Computing Vector-valued Modular Forms of Congruence Types and of Some Extension Types
Doctoral thesis, 2023
This thesis explores applications of vector-valued modular forms of congruence
and extension types to scalar-valued modular forms for congruence subgroups
with a character, higher order modular forms, and iterated Eichler-Shimura integrals
of depth one and two, including considerable generalizations thereof.
In \textsc{Paper I} (co-authored with Martin Raum), we present an algorithm
for computing bases for spaces of vector-valued modular forms of congruence type
and of weight at least $2$ in terms
of products of components of vector-valued Eisenstein series. Since the Fourier
series expansions of these Eisenstein series are available, our algorithm
can be used to compute Fourier series expansions of any vector-valued modular forms
belonging to these spaces. It complements two available algorithms
that (as opposed to ours) are limited to inductions of Dirichlet characters, and vector-valued modular
forms of Weil type. Our algorithm is based on a representation theoretical
interpretation of a theorem due to Raum and Xià. After a heuristic
evaluation of the time-complexity, we compare our algorithm to the two available ones,
highlighting the trade-offs between generality and performance.
In \textsc{Paper II} (co-authored with Martin Raum and Albin Ahlbäck), we show
that all Eichler integrals, and all ``generalized second order modular forms''
can be expressed as linear combinations of corresponding generalized second order
Eisenstein series with coefficients in classical modular forms. We compute
the Fourier series expansions of generalized second order Eisenstein series in
level one, and provide their tail estimates via convexity bounds for additively
twisted $L$-functions. As an application, we illustrate a bootstrapping procedure
that yields numerical evaluations of, for instance, Eichler integrals from merely the
associated cocycle.
Finally, in \textsc{Paper III} (co-authored with Martin Raum), we provide
an explicit vector-valued modular form whose top components are given by the
depth two iterated Eichler-Shimura integral $I_{f,g}$, where $f$ and $g$ are
cusp forms of weight $k\in\ZZ_{\geq 2}$. We show that this vector-valued modular
form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted
by $\cE_{f,g}$, that can be seen as a higher-depth generalization of the scalar-valued Eichler integral $\cE_f$
of depth one. As an aside, our argument provides an alternative explanation of
an orthogonality relation satisfied by period polynomials originally due to
Pa\c{s}ol-Popa. We show that $\cE_{f,g}$ can be expressed in terms of sums of
products of components of vector-valued Eisenstein series with classical modular
forms after multiplication with a suitable power of the discriminant modular form
$\Delta$. This allows for effective computation of $\cE_{f,g}$.
and extension types to scalar-valued modular forms for congruence subgroups
with a character, higher order modular forms, and iterated Eichler-Shimura integrals
of depth one and two, including considerable generalizations thereof.
In \textsc{Paper I} (co-authored with Martin Raum), we present an algorithm
for computing bases for spaces of vector-valued modular forms of congruence type
and of weight at least $2$ in terms
of products of components of vector-valued Eisenstein series. Since the Fourier
series expansions of these Eisenstein series are available, our algorithm
can be used to compute Fourier series expansions of any vector-valued modular forms
belonging to these spaces. It complements two available algorithms
that (as opposed to ours) are limited to inductions of Dirichlet characters, and vector-valued modular
forms of Weil type. Our algorithm is based on a representation theoretical
interpretation of a theorem due to Raum and Xià. After a heuristic
evaluation of the time-complexity, we compare our algorithm to the two available ones,
highlighting the trade-offs between generality and performance.
In \textsc{Paper II} (co-authored with Martin Raum and Albin Ahlbäck), we show
that all Eichler integrals, and all ``generalized second order modular forms''
can be expressed as linear combinations of corresponding generalized second order
Eisenstein series with coefficients in classical modular forms. We compute
the Fourier series expansions of generalized second order Eisenstein series in
level one, and provide their tail estimates via convexity bounds for additively
twisted $L$-functions. As an application, we illustrate a bootstrapping procedure
that yields numerical evaluations of, for instance, Eichler integrals from merely the
associated cocycle.
Finally, in \textsc{Paper III} (co-authored with Martin Raum), we provide
an explicit vector-valued modular form whose top components are given by the
depth two iterated Eichler-Shimura integral $I_{f,g}$, where $f$ and $g$ are
cusp forms of weight $k\in\ZZ_{\geq 2}$. We show that this vector-valued modular
form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted
by $\cE_{f,g}$, that can be seen as a higher-depth generalization of the scalar-valued Eichler integral $\cE_f$
of depth one. As an aside, our argument provides an alternative explanation of
an orthogonality relation satisfied by period polynomials originally due to
Pa\c{s}ol-Popa. We show that $\cE_{f,g}$ can be expressed in terms of sums of
products of components of vector-valued Eisenstein series with classical modular
forms after multiplication with a suitable power of the discriminant modular form
$\Delta$. This allows for effective computation of $\cE_{f,g}$.
eisenstein series
modular forms
iterated eichler-shimura integrals
vector-valued modular forms
Pascal, Chalmers Tvärgata 3
Opponent: Prof. Morten S. Risager, Department of Mathematical Sciences, University of Copenhagen, Denmark
Author
Tobias Magnusson
Chalmers, Mathematical Sciences, Algebra and geometry
Tobias Magnusson, Martin Raum - Scalar-valued depth two Eichler-Shimura Integrals of Cusp Forms
Albin Ahlbäck, Tobias Magnusson, Martin Raum - Eichler integrals and generalized second order Eisenstein series
Tobias Magnusson, Martin Raum - On the Computation of General Vector-valued Modular Forms
Modulära former är centrala objekt inom modern talteori. I deras enklaste form
kan de ses som komplexvärda funktioner på gitter som uppfyller en viss skalningsrelation
och som beter sig på ett kontrollerat sätt.
