Recent zbMATH articles in MSC 11Fhttps://zbmath.org/atom/cc/11F2021-04-16T16:22:00+00:00WerkzeugQuasi-pullback of Borcherds products.https://zbmath.org/1456.110832021-04-16T16:22:00+00:00"Ma, Shouhei"https://zbmath.org/authors/?q=ai:ma.shouheiSummary: Quasi-pullback of Borcherds products is an operation of renormalized restriction. It produces a meromorphic modular form on a lower dimensional symmetric domain which is again a Borcherds product. We give an explicit formula for the weakly holomorphic modular form of Weil representation type whose Borcherds lift is the quasi-pullback of the given Borcherds product.Rankin-Cohen brackets on Jacobi forms of several variables and special values of certain Dirichlet series.https://zbmath.org/1456.110822021-04-16T16:22:00+00:00"Jha, Abhash Kumar"https://zbmath.org/authors/?q=ai:jha.abhash-kumar"Sahu, Brundaban"https://zbmath.org/authors/?q=ai:sahu.brundabanGeometric Waldspurger periods. II.https://zbmath.org/1456.110912021-04-16T16:22:00+00:00"Lysenko, Sergey"https://zbmath.org/authors/?q=ai:lysenko.sergeySummary: In this paper we extend the calculation of the geometric Waldspurger periods from our paper [Part I, Compos. Math. 144, No. 2, 377--438 (2008; Zbl 1209.14010)] to the case of ramified coverings. We give some applications to the study of Whittaker coeffcients of the theta-lifting of automorphic sheaves from \(\operatorname{PGL}_2\) to the metaplectic group \(\widetilde{\operatorname{SL}}_2\); they agree with our conjectures from [``Geometric Whittaker models and Eisenstein series for \(\mathrm{Mp}_2\)'', Preprint, \url{arXiv:1221.1596}]. In the process of the proof, we construct some new automorphic sheaves for \({\operatorname{GL}_2}\) in the ramified setting. We also formulate stronger conjectures about Waldspurger periods and geometric theta-lifting for the dual pair \((\widetilde{\operatorname{SL}}_2, \operatorname{PGL}_2)\).Compatibility of Kisin modules for different uniformizers.https://zbmath.org/1456.110942021-04-16T16:22:00+00:00"Liu, Tong"https://zbmath.org/authors/?q=ai:liu.tong.1|liu.tongSummary: Let \(p\) be a prime and \(T\) a lattice inside a semi-stable representation \(V\). We prove that Kisin modules associated to \(T\) by selecting different uniformizers are isomorphic after tensoring a subring in \(W(R)\). As consequences, we show that several lattices inside the filtered \((\phi,N)\)-module of \(V\) constructed from Kisin modules are independent on the choice of uniformizers. Finally, we use a similar strategy to show that the Wach module can be recovered from the \((\phi,\hat{G})\)-module associated to \(T\) when \(V\) is crystalline and the base field is unramified.New \(P-Q\) mixed modular equations and their applications.https://zbmath.org/1456.110642021-04-16T16:22:00+00:00"Mahadeva Naika, M. S."https://zbmath.org/authors/?q=ai:mahadeva-naika.megadahalli-sidda"Hemanthkumar, B."https://zbmath.org/authors/?q=ai:hemanthkumar.b"Chandankumar, S."https://zbmath.org/authors/?q=ai:chandankumar.sathyanarayanaSummary: In his second notebook, Ramanujan recorded total of seven \(P\)-\(Q\) modular equations involving theta-function \(f(-q)\) with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained for higher orders. As a consequence, several values of quotients of theta-function are evaluated. The cubic singular modulus is evaluated at \(q=\exp (-2\pi \sqrt{n/3})\) for \(n\in \{5k, 1/5k, 5/k, k/5\}\), where \(k\in \{4,7,16\}\).Weierstrass coefficients of the canonical lifting.https://zbmath.org/1456.111052021-04-16T16:22:00+00:00"Finotti, Luís R. A."https://zbmath.org/authors/?q=ai:finotti.luis-r-aBounds for a spectral exponential