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Multiple Eisenstein series are holomorphic functions in the complex upper-half plane, which can be seen as a crossbreed between multiple zeta values and classical Eisenstein series. They were originally defined by Gangl-Kaneko-Zagier in 2006, and since then, many variants and regularizations of them have been studied. They give a natural bridge between the world of modular forms and multiple zeta values. In this note, we give a new algebraic interpretation of stuffle regularized multiple Eisenstein series based on the Hopf algebra structure of the harmonic algebra introduced by Hoffman.

In this defense I will describes three approaches to learn more about the relationship between the dynamics of black-holes and the distinctive signatures of a black hole systems: infinitesimal changes in the black hole background producing field excitations relating new fundamental black hole thermodynamic relations, mechanisms powering relativistic black hole jets and spontaneous symmetry breaking in five space-time dimensions, and physical signatures of black hole event horizons as conformal field theory duals (in both d=4,5 dimensional axisymmetric spacetimes).

Les formes modulars són objectes que provenen de l'anàlisi harmònica, però que juguen un paper fonamental en la teoria de nombres. Aquest projecte presenta la teoria de les formes modulars i la prova de la Fòrmula de la Valença, amb implicacions en l'estudi dels espais de formes modulars i les seves dimensions. A més, desenvolupa la formalització de les bases matemàtiques per enunciar la Fòrmula de la Valença utilitzant el llenguatge LEAN per demostrar teoremes amb la seva Mathlib. Las formas modulares son objetos que provienen de la análisis harmónica, pero que juegan un papel fundamental en teoría de números. Este proyecto presenta la teoría de formas modulares y la prueba de la Valence Formula, que tiene implicaciones en el estudio de los espacios de formas modulares i sus dimensiones. Además, desarrolla la formalización de las bases matemáticas para enunciar la Fórmula de la Valenza utilizando el lenguaje de programación LEAN para demostrar teoremas con su Mathlib. Modular forms are objects from harmonic analysis, but play an important role in number theory. This thesis presents the theory of modular forms and proof of the Valence Formula, which has implications in the study of the spaces of modular forms and their dimensions. During the thesis we have formalised the basis to state the Valence Formula using the LEAN theorem prover and its Mathlib.

Anadolu Üniversitesi ve Bilecik Şeyh Edebali Üniversitesi tarafından ortak yürütülen program. Modüler formlar ve özelinde Hecke eigenformlar sayılar teorisinde ilginç özellikleriyle ön plana çıkarlar. Örneğin matematikte son yılların en önemli sonucu olan Sato-Tate Teoremi sadece Hecke eigenformlar için geçerli bir sonuçtur. Hecke eigenformların sistematik seçimi problemi özellikle yarım tamsayı ağırlıklı Hecke eigenformlar üzerinde istatistiksel sonuçlar elde edilmesi adına kritik bir adımdır. Konuyla ilgili literatürde çalışmalar mevcuttur. Problemin bir başka çözüm yolu da Dedekind-eta çarpımları olabilir. Üç bölümden oluşan bu çalışmada Dedekind-eta çarpımlarından oluşan Hecke eigenformların sınıflandırılması problemi ele alınmıştır. İlk bölümde konunun temeli olan modüler formlar tanıtılmış ve temel özellikleri incelenmiştir. İkinci bölüm Dedekind-eta çarpımlarına ayrılmış olup, çalışmanın orijinal kısmını oluşturan üçüncü bölümde Dedekind-eta çarpımlarından oluşan Hecke eigenformların özel bir sınıflandırılması elde edilmiştir. Modular forms and, in particular, Hecke eigenforms come to the fore with their interesting properties in number theory. For example, the Sato-Tate Theorem, which is the most important result of recent years in mathematics, is valid only for Hecke eigenforms. The problem of systematic selection of Hecke eigenforms is a critical step in obtaining statistical results, especially on half-integral weight Hecke eigenforms. There are studies in the literature on the subject. Another solution to the problem can be Dedekind-eta products. In this study, which consists of three parts, the problem of classification of Hecke eigenforms consisting of Dedekind-eta products is discussed. In the first chapter, modular forms, which are the basis of the subject, are introduced and their basic features are examined. The second section is devoted to Dedekind-eta products, and in the third section, which constitutes the original part of the study, a special classification of Hecke eigenforms consisting of Dedekind-eta products is obtained.

志村多様体の小平次元は多くの人によって研究されてきた．Tai氏はAbel多様体のモジュライ空間，金銅氏，Gritsenko-Hulek-Sankaran諸氏，馬氏はBorcherdsリフトを用いて，K3曲面のモジュライ空間となる直交型志村多様体が一般型になるということを示した．一方で，Gritsenko-Hulek両氏はとある直交型志村多様体が単繊織的になるということを示した．本論文では上記の問題のユニタリ群類似に取り組む．具体的には，Borcherds形式や鏡映的保型形式を用いることによってユニタリ型志村多様体の小平次元を解析する．また，講演者と京都大学の尾高悠志氏により得られた，ユニタリ型志村多様体がより精密にFanoやCalabi-Yau, log canonical modelになるための判定法を紹介する．さらに条件を満たすエルミート格子すなわちユニタリ型志村多様体を具体的に構成する．

This thesis is about combinatorics and number theory. More precisely, this is based on the 3 papers I wrote during my time as a PhD student. The content of our thesis is on Higer Turan inequalities for plane partitions, introduction of a generalization of spherical t-designs, which we call ellipsoidal t-design, and providing inversion formulae for j-function around elliptic points, out of which the later work is a joint work with, my fellow PhD student at University of Virginia, Alejandro De Las Penas Castano.

