statement holds for associators. In particular, these coordinates encode algebraic relations between multiple zeta values, and enable one to compress the currently used tables for relations between multiple zeta values in, for example, weights $\leq 13$, already by a factor of a thousand. The Lie algebras and groups studied here form part of a large algebraic structure which is related to the work of Ecalle on the calculus of moulds, and also related to the theory of universal mixed elliptic motives, and modular forms for the full modular group.

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arXiv2017

Multiple Modular Values and the relative completion of the fundamental group of $M_{1,1}$

Multiple modular values are a common generalisation of multiple zeta values and periods of modular forms, and are periods of a hypothetical Tannakian category of mixed modular motives. They are given by regularised iterated integrals on the upper half plane generalising the iterated Shimura integrals of Manin. In this paper, some first properties of the underlying theory are established in the case of the full modular group: in particular, the relationship with special values of L-functions of modular forms at all positive integers; and the action of the conjectural motivic Galois group via a certain group of automorphisms.

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arXiv2017

Integral points on curves, the unit equation, and motivic periods

This paper recasts some of the recent literature on Kim's extension of Chabauty's method for bounding points on curves in the language of motivic periods. A variant of the higher Albanese manifolds is defined which is equipped with a canonical de Rham path.

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arXiv2017

Notes on Motivic Periods

The second part of a set of notes based on lectures given at the IHES in 2015 on Feynman amplitudes and motivic periods.

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arXiv2015

Periods and Feynman amplitudes

Feynman amplitudes in perturbation theory form the basis for most predictions in particle collider experiments. The mathematical quantities which occur as amplitudes include values of the Riemann zeta function and relate to fundamental objects in number theory and algebraic geometry. This talk reviews some of the recent developments in this field, and explains how new ideas from algebraic geometry have led to much progress in our understanding of amplitudes. In particular, the idea that certain transcendental numbers, such as $π$, can be viewed as a representation of a group, provides a powerful framework to study amplitudes which reveals many hidden structures.

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arXiv2015

Zeta elements in depth 3 and the fundamental Lie algebra of a punctured elliptic curve

This paper draws connections between the double shuffle equations and structure of associators; universal mixed elliptic motives as defined by Hain and Matsumoto; and the Rankin-Selberg method for modular forms for $SL_2(\mathbb{Z})$. We write down explicit formulae for zeta elements $σ_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the Broadhurst-Kreimer conjecture, and completely solve the double shuffle equations for multiple zeta values in depths two and three.

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arXiv2014

Irrationality proofs for zeta values, moduli spaces and dinner parties

A simple geometric construction on the moduli spaces $\mathcal{M}_{0,n}$ of curves of genus $0$ with $n$ ordered marked points is described which gives a common framework for many irrationality proofs for zeta values. This construction yields Apéry's approximations to $ζ(2)$ and $ζ(3)$, and for larger $n$, an infinite family of small linear forms in multiple zeta values with an interesting algebraic structure. It also contains a generalisation of the linear forms used by Ball and Rivoal to prove that infinitely many odd zeta values are irrational.

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arXiv2013

Single-valued periods and multiple zeta values

The values at 1 of single-valued multiple polylogarithms span a certain subalgebra of multiple zeta values. In this paper, the properties of this algebra are studied from the point of view of motivic periods.

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arXiv2013

Modular forms in Quantum Field Theory

The amplitude of a Feynman graph in Quantum Field Theory is related to the point-count over finite fields of the corresponding graph hypersurface. This article reports on an experimental study of point counts over F_q modulo q^3, for graphs up to loop order 10. It is found that many of them are given by Fourier coefficients of modular forms of weights <=8 and levels <=17.

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