sum.https://zbmath.org/1456.110922021-04-16T16:22:00+00:00"Balkanova, Olga"https://zbmath.org/authors/?q=ai:balkanova.olga-g"Frolenkov, Dmitry"https://zbmath.org/authors/?q=ai:frolenkov.dmitrii-aSummary: We prove new upper bounds for a spectral exponential sum by refining the process by which one evaluates mean values of \(L\)-functions multiplied by an oscillating function. In particular, we introduce a method which is capable of taking into consideration the oscillatory behaviour of the function. This gives an improvement of the result of \textit{W. Luo} and \textit{P. Sarnak} [Publ. Math., Inst. Hautes Étud. Sci. 81, 207--237 (1995; Zbl 0852.11024)] when \(T\geqslant X^{1/6+2\theta/3+\epsilon}\). Furthermore, this proves the conjecture of \textit{Y. N. Petridis} and \textit{M. S. Risager} [Int. Math. Res. Not. 2018, No. 16, 4942--4968 (2018; Zbl 1451.11045)] in some ranges. Finally, this allows obtaining a new proof of the Soundararajan-Young error estimate in the prime geodesic theorem.Examples of genuine QM abelian surfaces which are modular.https://zbmath.org/1456.110692021-04-16T16:22:00+00:00"Schembri, Ciaran"https://zbmath.org/authors/?q=ai:schembri.ciaranSummary: Let \(K\) be an imaginary quadratic field. Modular forms for GL(2) over \(K\) are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over \(K\) or an associated abelian surface with quaternionic multiplication over \(K\). We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to \textit{genuine} Bianchi newforms, which answers a question posed by \textit{J. E. Cremona} [J. Lond. Math. Soc., II. Ser. 45, No. 3, 404--416 (1992; Zbl 0773.14023)] as to whether this phenomenon can happen.Higher dimensional Dedekind sums and twisted mean values of Dirichlet \(L\)-series.https://zbmath.org/1456.111532021-04-16T16:22:00+00:00"Das, Mithun Kumar"https://zbmath.org/authors/?q=ai:das.mithun-kumar"Juyal, Abhishek"https://zbmath.org/authors/?q=ai:juyal.abhishekSummary: We provide an identity for evaluating products of trigonometric functions of the form \(\mathrm{sec}^m2x\mathrm{cot}^{2n}x\), where \(m,n\) are positive integers. Using this identity, we are able to give a partial answer of a question raised by \textit{A. Straub} [Ramanujan J. 41, No. 1--3, 269--285 (2016; Zbl 1418.11067)]. As an application of this identity, we evaluated formulas for special values of Don Zagier's higher dimensional Dedekind sums. We also study the mean values
\[
\frac{2}{\phi(q)}\sum_{\substack{\chi\pmod q\\ \chi(-1)=(-1)^{m}}}\chi (c)L(m,\chi )L(n,\bar{\chi}),
\]
where \(\chi\) is a Dirichlet character modulo an odd integer \(q, c\) a positive integer coprime to \(q\), \(\phi(\cdot)\) is the Euler's \(\phi\)-function and \(m,n\) are positive integers. Moreover, we express the mean values in terms of higher dimensional Dedekind sums. For odd \(q\) and \(c=1,2,4\), we determine the mean values explicitly for some integers \(m,n\), which generalize results of
\textit{H. Liu} [J. Number Theory 147, 172--183 (2015; Zbl 1381.11069)].The second moment for counting prime geodesics.https://zbmath.org/1456.111662021-04-16T16:22:00+00:00"Kaneko, Ikuya"https://zbmath.org/authors/?q=ai:kaneko.ikuyaSummary: A brighter light has freshly been shed upon the second moment of the Prime Geodesic Theorem. We work with such moments in the two and three dimensional hyperbolic spaces. Letting \(E_{\Gamma}(X)\) be the error term arising from counting prime geodesics associated to \(\Gamma = \text{PSL}_2(\mathbb{Z}[i])\), the bound \(E_{\Gamma}(X) \ll X^{3/2+\epsilon}\) is proved in a square mean sense. Our second moment bound is the pure counterpart of the work of