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory., 100 pages, 6 figures

This article is an extended version of a talk delivered at RJMS conference on "Problems and Prospects in Analytic Number Theory" held in November, 2020. In this note, we give a brief overview of the theme 'values of logarithmic derivatives of £-functions and zeta functions and its related topics'. We end by providing an outline of a recent work on values of logarithmic derivatives of £-functions attached to cuspidal elliptic Hecke eigenforms of integral weight.

志村多様体上の代数的サイクルを係数とした形式的べき級数がコホモロジ一係数のHilbert-Siegel保型形式のFourier展開になるという現象は1990年代にKudla-Millsonにより観察されており，一連の研究は“Kudlaのプログラム”の始まりとされている．近年，Yuan-Zhang-Zhangにより，Chow群係数の形式的べき級数が研究され，ある種の直交型志村多様体の場合に保型性が示された．本稿では，より一般の直交型志村多様体に対するYuan-Zhang-Zhangの結果の拡張について紹介する．

In this survey article, we discuss our recent work [KKS20], [KN20] on the numerical verification of the Iwasawa main conjecture for modular forms of weight two at good primes and elliptic curves with potentially good reduction. The criterion is based on the Euler system method and the equality of the main conjecture can be checked via the non-vanishing of Kurihara numbers. We also discuss further arithmetic applications of Kurihara numbers to study the structure of Selmer groups following the philosophy of refined Iwasawa theory `a la Kurihara. In the appendix by Alexandru Ghitza, the SageMath code for an effective computation of Kurihara numbers is illustrated., Algebraic Number Theory and Related Topics 2018. November 26-30, 2018. edited by Takao Yamazaki and Shuji Yamamoto. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

This Volume is a continuation of the first volume of the 8T.TheGrand Unified Theory of Physics.V2.6Inserts: (Pages 233-234) Certainties Elimination - Breakthrough pg-234.Similar tothe first volume, the second volume is written by the solely by Manor Ohad. The second volume contains three parts. Firstpart - Reflections on the open questions of the 8T. Second Part - Reflections on Quantumphenomena using Number theory.Thirdpart - Reflections on the 8T framework and the Beauty of the finallaws of Nature. The authorincluded a collection ofadditional proofs to the most famous problem in number theory. The counting begins with the second proof as the first already presented in the first volume, Page 96-99 in "Classics". The Purpose of This volume is to Eliminate Certainties.&nbsp

In this note we realize seven small simple groups as Galois groups over Q. The technique that we employ is the determination of the images of Galois representations attached to modular and automorphic forms, relying in two cases on recent results of Scholze on the existence of Galois representations attached to non-selfdual automorphic forms.

Given Jacobi forms, we determine associated Jacobi-like forms, whose coefficients are quasimodular forms. We then use these quasimodular forms to construct differential operators on modular forms, which are expressed in terms of the Fourier coefficients of the given Jacobi forms.

En aquesta tesi presentem les propietats principals de les superfícies modulars de Hilbert i les formes modulars associades. La més remarcable és que poden ser vistes com a varietats modulars associades al grup ortogonal d'un espai quadràtic de tipus (2,2). Aquesta propietat dona una font de formes modulars, que estudiarem, posant un especial èmfasi al Borcherds lift i el Doi-Naganuma lift. Una vegada els fonaments per les superfícies modulars de Hilbert hagin estat establerts, introduirem la teoria de la Multiplicació Complexa, començant per alguns fets bàsics en el cas de corbes el·líptiques que servirà com a introducció per al cas de Multiplicació Complexa per a superfícies modulars de Hilbert. Mostrarem com obtenir els anomenats punts CM a la superfície modular de Hilbert i com avaluar el Borcherds lift en aquests punts. També veurem que aquests valors són nombres algebraics que pertanyen a cossos concrets i que quan avaluem una funció modular en tot un cicle CM obtenim nombres racionals amb múltiples factors primers. Donem diversos exemples de càlculs numèrics fets amb SageMath per confirmar els resultats teòrics. En esta tesis presentamos las propiedades principales de las superficies modulares de Hilbert y las formas modulares asociadas. La más remarcable es que pueden ser vistas como variedades modulares asociadas al grupo ortogonal de un espacio cuadrático de tipo (2,2). Esta propiedad nos da una fuente de formas modulares, que estudiaremos, poniendo especial énfasis en el Borcherds lift y el Doi-Naganuma lift. Una vez hayamos establecido los fundamentos de las superficies modulares de Hilbert, introduciremos la teoría de la Multiplicación Compleja, empezando por algunos hechos básicos en el caso de curvas elípticas que nos servirá como introducción para el caso de Multiplicación Compleja para superficies modulares de Hilbert. Mostraremos cómo obtener los llamados puntos CM en la superficie modular de Hilbert y cómo evaluar el Borcherds lift en esos puntos. También veremos que esos valores son números algebraicos pertenecientes a unos cuerpos concretos y que cuando evaluamos una función modular en todo el ciclo CM obtenemos números racionales con muchos factores primos. Damos varios ejemplos de los cálculos numéricos realizados con SageMath para respaldar los resultados teóricos. In this thesis we present the main properties of Hilbert modular surfaces and their associated modular forms. The most remarkable one is that they can be viewed as modular varieties associated to the orthogonal group of a quadratic space of type (2,2). This property provides a source of modular forms, which we will study, with a special focus on the so-called Borcherds lift and the Doi-Naganuma lift. Once the foundations of Hilbert modular surfaces and modular forms are established, we introduce the theory of Complex Multiplication, starting with some basic facts for elliptic curves that will serve as an introduction to the Theory of Complex Multiplication for Hilbert modular surfaces. We will show how to obtain the so-called CM points on the Hilbert Modular surface and how to evaluate Borcherds lifts on them. We will also see that those values are nice algebraic numbers in some concrete fields and that when we evaluate our modular function on a full CM cycle we get rational numbers with several prime factors. We provide several examples of those numerical computations on SageMath to support the theoretical results. Outgoing