Intresset för modulära former kommer till stor del från den inverkan de har haft
på undersökandet av ett flertal viktiga problem inom talteori, däribland
Fermats sista sats och Birch och Swinnerton-Dyers förmodan. Utöver detta dyker
ofta modulära former upp i närmast oväntade sammanhang, där de ofta avgör hur
olika talteoretiskt intressanta funktioner beter sig. Till exempel är den så kallade
funktionalekvationen för Riemanns $\zeta$-funktion starkt relaterad till en speciell
typ av modulär form -- en så kallad $\theta$-funktion. Riemanns $\zeta$-funktion
är i sin tur relaterad till primtalens distribution, och är centrum i talteorins
kanske mest kända förmodan -- Riemannhypotesen.
I denna avhandling undersöks en familj av generaliserade modulära former vars
värden är komplexa vektorer, istället för komplexa tal. Vi visar hur man
kan beräkna sådana modulära former med hjälp av Eisensteinserier och hur de kan
användas för att bättre förstå komplexvärda modulära former och andra närbesläktade
objekt. Till exempel undersöker vi med hjälp av vektorvärda modulära former
en klass av funktioner som kallas för itererade Eichler-Shimura integraler.
De har i sin tur visat sig vara centrala inom andra delar av talteorin och inom
delar av strängteori.
kan de ses som komplexvärda funktioner på gitter som uppfyller en viss skalningsrelation
och som beter sig på ett kontrollerat sätt.
Intresset för modulära former kommer till stor del från den inverkan de har haft
på undersökandet av ett flertal viktiga problem inom talteori, däribland
Fermats sista sats och Birch och Swinnerton-Dyers förmodan. Utöver detta dyker
ofta modulära former upp i närmast oväntade sammanhang, där de ofta avgör hur
olika talteoretiskt intressanta funktioner beter sig. Till exempel är den så kallade
funktionalekvationen för Riemanns $\zeta$-funktion starkt relaterad till en speciell
typ av modulär form -- en så kallad $\theta$-funktion. Riemanns $\zeta$-funktion
är i sin tur relaterad till primtalens distribution, och är centrum i talteorins
kanske mest kända förmodan -- Riemannhypotesen.
I denna avhandling undersöks en familj av generaliserade modulära former vars
värden är komplexa vektorer, istället för komplexa tal. Vi visar hur man
kan beräkna sådana modulära former med hjälp av Eisensteinserier och hur de kan
användas för att bättre förstå komplexvärda modulära former och andra närbesläktade
objekt. Till exempel undersöker vi med hjälp av vektorvärda modulära former
en klass av funktioner som kallas för itererade Eichler-Shimura integraler.
De har i sin tur visat sig vara centrala inom andra delar av talteorin och inom
delar av strängteori.
Modular forms are central to modern number theory. In the simplest case,
they can be viewed as complex-valued functions on lattices that satisfy a certain
scaling relation and have a controlled behavior.
The interest in modular forms can in large part be attributed to the impact that they have
had on the investigation of several important problems in number theory,
including Fermat's Last Theorem and Birch and Swinnerton-Dyer's conjecture.
In addition to this, modular forms often appear in somewhat unexpected situations,
where they often determine how various number-theoretically interesting
functions behave. For example, the functional equation of Riemann's $\zeta$-function
is strongly related to a special type of modular form -- a so-called $\theta$-function.
Riemann's $\zeta$-function is in turn related to the distribution of prime
numbers and is the topic of one of the most famous open problems in number theory
-- the Riemann hypothesis.
This thesis investigates a family of generalized modular forms whose values
are complex vectors, rather than complex numbers. We demonstrate how such modular
forms can be computed in terms of Eisenstein series and how they can be used to
better understand complex-valued modular forms and other closely related objects.
As an example, using vector-valued modular forms, we investigate a class of
functions that are known as iterated Eichler-Shimura integrals. They have shown
to be important in other areas of number theory and within parts of string theory.
they can be viewed as complex-valued functions on lattices that satisfy a certain
scaling relation and have a controlled behavior.
The interest in modular forms can in large part be attributed to the impact that they have
had on the investigation of several important problems in number theory,
including Fermat's Last Theorem and Birch and Swinnerton-Dyer's conjecture.
In addition to this, modular forms often appear in somewhat unexpected situations,
where they often determine how various number-theoretically interesting
functions behave. For example, the functional equation of Riemann's $\zeta$-function
is strongly related to a special type of modular form -- a so-called $\theta$-function.
Riemann's $\zeta$-function is in turn related to the distribution of prime
numbers and is the topic of one of the most famous open problems in number theory
-- the Riemann hypothesis.
This thesis investigates a family of generalized modular forms whose values
are complex vectors, rather than complex numbers. We demonstrate how such modular
forms can be computed in terms of Eisenstein series and how they can be used to
better understand complex-valued modular forms and other closely related objects.
As an example, using vector-valued modular forms, we investigate a class of
functions that are known as iterated Eichler-Shimura integrals. They have shown
to be important in other areas of number theory and within parts of string theory.
Subject Categories
Mathematics
Roots
Basic sciences
ISBN
978-91-7905-843-2
Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 5309
Publisher
Chalmers
Pascal, Chalmers Tvärgata 3
Opponent: Prof. Morten S. Risager, Department of Mathematical Sciences, University of Copenhagen, Denmark