\textit{A. Balog} et al. [J. Number Theory 198, 239--249 (2019; Zbl 07005769)] for \(\Gamma = \text{PSL}_2(\mathbb{Z})\), and the main innovation entails the delicate analysis of sums of Kloosterman sums. We also infer pointwise bounds from the standpoint of the second moment. Finally, we announce the pointwise bound \(E_{\Gamma}(X) \ll X^{67/42+\varepsilon}\) for \(\Gamma = \text{PSL}_2(\mathbb{Z}[i])\) by an application of the Weyl-type subconvexity.Hybrid bounds for twists of \(\mathrm{GL}(3)\) \(L\)-functions.https://zbmath.org/1456.110862021-04-16T16:22:00+00:00"Sun, Qingfeng"https://zbmath.org/authors/?q=ai:sun.qingfengSummary: Let \(\pi\) be a Hecke-Maass cusp form for \(\mathrm{SL}(3,\mathbb{Z})\) and \(\chi=\chi_1\chi_2\) a Dirichlet character with \(\chi_i\) primitive modulo \(M_i\). Suppose that \(M_1, M_2\) are primes such that \(\max\{(M|t|)^{1/3+2\delta/3},M^{2/5}|t|^{-9/20}, M^{1/2+2\delta}|t|^{-3/4+2\delta}\}(M|t|)^{\varepsilon}<M_1< \min\{ (M|t|)^{2/5}, (M|t|)^{1/2-8\delta}\}(M|t|)^{-\varepsilon}\) for any \(\varepsilon>0\), where \(M=M_1M_2\), \(|t|\geq 1\), and \(0<\delta< 1/52\). Then we have \[L\left(\frac{1}{2}+it,\pi\otimes \chi\right)\ll_{\pi,\varepsilon} (M|t|)^{3/4-\delta+\varepsilon}. \]A formula for generating weakly modular forms with weight 12.https://zbmath.org/1456.110512021-04-16T16:22:00+00:00"Aygunes, Aykut Ahmet"https://zbmath.org/authors/?q=ai:aygunes.aykut-ahmetSummary: In this short paper, generally, we define a family of functions \(f_k\) that depends on the Eisenstein series with weight \(2k\), for \(k\in\mathbb N\). In more detail, by considering the function \(f_k\), we define a derivative formula for generating weakly modular forms with weight 12. As a result, we claim that this formula gives an advantage to find the special solutions of some differential equations.An extension of Macdonald's identity for \(\mathfrak{sl}_{n}\).https://zbmath.org/1456.110602021-04-16T16:22:00+00:00"Gazda, Quentin"https://zbmath.org/authors/?q=ai:gazda.quentinSummary: Let \(n\) be an odd positive integer. In this short elementary note, we slightly extend Macdonald's identity for \(\mathfrak{sl}_{n}\) into a two-variables identity in the spirit of Jacobi forms. The peculiarity of this work lies in its proof which uses Wronskians of vector-valued \(\theta\)-functions. This complements the work of \textit{A. Milas} [Lond. Math. Soc. Lect. Note Ser. 372, 330--357 (2010; Zbl 1227.11065)] towards modular Wronskians and denominator identities.A note on Hardy type sums and Dedekind sums.https://zbmath.org/1456.110592021-04-16T16:22:00+00:00"Cetin, Elif"https://zbmath.org/authors/?q=ai:cetin.elifSummary: In [Adv. Difference Equ. 2014, Paper No. 283, 18 p. (2014; Zbl 1417.11038)], the author et al. defined a new special finite sum which is denoted by \(C_1(h,k)\). In this paper, with the help of two-term polynomial relation, we will give the explicit values of the sum \(C_1(h,k)\): We will see that for the odd values of \(h\) and \(k\), this sum only depends on one variable. After that we will give many properties of this sum and connections with other well-known finite sums such as the Dedekind sums, Hardy sums and Simsek sums \(Y(h,k)\): By using the Fibonacci numbers and two-term polynomial relation, we will also give some relations for these sums.The first negative eigenvalue of Yoshida lifts.https://zbmath.org/1456.110792021-04-16T16:22:00+00:00"Das, Soumya"https://zbmath.org/authors/?q=ai:das.soumya"Pal, Ritwik"https://zbmath.org/authors/?q=ai:pal.ritwikSummary: We prove that for any given \(\epsilon >0\), the first negative eigenvalue of the Yoshida lift \(F\) of a pair of elliptic cusp forms \(f\), \(g\) having square-free levels (where \(g\) has weight 2 and satisfies \((\log Q_{g})^2 \ll \log Q_f\)), occurs before \(c_{\epsilon } \cdot Q_F^{1/2-2 \theta + \epsilon }\); where \(Q_F,Q_f,Q_g\) are the analytic conductors of \(F\), \(f\), \(g\) respectively, \(\theta < 1/4\), and \(c_{\epsilon }\) is a constant depending only on \(\epsilon \).On the work of Peter Scholze.https://zbmath.org/1456.140012021-04-16T16:22:00+00:00"Wedhorn, T."https://zbmath.org/authors/?q=ai:wedhorn.torstenThe article surveys the mathematical work of Peter Scholze giving an exposition suitable for non-experts. So, no previous knowledge of the subjects touched by the paper is necessary but for basic knowledge of algebraic geometry. Because of the vastness of the topics of Scholze's research, it is impossible to introduce all of them in detail in a short paper, therefore only basic definitions are recalled and more advanced notions are described informally and references are given to specialized literature for details.
The paper is divided into five parts. The first part surveys some basic notions of arithmetic geometry mainly related to the interplay between geometry over fields of characteristic zero and fields over characteristic \(p\), for \(p\) a prime number. The second part describes how the theory of perfectoid spaces, introduced by Peter Scholze, permits a novel way of passing between characteristic \(0\) and \(p\). The main applications of the theory of perfectoid spaces are also discussed: the proof of new important cases of the weight monodromy conjecture, obtained by Scholze himself, and the full proof of Hochster's conjecture obtained by Yves Anré using perfectoid spaces. The third part deals with the work of Scholze on \(p\)-adic Hodge theory until the most recent developments of prismatic cohomology. The fourth part is about the work of Scholze on the Langlands program. This part is the one that relies more on references to the literature as the Langlands program is such an intricate web of ideas, most of which of high technical nature, that it is impossible to summarize them in few pages. So, the author bounds the discussion in describing the main ideas of Scholze on the topic giving precise references for those who want to understand more. The last part briefly describes other topics of Scholze's research: topological Hochschild homology and the new topic of condensed mathematics.
Reviewer: Federico Bambozzi (Oxford)Hida duality and the Iwasawa main conjecture.https://zbmath.org/1456.110722021-04-16T16:22:00+00:00"Lafferty, Matthew J."https://zbmath.org/authors/?q=ai:lafferty.matthew-jBased on authors' abstract: The central result of this paper is a refinement of Hida's duality theorem between ordinary \(\Lambda\)-adic modular forms and the universal ordinary Hecke algebra. In particular, the author gives a sufficient condition for this duality to be integral with respect to particular submodules of the space of ordinary \(\Lambda\)-adic modular forms. This refinement allows us to give a simple proof that the universal ordinary cuspidal Hecke algebra modulo Eisenstein ideal is isomorphic to the Iwasawa algebra modulo an ideal related to the Kubota-Leopoldt \(p\)-adic \(L\)-function. The motivation behind these results stems from a proof of the Iwasawa main conjecture over \(\mathbb{Q}\) by \textit{M. Ohta} [Ann. Sci. Éc. Norm. Supér. (4) 36, No. 2, 225--269 (2003; Zbl 1047.11046)]. Ohta's argument in [loc. cit.] employs results on congruence modules that require a Euler's totient function condition and a non-exceptional hypotheses. While simple and elegant, Ohta's proof requires some restrictive hypotheses which we could be removed using the author's results. The results in this paper were obtained in an effort to extend Ohta's proof in [loc. cit.] by circumventing the obstructions arising from his congruence module argument.