This is a survey of a part of an article of the author and Florian Sprung. Sprung con- structed #/♭ p-adic L-functions L#p and L♭p for elliptic curves defined over Q in [10] and for modular forms of weight 2 in [11] in non ordinary case. In the article, we generalize p-adic L-functions L#p and L♭p for higher weight modular forms by an analogous method to that of Sprung. We assume that p ≥ k-1., "Algebraic Number Theory and Related Topics 2016". November 28 - December 2, 2016. edited by Yasuo Ohno, Hiroshi Tsunogai and Toshiro Hiranouchi. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

$t$-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on $t$-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on $t$-cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of $t$-cores in number theoretic aspects of partition theory. This includes the modularity of $t$-core partition generating functions, the existence of $t$-core partitions, asymptotic formulas and arithmetic properties of $t$-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of $t$-core partitions, which include relations between core partitions and self-conjugate core partitions, a $t$-core crank explaining Ramanujan's partition congruences, and relations with class numbers.

We prove a geometric formula for the cycle integrals of Parson’s weight 2k modular integrals in terms of the intersection angles of geodesics on modular curves. Our result is an analog for modular integrals of a classical formula for the cycle integrals of certain hyperbolic Poincaré series, due to Katok. On the other hand, it extends a recent geometric formula of Matsusaka and of Duke, Imamoḡlu, and Tóth for the cycle integrals of weight 2 modular integrals. ISSN:0065-1036 ISSN:1730-6264

Αντικείμενο της εργασίας είναι το πρόβλημα του sphere packing στις οκτώ διαστάσεις. Αυτό αφορά στην εύρεση της πυκνότερης ένωσης ξένων ισοδύναμων μπαλών στον 8-διάστατο Ευκλείδειο χώρο. Με εξαίρεση τη μία διάσταση , όπου η απάντηση δίνεται σχεδόν τετριμμένα, το πρόβλημα αυτό δεν έχει καθόλου προφανείς λύσεις. Ο λόγος για τον οποίο ασχολούμαστε με τις οκτώ διαστάσεις είναι οι αξιοσημείωτες ιδιότητες του sphere packing Ε8. Το καθοριστικό θεώρημα των Cohn και Elkies σε συνδυασμό με τον τύπο άθροισης του Poisson, καθώς και δύο είδη modular forms οδήγησαν στην επίλυση του προβλήματος. Όλα αυτά παρουσιάζονται αναλυτικά στο κείμενο, ανοίγοντας το δρόμο για την ανάδειξη του Ε8 ως το επιθυμητό sphere packing, This master’s thesis focuses on the problem of sphere packing in dimension 8, which is about finding the densest union of congruent nonoverlapping balls in the 8-dimensional Euclidean space. Except for the dimension 1, where the answer is trivial, the solution to this problem is not obvious at all. The reason why dimension 8 stands out is the remarkable properties of the sphere packing E8. The valuable theorem of Cohn and Elkies combined with the Poisson summation formula, as well as two kinds of modular forms provided a solution to the problem. All the above are thoroughly discussed in the thesis, leading to the proof that E8 is the proper sphere packing

