Some Cases of Kudla’s Modularity Conjecture for Unitary Shimura Varieties
Doctoral thesis, 2021

A common theme of the thesis is the interplay of symmetry and rigidity, which is a general phenomenon in mathematics. Symmetry is a notion related to the degree to which an object remains unchanged under transformations, and rigidity is a notion that in terms of physics can be thought of as a lack of freedom, which leads to stronger properties of an object than we normally expect. An object of higher symmetry often also exhibits a higher extent of rigidity, and vice versa.

In the introduction of the thesis, we provide some background on modular forms, number theory, and geometry in a way that does not require familiarity with these subjects. The contributions of this thesis are presented in three articles.

In Article I, we establish the existence of rational geometric designs for rational polytopes via the circle method and convex geometry, and discuss the existence of rational spherical designs which relates to Lehmer's conjecture on the Ramanujan tau function.

In Article II, we break the barrier of expressing weight-2 modular forms of higher level whose central L-values vanish by products of at most two Eisenstein series. This work shows the power of Rankin--Selberg method and also contributes to the computation of elliptic modular forms.

In Preprint III, we prove unconditionally some cases of Kudla's conjecture on the modularity of generating functions of special cycles on unitary Shimura varieties, for norm-Euclidean imaginary quadratic fields. Our method is based on a result of Liu and work of Bruinier--Raum, who confirmed the orthogonal Kudla conjecture over Q.

special cycles

Eisenstein series

spherical designs

central L-values

unitary Shimura varieties

Jacobi forms

rational points

generating functions

the circle method

Kudla's modularity conjecture

theta series

Rankin--Selberg method

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Opponent: Prof. Wadim Zudilin, Radboud University Nijmegen, the Netherlands.

Author

Jiacheng Xia

Chalmers, Mathematical Sciences, Algebra and geometry

Rational designs

Advances in Mathematics,; Vol. 352(2019)p. 541-571

Journal article

All modular forms of weight 2 can be expressed by Eisenstein series

Research in Number Theory,; Vol. 6(2020)

Journal article

Jiacheng Xia: Some cases of Kudla’s modularity conjecture for unitary Shimura varieties

Talteori och geometri tillhör de äldsta grenarna inom matematiken. Talteori rör frågor om heltal och geometri är studiet av former. De två utvecklades nästintill åtskilda under tusentals år, fram till att den moderna talteorin utvecklades under förra seklet, med syfte att förena dessa två läror och låta dem dra nytta av varandra.

Denna avhandling utgår från denna förening mellan talteori och geometri, och ett av dess centrala teman är samspelet mellan symmetri och rigiditet, vilket är ett generellt fenomen inom matematik. Symmetri är ett begrepp som handlar om till vilken grad ett objekt förblir oförändrat under olika transformationer. Begreppet rigiditet härstammar från fysiken och har att göra med en brist på frihet, vilket leder till att ett rigitt objekt tenderar att uppvisa fler gynnsamma egenskaper än förväntat. Ett objekt som har en högre grad av symmetri visar ofta även en högre grad av rigiditet, och vice versa.

Detta samspel hjälper oss att på ett precist sätt förutsäga en mängd relationer mellan objekt av talteoretisk och geometrisk natur, till exempel så kallade speciella cykler. Speciella cykler utgör en familj geometriska objekt som är mycket viktiga inom modern talteori. Kudlas förmodan som nämns i titeln säger, om den skulle bekräftas, att dessa speciella cykler är väldigt rigida och dessutom lämpliga för beräkningar via så kallade modulära former. Dessa är ett slags funktioner som uppvisar en hög grad av symmetri och rigiditet. De utgör därmed ett ypperligt verktyg för beräkningar och sammanlänkar olika intressanta objekt i modern talteori.

I denna avhandling bekräftar vi några fall av Kudlas förmodan och studerar andra aspekter av modulära former, inklusive beräkningsaspekter och deras relation till så kallade sfäriska designer, som även de är väldigt symmetriska och rigida objekt.

Number theory and geometry are among the oldest branches in mathematics. Number theory is concerned with the study of whole numbers, and geometry that of shapes. They were developed almost separately for thousands of years, until the birth of modern number theory in the last century, which aims to unite them and benefit them from each other.

This thesis is based on such a standpoint unifying number theory and geometry, and a central theme is the interplay of symmetry and rigidity, which is a general phenomenon in mathematics. Symmetry is a notion related to the degree to which an object remains unchanged under transformations, and rigidity is a notion that in terms of physics can be thought of as a lack of freedom, which leads to stronger properties of objects than we normally expect. An object of higher symmetry often also exhibits a higher extent of rigidity, and vice versa.

This interplay helps us to precisely predict several relations among objects of number theoretic and geometric nature, for instance relations among special cycles. Special cycles form a family of highly symmetric geometric objects that are of great importance to modern number theory. The Kudla conjecture in the title, if confirmed, tells us that these special cycles are also very rigid and in addition accessible to computation via a tool called modular forms. The latter are a kind of function featuring both high symmetry and high rigidity. They therefore provide us with an ideal tool for computation and they also connect various interesting objects in modern number theory.

In this thesis, we confirm some cases of the Kudla conjecture and study other aspects of modular forms, which include the computational aspect and a relation to spherical designs, another kind of object of high symmetry and rigidity.

Subject Categories

Mathematics

ISBN

978-91-7905-464-9

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4931

Publisher

Chalmers

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Opponent: Prof. Wadim Zudilin, Radboud University Nijmegen, the Netherlands.

More information

Latest update

3/23/2022