Reviewer: Wei Feng (Beijing)Automorphic Schwarzian equations.https://zbmath.org/1456.110462021-04-16T16:22:00+00:00"Sebbar, Abdellah"https://zbmath.org/authors/?q=ai:sebbar.abdellah"Saber, Hicham"https://zbmath.org/authors/?q=ai:saber.hichamSummary: This paper concerns the study of the Schwartz differential equation \(\{h,\tau\}=s\operatorname{E}_4(\tau)\), where \(\operatorname{E}_4\) is the weight 4 Eisenstein series and \(s\) is a complex parameter. In particular, we determine all the values of \(s\) for which the solutions \(h\) are modular functions for a finite index subgroup of \(\operatorname{SL}_2({\mathbb{Z}})\). We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of \(\operatorname{SL}_2({\mathbb{Z}})\). This also leads to the solutions to the Fuchsian differential equation \(y^{\prime\prime}+s\operatorname{E}_4y=0\).General Serre weight conjectures.https://zbmath.org/1456.110932021-04-16T16:22:00+00:00"Gee, Toby"https://zbmath.org/authors/?q=ai:gee.toby"Herzig, Florian"https://zbmath.org/authors/?q=ai:herzig.florian"Savitt, David"https://zbmath.org/authors/?q=ai:savitt.davidSummary: We formulate a number of related generalisations of the weight part of Serre's conjecture to the case of \(\mathrm{GL}_n\) over an arbitrary number field, motivated by the formalism of the Breuil-Mézard conjecture. We give evidence for these conjectures, and discuss their relationship to previous work. We generalise one of these conjectures to the case of connected reductive groups which are unramified over \(\mathbb{Q}_p\), and we also generalise the second author's previous conjecture for \(\mathrm{GL}_n/\mathbb{Q}\) to this setting, and show that the two conjectures are generically in agreement.On an analogue of prime vectors among integer lattice points in ellipsoids for automorphic forms.https://zbmath.org/1456.111502021-04-16T16:22:00+00:00"Jiang, Yujiao"https://zbmath.org/authors/?q=ai:jiang.yujiao"Lü, Guangshi"https://zbmath.org/authors/?q=ai:lu.guangshiTraces of CM values and cycle integrals of polyharmonic Maass forms.https://zbmath.org/1456.110762021-04-16T16:22:00+00:00"Matsusaka, Toshiki"https://zbmath.org/authors/?q=ai:matsusaka.toshikiSummary: \textit{J. C. Lagarias} and \textit{R. C. Rhoades} [Ramanujan J. 41, No. 1--3, 191--232 (2016; Zbl 1418.11077)] generalized harmonic Maass forms by considering forms which are annihilated by a number of iterations of the action of the \(\xi \)-operator. In our previous work, we considered polyharmonic weak Maass forms by allowing the exponential growth at cusps, and constructed a basis of the space of such forms. This paper focuses on the case of half-integral weight. We construct a basis as an analogue of our work, and give arithmetic formulas for the Fourier coefficients in terms of traces of CM values and cycle integrals of polyharmonic weak Maass forms. These results put the known results into a common framework.Precise estimates for the solution of Ramanujan's generalized modular equation.https://zbmath.org/1456.110472021-04-16T16:22:00+00:00"Wang, Miao-Kun"https://zbmath.org/authors/?q=ai:wang.miaokun"Chu, Yu-Ming"https://zbmath.org/authors/?q=ai:chu.yuming"Zhang, Wen"https://zbmath.org/authors/?q=ai:zhang.wen.3Summary: In the article, we present several monotonicity theorems and inequalities for the modular equation functions \(m_{a}(r)\) and \(\mu _{a}(r),\) and find the infinite-series formulas for \(m_{1/3}(r)\) and \(m_{1/4}(r)\) which depend only on \(r\). As