Anadolu Üniversitesi ve Bilecik Şeyh Edebali Üniversitesi tarafından ortak yürütülen program. Modüler formlar matematiğin özellikle de sayılar teorisinin önemli bir konusu olup yoğun bir şekilde çalışılmaktadır. Birçok anabilim dalını bir araya getirmesi nedeniyle de birçok matematikçi için ‘modüler formlar her yerde’ bulunur. Altı bölümden oluşan bu çalışmanın 1. Bölüm'ünde modüler formlar tanımlanıp temel özellikleri incelenecek ve böylece tezde ihtiyaç duyulan alt yapı oluşturulacaktır. Tezin özgün kısımlarından ilkini oluşturan 2. Bölüm'de yarım tamsayı ağırlıklı Hecke eigenformların sistematik seçimi probleminin çözümü verilecektir. 3. Bölüm'de ise 21. yüzyılın matematikteki en önemli başarılardan birisi olan Sato-Tate Konjektürü tanıtılacak ve Bruinier-Kohnen Konjektürü üzerine bir uygulaması verilecektir. Özgün kısmın ikinci parçası olan 4. Bölüm'de ise Ramanujan-Petersson Konjektürü tarafından normalleştirilen yarım tamsayı ağırlıklı modüler formların Fourier katsayılarının dağılımı konusu üzerinde durulacak, bir açık soru ortaya konulup mümkün olan tüm verilerle iddia desteklenecektir. Özgün kısmın son parçası olan 5. Bölüm'de Bruinier-Kohnen Konjektürü güçlendirilerek ifade edilecektir. Altıncı ve son bölüm ise tartışma, sonuç ve gözlemlerden oluşmaktadır. Modular forms are an important subject of mathematics, especially number theory, and they are studied intensively. Because it brings together many branches of science, 'modular forms are everywhere' for many mathematicians. In the first part of this six-part study, modular forms will be defined and their basic properties will be examined, thus creating the background needed in the thesis. In Chapter 2, which is the first of the original parts of the thesis, the solution of the systematic selection problem of half-integral weight Hecke eigenforms will be given. In Chapter 3, Sato-Tate Conjecture, one of the most important achievements in mathematics of the 21st century, will be introduced and an application on the Bruinier-Kohnen Conjecture will be given. In the second part of the original part, Chapter 4, the distribution of Fourier coefficients of half-integral weight modular forms normalized by the Ramanujan-Petersson Conjecture will be discussed, an open question will be raised and the claim will be supported with all possible data. The last part of the original part, In Chapter 5, the Bruinier-Kohnen Conjecture will be strengthened and expressed. The sixth and last part consists of discussion, conclusion and observations. Türkiye Bilimsel ve Teknolojik Araştırma Kurumu (TÜBİTAK) - 118F148. The Scientific and Technological Research Council of Turkey - (TUBITAK) - 118F148.

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2021, Director: Xavier Guitart Morales, [en] In this Bachelor’s Final Project we present simply and understandably the basic properties of modular forms for different weights and levels. Furthermore, we define and treat Hecke operators, the Petersson inner product and the Atkin–Lehner theory. In the last section, we introduce modular symbols and we study their relation with modular forms.

Dissertation, RWTH Aachen University, 2022; Aachen : RWTH Aachen University 1 Online-Ressource (2022). = Dissertation, RWTH Aachen University, 2022, In this thesis we study various scalar-valued Eisenstein series for the orthogonal group O(2,n) where the underlying lattice contains two hyperbolic planes. We consider the Fourier coefficients of the Eisenstein series defined in analogy to elliptic Eisenstein series, Jacobi Eisenstein series, and Siegel Eisenstein series, respectively. The main focus is on the Fourier coefficients of Eisenstein series of Siegel type for the standard cusp. For maximal lattices, we prove the rationality of the Fourier coefficients, as well as that they belong to the Maaß space. Moreover, we prove that the results hold for all localizations of the lattice which are maximal ones. Thus, this also holds true for non-maximal lattices except for finitely many local places. The Fourier-Jacobi coefficient of index 1 turns out to be a Jacobi-Eisenstein series. We give explicit formulas for the Fourier expansion in some cases and links to other known Eisenstein series. We also consider Eisenstein series of Klingen type, prove their absolute convergence, observe their behavior under the Petersson inner product and prove that they generate the space of non-cusp forms, at least whenever the lattice is Euclidean. Lastly, we deal with Hecke theory for the O(2,n+2) and its applications to Eisenstein series which are Hecke eigenforms. We show that the Eisenstein series are the only non-cusp forms which are eigenforms of some Hecke operators. Finally, using the methods of Heim and Krieg (2020), the Maaß relations of the Eisenstein series hold for non-maximal lattices, too., Published by RWTH Aachen University, Aachen

Given a primitive, non-CM, holomorphic cusp form $f$ with normalized Fourier coefficients $a(n)$ and given an interval $I\subset [-2, 2]$, we study the least prime $p$ such that $a(p)\in I$ . This can be viewed as a modular form analogue of Vinogradov's problem on the least quadratic non-residue. We obtain strong explicit bounds on $p$, depending on the analytic conductor of $f$ for some specific choices of $I$., Comment: 14 pages

In this note, we exhibit a weakly holomorphic modular form for use in constructing a Fourier eigenfunction in four dimensions. Such auxiliary functions may be of use to the D4 checkerboard lattice and the four dimensional sphere packing problem., 6 pages, comments welcome

Статья посвящена особенностям проектирования конструктивной игрушки с использованием модульных элементов. Проведён анализ работ учёных, художников, мастеров игрушки. Предлагаются пути совершенствования специфики модульных элементов в проектировании конструктивной игрушки, связанные с эскизной и поисковой деятельностью, вариативностью графического и формообразующего решения композиции. Рассматриваются разные композиционные способы создания образа в проектировании конструктивной игрушки, способствующие формированию целостного видения композиции. Изучена история возникновение модульных элементов в проектировании конструктивной игрушки и их связь с природными аналогами, определена их роль в совершенствовании учебно-методического процесса., The article is devoted to the specifics of designing a constructive toy using modular elements. The analysis of the works of scientists, artists and toy makers is carried out. The ways of improvement and the specifics of modular elements in the design of a constructive toy are proposed related to the sketching and search activities, the variability of the graphic and formative solutions of the composition. Various compositional ways of creating an image in the design of a constructive toy are considered, which contribute to the formation of a holistic vision of the composition. The history of the emergence of modular elements in the design of a constructive toy and their connection with natural analogues is studied, their role in improving the educational and methodological process is determined.