applications, we find several precise explicit estimates for the solution of Ramanujan's generalized modular equation.On coefficients of Poincaré series and single-valued periods of modular forms.https://zbmath.org/1456.110672021-04-16T16:22:00+00:00"Fonseca, Tiago J."https://zbmath.org/authors/?q=ai:fonseca.tiago-jSummary: We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level \(\varGamma_0(N)\) and integral weight \(k\ge 2\) coincides with the field generated by the single-valued periods of a certain motive attached to \(\varGamma_0(N)\). This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of
\textit{F. Brown} [Res. Math. Sci. 5, No. 3, Paper No. 34, 36 p. (2018; Zbl 1440.11071)] and Acres-Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann-Ono's construction of harmonic lifts of Poincaré series [\textit{K. Bringmann} and \textit{K. Ono}, Proc. Natl. Acad. Sci. USA 104, No. 10, 3725--3731 (2006; Zbl 1191.11013)].On the theory of Maass wave forms.https://zbmath.org/1456.110022021-04-16T16:22:00+00:00"Mühlenbruch, Tobias"https://zbmath.org/authors/?q=ai:muhlenbruch.tobias"Raji, Wissam"https://zbmath.org/authors/?q=ai:raji.wissamThe textbook under review consists of 7 chapters and background material as an appendix. The main motivation of the authors is that they only focus on Maass wave forms as the main objects of interest and strive to give Maass wave forms a firm analytical treatment, only briefly pointing out connections with other fields in mathematics like representation theory, in contrast with other approaches exist in the literature.
Surprisingly, the book starts with an ``Introductory roadmap'' section that sketches the content of this textbook through simple examples and brief explanations of certain concepts.
Chapters 1 and 2 give a brief introduction to the theory of classical modular forms and period polynomials. Theorems are given without proofs and necessary citations are provided. There are enough examples to understand and reinforce well the concepts.
Chapter 3 is reserved for Maass wave forms of real weight. After defining multiplier systems, the authors discuss three differential operators: the hyperbolic Laplace operator and the two Maass operators. Then, they introduce Maass wave forms of real weight and give some examples. They end this chapter up by discussing Hilbert spaces of Maass forms, Hecke operators, and the Friedrichs extension of the hyperbolic Laplace operator, concluding with a few remarks on the associated Selberg's conjecture.
Chapters 4 and 5 are the main chapters of this textbook. They introduce the concept of families of Maass wave forms and discuss the associated \(L\)-series in Chapter 4. Chapter 5 introduces period functions associated with (families of) Maass wave forms. The above objects are presented in analogy with classical modular forms and their associated \(L\)-series and period polynomials.
Chapter 6 connects Maass cusp forms (of weight 0) with discrete dynamical systems associated with the Artin billiard.
The last chapter, Chapter 7, introduces weak harmonic Maass wave forms and their associated objects like mock modular period functions and regularized \(L\)-series.
The textbook under review ends with an appendix where some theorems and results are briefly presented with helpful references in case more details are desired.
The level of the textbook is suitable for master students and so on, but some proofs can easily be understood by advanced undergraduate students.
Reviewer: Ilker Inam (Bilecik)Hyper-algebras of vector-valued modular forms.https://zbmath.org/1456.110802021-04-16T16:22:00+00:00"Raum, Martin"https://zbmath.org/authors/?q=ai:raum.martinSummary: We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of \(\mathbb{Q}\), acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein series. As a byproduct, we obtain inclusions of cuspidal automorphic representations into the tensor product of global principal series.Arithmetic Teichmuller theory.https://zbmath.org/1456.110952021-04-16T16:22:00+00:00"Rastegar, Arash"https://zbmath.org/authors/?q=ai:rastegar.arashSummary: By Grothendieck's anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number-fields encode all the arithmetic information of these curves. The goal of this paper is to develop an arithmetic Teichmuller theory, by which we mean, introducing arithmetic objects summarizing the arithmetic information coming from all curves of the same topological type defined over number-fields. We also introduce the Hecke-Teichmüller Lie algebra which plays the role of Hecke algebra in the anabelian framework.Explicit formulas for the spectral side of the trace formula of \(\mathrm{SL}(2)\).https://zbmath.org/1456.111672021-04-16T16:22:00+00:00"Wong, Tian An"https://zbmath.org/authors/?q=ai:wong.tian-anSummary: The continuous spectrum to the spectral side of the Arthur-Selberg trace formula is described in terms of intertwining operators, whose normalising factors involve quotients of \(L\)-functions. In this paper, we derive two expressions in the case of SL(2) over a number field in terms of the Riemann-Weil explicit formula: as a sum over zeroes of the associated \(L\)-functions, and as a sum of adelic distributions on Weil groups. As an application, we obtain an expression for a lower bound for the sums over zeroes with respect to the truncation parameter for Eisenstein series.Patterns of primes in the Sato-Tate conjecture.https://zbmath.org/1456.111742021-04-16T16:22:00+00:00"Gillman, Nate"https://zbmath.org/authors/?q=ai:gillman.nate"Kural, Michael"https://zbmath.org/authors/?q=ai:kural.michael"Pascadi, Alexandru"https://zbmath.org/authors/?q=ai:pascadi.alexandru"Peng, Junyao"https://zbmath.org/authors/?q=ai:peng.junyao"Sah, Ashwin"https://zbmath.org/authors/?q=ai:sah.ashwinSummary: Fix a non-CM elliptic curve \(E/\mathbb{Q} \), and let \(a_E(p) = p + 1 - \#E(\mathbb{F}_p)\) denote the trace of Frobenius at \(p\). The Sato-Tate conjecture gives the limiting distribution \(\mu_{ST}\) of \(a_E(p)/(2\sqrt{p})\) within \([-1, 1]\). We establish bounded gaps for primes in the context of this distribution. More precisely, given an interval \(I\subseteq [-1, 1]\), let \(p_{I,n}\) denote the \(n\)th prime such that \(a_E(p)/(2\sqrt{p})\in I\). We show \(\liminf_{n\rightarrow \infty }(p_{I,n+m}-p_{I,n}) < \infty\) for all \(m\ge 1\) for ``most'' intervals, and in particular, for all \(I\) with \(\mu_{ST}(I)\ge 0.36\). Furthermore, we prove a common generalization of our bounded gap result with the Green-Tao theorem. To obtain these results, we demonstrate a Bombieri-Vinogradov type theorem for Sato-Tate primes.Fourier coefficients of half-integral weight cusp forms and Waring's problem.https://zbmath.org/1456.110682021-04-16T16:22:00+00:00"Waibel, Fabian"https://zbmath.org/authors/?q=ai:waibel.fabianSummary: Extending the approach of \textit{H. Iwaniec} [Invent. Math. 87, 385--401 (1987; Zbl 0606.10017)] and
\textit{W. Duke} [Invent. Math. 92, No. 1, 73--90 (1988; Zbl 0628.10029)], we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level \(N\). As an application, we consider a Waring-type problem with sums of mixed powers.A framework for modular properties of false theta functions.https://zbmath.org/1456.110742021-04-16T16:22:00+00:00"Bringmann, Kathrin"https://zbmath.org/authors/?q=ai:bringmann.kathrin"Nazaroglu, Caner"https://zbmath.org/authors/?q=ai:nazaroglu.canerSummary: False theta functions closely resemble ordinary theta functions; however, they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal sequences via the circle method and extend earlier work on their asymptotic properties. Secondly, we show how quantum modular properties of the limits of false theta functions can be rederived directly from the modular completion of false theta functions proposed in this paper.On the kernel of the theta operator mod \(p\).https://zbmath.org/1456.110702021-04-16T16:22:00+00:00"Böcherer, Siegfried"https://zbmath.org/authors/?q=ai:bocherer.siegfried"Kodama, Hirotaka"https://zbmath.org/authors/?q=ai:kodama.hirotaka"Nagaoka, Shoyu"https://zbmath.org/authors/?q=ai:nagaoka.shoyuThe authors use theta series attached to positive definite quadratic forms to construct many examples of level one Siegel modular forms in the kernel of theta operators \(\text{mod\,}p\). They use an earlier paper of the first and third authors [Manuscr. Math. 132, No. 3--4, 501--515 (2010; Zbl 1200.11032)] and correct an error in this paper.