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

We study an analog of Serre's modularity conjecture for projective representations [Formula: see text], where K is a totally real number field. We prove cases of this conjecture when [Formula: see text].

Significance The connection between modular forms and Galois representations plays a significant role in modern algebraic number theory. J.-P. Serre made an influential conjecture relating mod p modular forms and mod p representations of the absolute Galois group of Q. Such a relationship has consequences for classical Diophantine questions, for example implying Fermat’s Last Theorem, and is also a mod p analogue of the Langlands program. It is thus important to study analogues of Serre’s conjecture in the broadest possible context. Serre’s modularity conjecture, definitively stated in 1986, was proved by Khare–Wintenberger in 2009. In this paper we prove new cases of extensions of Serre’s conjecture to mod p representations of absolute Galois groups of totally real number fields., We study an analog of Serre’s modularity conjecture for projective representations ρ¯:Gal(K¯/K)→PGL2(k), where K is a totally real number field. We prove cases of this conjecture when k=F5.

We study an analog of Serre's modularity conjecture for projective representations [Formula: see text], where K is a totally real number field. We prove cases of this conjecture when [Formula: see text].

We study an analog of Serre's modularity conjecture for projective representations [Formula: see text], where K is a totally real number field. We prove cases of this conjecture when [Formula: see text].

There are many families of functions on partitions, such as the shifted symmetric functions, for which the corresponding q-brackets—certain normalized generating series—are quasimodular forms. This provides a tool for enumerative geometers to show that certain generating series of Gromov–Witten invariants or Hurwitz numbers are quasimodular forms. In this thesis, our aim is to study graded algebras of functions on partitions such that all homogeneous elements of the algebra have quasimodular forms as q-brackets. That is, we give explicit constructions answering the following three main questions in the affirmative: (I) Are there other graded algebras than the algebra of shifted symmetric functions such that the q-brackets of its elements are quasimodular forms? (II) Given a congruence subgroup, is there an (even larger) algebra of functions for which the q-bracket is a quasimodular form for this subgroup? (III) What is the class of functions for which the q-bracket is not only a quasimodular form, but even a modular form? The answer to the second and third question follows by studying the following question of independent interest: (IV) What is the modular or quasimodular behavior of the Taylor coefficients of meromorphic quasi-Jacobi forms? This question brings us back to the origin of the results on the q-bracket in enumerative geometry. Namely, we show that the solutions to a differential equation originating from the study of K3 surfaces are quasi-Jacobi forms and describe their transformation.

We use covariants of binary sextics to describe the structure of modules of scalar-valued or vector-valued Siegel modular forms of degree 2 with character, over the ring of scalar-valued Siegel modular forms of even weight. For a modular form defined by a covariant we express the order of vanishing along the locus of products of elliptic curves in terms of the covariant., Comment: 18 pages

In this paper we study the values of modular functions at the Markov quadratics which are defined in terms of their cycle integrals along the associated closed geodesics. These numbers are shown to satisfy two properties that were conjectured by Kaneko. More precisely we show that the values of a modular function [math] , along any branch [math] of the Markov tree, converge to the value of [math] at the Markov number which is the predecessor of the tip of [math] . We also prove an interlacing property for these values.

Modifying a method of Jutila, we prove a t aspect subconvexity estimate for L-functions associated to primitive holomorphic cusp forms of arbitrary level that is of comparable strength to Good's bound for the full modular group, thus resolving a problem that has been open for 35 years. A key innovation in our proof is a general form of Voronoi summation that applies to all fractions, even when the level is not squarefree., Comment: minor revisions; to appear in Adv. Math.; 30 pages

This is a survey of author's article of the same title. In the article, we construct infinitely many new examples of (not necessarily geometrically simply‐connected) Calabi-Yau varieties defined over mathbb{Q} with the middle L‐functions related to the L‐functions of modular forms. These varieties give new affirmative examples for the problem proposed independently by B. Mazur and D. van Straten., "Algebraic Number Theory and Related Topics 2015". November 30 - December 4, 2015. edited by Hiroki Takahashi, Yasuo Ohno and Takahiro Tsushima. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.

The minimal excludant or "mex" function for an integer partition π of a positive integer n, mex(π), is the smallest positive integer that is not a part of π. Andrews and Newman introduced σmex(n) to be the sum of mex(π) taken over all partitions π of n. Ballantine and Merca generalized this combinatorial interpretation to σrmex(n), as the sum of least r-gaps in all partitions of n. In this article, we study the arithmetic density of σ_2 mex(n) and σ_3 mex(n) modulo 2^k for any positive integer k.

Thispaper present threetheorems on a Lorentz manifoldwhich yield an on pointprediction ofthefine structureconstant, and as the result theawaitedequation of coupling magnitudes.this paper than allow us tounify QM with GR. The equation also predicting the magnitude of any additional element in the series, the next element should stand at ratio of 1/850 compared to the strong interaction. Updated to 7.9.21 &nbsp

In this paper we give an overview of some aspects of Iwasawa theory for modular forms. We start with the classical formulation in terms of $p$-adic $L$-functions in the ordinary case and the $\pm$-formulation for supersingular elliptic curves. Then we discuss some recent progresses in the proof of the corresponding Iwasawa main conjectures formulated by Kato (Conjecture 4.1), which relates the index of his zeta element to the characteristic ideal of the strict Selmer groups.

Our first goal in this article is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson–Kato classes follows as an easy consequence of the Iwasawa main conjectures. We also explain that the refined form of this conjecture in the $p$-supersingular case also follows from the classical Gross–Zagier formula and Kobayashi's $p$-adic Gross–Zagier formula combined with this simple observation. Our second goal is to set up a conceptual framework in the context of $\Lambda$-adic Kolyvagin systems to treat analogues of Perrin-Riou's conjectures for motives of higher rank. We apply this general discussion in order to establish a link between Heegner points on a general class of CM abelian varieties and the (conjectural) Coleman–Rubin–Stark elements we introduce here. This can ben thought of as a higher dimensional version of Rubin's results on rational points on CM elliptic curves.

We obtain new Fourier interpolation and -uniqueness results in all dimensions, extending methods and results by the first author and M. Sousa, and by the second author. We show that the only Schwartz function which, together with its Fourier transform, vanishes on surfaces close to the origin-centered spheres whose radius are square roots of integers, is the zero function. In the radial case, these surfaces are spheres with perturbed radii, while in the non-radial case, they can be graphs of continuous functions over the sphere. As an application, we translate our perturbed Fourier uniqueness results to perturbed Heisenberg uniqueness for the hyperbola, using the interrelation between these fields introduced and studied by Bakan, Hedenmalm, Montes-Rodriguez, Radchenko and Viazovska., Comment: 22 pages

Dissertation, RWTH Aachen University, 2021; Aachen : RWTH Aachen University 1 Online-Ressource : Illustrationen (2021). = Dissertation, RWTH Aachen University, 2021, In the first part of the thesis a statement of Böcherer and Kohnen (2016) about the growth of Fourier coefficients of cusp forms and non-cusp forms is transferred from Siegel modular forms to Hermitian modular forms and orthogonal modular forms. The second part of the thesis constructs the exceptional isomorphism between the symplectic group with respect to Hamiltonian quaternions and the orthogonal group SO(2,6). A method is given to calculate it explicitly. The third and last part deals with Hermitian Maaß-forms of degree 2. In the first half, possible algebraic dependencies between Hermitian Eisenstein series to arbitrary discriminant are investigated using the Fourier expansion. In the second half, a construction of Hermitian measure forms to odd weight and the nontrivial character is given., Published by RWTH Aachen University, Aachen

This paper contains the written notes of a course the author gave at the VIASM of Hanoi in the Summer 2018. It provides an elementary introduction to the analytic naive theory of Drinfeld modular forms for the simplest ‘Drinfeld modular group’ \(\operatorname {GL}_2(\mathbb {F}_q[\theta ])\) also providing some perspectives of development, notably in the direction of the theory of vector modular forms with values in certain ultrametric Banach algebras.

This thesis deals with the study of the cohomology and the Kodaira dimension of some moduli spaces. In the first part we compute the intersection Betti numbers of the GIT models of two moduli spaces. They parametrize non-hyperelliptic Petri-general curves of genus four and numerically polarized Enriques surfaces of degree two respectively. In both cases, the strategy of the cohomological calculation relies on a general method developed by Kirwan to compute the cohomology of GIT quotients of projective varieties. This procedure is based on the equivariantly perfect stratification of the unstable points studied by Hesselink, Kempf, Kirwan and Ness, and a partial resolution of singularities, called the Kirwan blow-up. In the second part of the thesis, we study the moduli spaces of elliptic K3 surfaces of Picard number at least three, i.e.\ $U\oplus \langle -2k \rangle$-polarized K3 surfaces. Such moduli spaces are proved to be of general type for $k\geq 220$. The proof relies on the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran.

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2021, Director: Xavier Guitart Morales, [en] This master’s thesis studies the modularity of elliptic curves over the rationals and two generalizations. The first is a theorem of Ribet based on Serre’s modularity conjecture, asserting that all abelian varieties of $\mathrm{G} \mathrm{L}_{2}$-type come from the Eichler-Shimura construction. The second is the Paramodularity Conjecture, which says that all abelian surfaces with trivial endomorphism ring have an associated Siegel paramodular form with coinciding $L$-function. We give background on abelian varieties, Galois representations and classical modular forms, all necessary to state modularity. Further, we explain the Eichler-Shimura construction and relation. We then study the basic theory of Siegel modular forms with respect to the paramodular group. The final chapter gives the statement of the Paramodularity Conjecture, along with a commentary of what a generalization to $\mathrm{GL}_{4}$-type abelian varieties could look like. An important part of this project is centered on explicit computation of Fourier-Siegel coefficients, and special care has been taken to present computational principles which are scattered across the literature. We also provide the first public implementation of the specialization method that was used to prove the first instance of the Paramodularity Conjecture.

Anadolu Üniversitesi ve Bilecik Şeyh Edebali Üniversitesi tarafından ortak yürütülen program. Modüler formlar Matematik’in birçok dalını bir araya getirmesi, uygulamalarının Fizik'e kadar uzanması nedeniyle yüzyılı aşan süredir popülerliğini kaybetmemiştir ve halen yaygın olarak çalışılmaktadır. Bu çalışmada modüler formların Analiz ve Fonksiyonlar Teorisi ile Sayılar Teorisi’nin ara kesitinde kalan Rankin-Cohen operatörleri (parantezi) konusu çalışılmıştır. 7 bölüm ve bir ekten oluşan bu tezin ilk bölümü ön bilgilere ve ileride kullanılacak olan kavramların tanıtılmasına ayrılmıştır. İkinci bölümde modüler formlar üzerinde tanımlı diferansiyel operatörler verilmiş olup üçüncü bölümde tezin ana kısmını oluşturan Rankin-Cohen parantezi kavramı tanıtılmış ve temel özellikleri incelenmiştir. Dördüncü bölümde Rankin-Cohen parantezlerinin modüler formlar üzerinde tanımlı Petersson iç çarpımı ile olan ilişkisi ele alınmıştır. Beşinci bölümde Choie ve Lee tarafından 2011'de yayımlanan bir makale incelenmiştir. Altıncı bölümde Lanphier ve Takloo-Bighash tarafından verilen Rankin-Cohen parantezleri ile ilgili ilginç bir sonuç verilecektir. Yedinci ve tezin orijinal kısmını oluşturan Rankin-Cohen parantezinin bir uygulaması olarak Kohnen plus uzayında yer alan yarım tam sayı ağırlıklı iki Hecke eigenform, Eisenstein serisi ve klasik teta serisinin birinci mertebeden Rankin-Cohen parantezi cinsinden ifade edilmiştir. Tezde ek olarak Pari-GP yazılımında mf-paketinde yer alan Rankin-Cohen parantezini de kapsayan bazı hesaplamaların ekran görüntüleri yer almaktadır. Modular forms have not lost popularity for more than a hundred years and are still widely studied, as they bring together many branches of mathematics and their applications extend to physics. In this study, the subject of Rankin-Cohen operators (brackets), which is located at the intersection of analysis and function theory and modular forms. The first part of this thesis, which consists of 7 chapters and an appendix, is devoted to preliminaries and the introduction of topics that will be used in the thesis. In the second part, differential operators defined on modular forms are given. In the third part, the concept of Rankin-Cohen bracket, which forms the main part of the thesis, is introduced and its basic features are examined. In the fourth section, the relation of Rankin-Cohen bracket with Petersson inner product defined on modular forms is discussed. In the fifth chapter, the article published in 2011 by Choie and Lee is examined. In the sixth chapter, an interesting result about the Rankin-Cohen parenthesis given by Lanphier and Takloo-Bighash is discussed. As an application of the Rankin-Cohen bracket, which forms the seventh and original part of the thesis, the two Hecke eigenforms in the Kohnen-plus space are expressed via the first order Rankin-Cohen parenthesis in terms of the classical theta series and Eisenstein series are expressed. In addition to the thesis, there are screenshots of some calculations including the Rankin-Cohen bracket included in the mf-package in Pari-GP software.

In this note, we propose a new construction of cyclotomic p-adic L-functions that are attached to classical modular cuspidal eigenforms. This allows for us to cover most known cases to date and provides a method which is amenable to generalizations to automorphic forms on arbitrary groups. In the classical setting of GL2 over Q, this allows for us to construct the p-adic L-function in the so far uncovered extremal case, which arises under the unlikely hypothesis that p-th Hecke polynomial has a double root. Although Tate&rsquo, s conjecture implies that this case should never take place for GL2/Q, the obvious generalization does exist in nature for Hilbert cusp forms over totally real number fields of even degree, and this article proposes a method that should adapt to this setting. We further study the admissibility and the interpolation properties of these extremal p-adic L-functionsLpext(f,s), and relate Lpext(f,s) to the two-variable p-adic L-function interpolating cyclotomic p-adic L-functions along a Coleman family.

Using half-integral weight modular forms we give a criterion for the existence of real quadratic $p$-rational fields. For $p=5$ we prove the existence of infinitely many real quadratic $p$-rational fields., 10 pages

A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic $\mathbb Z_p$-extensions at good ordinary primes $p$. We extend Greenberg's result to more general $p$-adic Galois representations, including a large subclass of those attached to $p$-ordinary modular forms of level $\Gamma_0(N)$ with $p\nmid N$., Comment: 21 pages

Eisenstein's symbols, defined by Beilinson in the 1980s, are special elements in the motivic cohomology of modular curves. Beilinson has shown that their integrals are special values of L functions of modular forms of weight 2.Part of this thesis concerns the regularized integrals of Beilinson elements. Using a new method of Rogers and Zudilin, we show that these integrals gives us L values of modular forms at negative integers. This generalize the recent work by Zudilin and Brunault. Our formula is in accordance with Beilinson's conjectures.In this computation, we systematically study the generalized Mellin transform. In addition, we extend it to certain functions with exponential growth. We also use the extended modular symbols by Stevens to formulate our results. The Rogers and Zudilin method is also a powerful tool for evaluating double L functions. Inspired by an example from Shinder and Vlasenko, we show more generally that many values of double L functions of Eisenstein series can be expressed in terms of L functions of modular forms. Tornheim and Mordell have defined certain double zeta series. In collaboration with Zhang, we define and study the Mordell-Tornheim Eisenstein series. Using Cohen's series, we give explicit formulas for these series.; Les symboles d'Eisenstein, construits par Beilinson dans les années 1980, sont des élémentsspéciaux dans la cohomologie motivique des courbes modulaires. Beilinson a montré que leursintégrales sont des valeurs spéciales de fonctions L de formes modulaires de poids 2.Une partie de cette thèse concerne les intégrales régularisées des éléments de Beilinson. En utilisant une nouvelle méthode de Rogers et Zudilin, nous montrons que ces intégrales sont des valeurs de fonctions L de formes modulaires aux entiers négatifs, généralisant ainsi des travaux récents de Zudilin et Brunault. Notre formule est en accord avec les conjectures générales de Beilinson.Pour ce faire, nous étudions systématiquement la transformée de Mellin généralisée. Par ailleurs, nous l'étendons à certaines fonctions à croissance exponentielle. Nous utilisons également les symboles modulaires étendus de Stevens pour formuler nos résultats. La méthode de Rogers et Zudilin est également un outil puissant pour évaluer les fonctions L doubles. Inspirés par un exemple de Shinder et Vlasenko, nous montrons plus généralement que denombreuses valeurs de fonctions L doubles de séries d’Eisenstein peuvent s'exprimer en termes de fonctions L de formes modulaires. Tornheim et Mordell ont défini certaines séries zêta doubles. En collaboration avec Zhang, nousdéfinissons et étudions les séries d’Eisenstein de type Mordell-Tornheim. En utilisant la théorie desséries de Cohen, nous donnons des formules explicites pour ces séries.

Eisenstein's symbols, defined by Beilinson in the 1980s, are special elements in the motivic cohomology of modular curves. Beilinson has shown that their integrals are special values of L functions of modular forms of weight 2.Part of this thesis concerns the regularized integrals of Beilinson elements. Using a new method of Rogers and Zudilin, we show that these integrals gives us L values of modular forms at negative integers. This generalize the recent work by Zudilin and Brunault. Our formula is in accordance with Beilinson's conjectures.In this computation, we systematically study the generalized Mellin transform. In addition, we extend it to certain functions with exponential growth. We also use the extended modular symbols by Stevens to formulate our results. The Rogers and Zudilin method is also a powerful tool for evaluating double L functions. Inspired by an example from Shinder and Vlasenko, we show more generally that many values of double L functions of Eisenstein series can be expressed in terms of L functions of modular forms. Tornheim and Mordell have defined certain double zeta series. In collaboration with Zhang, we define and study the Mordell-Tornheim Eisenstein series. Using Cohen's series, we give explicit formulas for these series.; Les symboles d'Eisenstein, construits par Beilinson dans les années 1980, sont des élémentsspéciaux dans la cohomologie motivique des courbes modulaires. Beilinson a montré que leursintégrales sont des valeurs spéciales de fonctions L de formes modulaires de poids 2.Une partie de cette thèse concerne les intégrales régularisées des éléments de Beilinson. En utilisant une nouvelle méthode de Rogers et Zudilin, nous montrons que ces intégrales sont des valeurs de fonctions L de formes modulaires aux entiers négatifs, généralisant ainsi des travaux récents de Zudilin et Brunault. Notre formule est en accord avec les conjectures générales de Beilinson.Pour ce faire, nous étudions systématiquement la transformée de Mellin généralisée. Par ailleurs, nous l'étendons à certaines fonctions à croissance exponentielle. Nous utilisons également les symboles modulaires étendus de Stevens pour formuler nos résultats. La méthode de Rogers et Zudilin est également un outil puissant pour évaluer les fonctions L doubles. Inspirés par un exemple de Shinder et Vlasenko, nous montrons plus généralement que denombreuses valeurs de fonctions L doubles de séries d’Eisenstein peuvent s'exprimer en termes de fonctions L de formes modulaires. Tornheim et Mordell ont défini certaines séries zêta doubles. En collaboration avec Zhang, nousdéfinissons et étudions les séries d’Eisenstein de type Mordell-Tornheim. En utilisant la théorie desséries de Cohen, nous donnons des formules explicites pour ces séries.