Reviewer: Meinhard Peters (Münster)\(r\)-tuple error functions and indefinite theta series of higher-depth.https://zbmath.org/1456.110652021-04-16T16:22:00+00:00"Nazaroglu, Caner"https://zbmath.org/authors/?q=ai:nazaroglu.canerSummary: Theta functions for definite signature lattices constitute a rich source of modular forms. A natural question is then their generalization to indefinite signature lattices. One way to ensure a convergent theta series while keeping the holomorphicity property of definite signature theta series is to restrict the sum over lattice points to a proper subset. Although such series do not generally have the modular properties that a definite signature theta function has, as shown by Zwegers for signature \((1, n-1)\) lattices, they can be completed to a function that has these modular properties by compromising on the holomorphicity property in a certain way. This construction has recently been generalized to signature \((2, n-2)\) lattices by \textit{S. Alexandrov} et al. [Sel. Math., New Ser. 24, No. 5, 3927--3972 (2018; Zbl 1420.11078)]. A crucial ingredient in this work is the notion of double error functions which naturally lends itself to generalizations. In this work we study the properties of such error functions which we will call \(r\)-tuple error
functions. We then construct an indefinite theta series for signature \((r, n-r)\) lattices and show they can be completed to modular forms by using these \(r\)-tuple error functions.Modular polynomials on Hilbert surfaces.https://zbmath.org/1456.110782021-04-16T16:22:00+00:00"Milio, Enea"https://zbmath.org/authors/?q=ai:milio.enea"Robert, Damien"https://zbmath.org/authors/?q=ai:robert.damienSummary: We describe an evaluation/interpolation approach to compute modular polynomials on a Hilbert surface, which parametrizes abelian surfaces with maximal real multiplication. Under some heuristics we obtain a quasi-linear algorithm. The corresponding modular polynomials are much smaller than the ones on the Siegel threefold. We explain how to compute even smaller polynomials by using pullbacks of theta functions to the Hilbert surface.Apéry-like numbers and families of newforms with complex multiplication.https://zbmath.org/1456.110532021-04-16T16:22:00+00:00"Gomez, Alexis"https://zbmath.org/authors/?q=ai:gomez.alexis"McCarthy, Dermot"https://zbmath.org/authors/?q=ai:mccarthy.dermot"Young, Dylan"https://zbmath.org/authors/?q=ai:young.dylanSummary: Using Hecke characters, we construct two infinite families of newforms with complex multiplication, one by \({\mathbb{Q}}(\sqrt{-3})\) and the other by \({\mathbb{Q}}(\sqrt{-2})\). The values of the \(p\)-th Fourier coefficients of all the forms in each family can be described by a single formula, which we provide explicitly. This allows us to establish a formula relating the \(p\)-th Fourier coefficients of forms of different weights, within each family. We then prove congruence relations between the \(p\)-th Fourier coefficients of these newforms at all odd weights and values coming from two of Zagier's sporadic Apéry-like sequences.Monomials of Eisenstein series.https://zbmath.org/1456.110542021-04-16T16:22:00+00:00"Griffin, Trevor"https://zbmath.org/authors/?q=ai:griffin.trevor"Kenshur, Nathan"https://zbmath.org/authors/?q=ai:kenshur.nathan"Price, Abigail"https://zbmath.org/authors/?q=ai:price.abigail"Vandenberg-Daves, Bradshaw"https://zbmath.org/authors/?q=ai:vandenberg-daves.bradshaw"Xue, Hui"https://zbmath.org/authors/?q=ai:xue.hui"Zhu, Daozhou"https://zbmath.org/authors/?q=ai:zhu.daozhouLet \(E_{k}(z)\) be the normalized Eisenstein series of weight \(k\) for the modular group \(\mathrm{SL}(2,\mathbb{Z})\). In this work, the authors consider the
zeros of \(E_{k}\) to show that the equation
\[
\prod\limits_{i=1}^{n}E_{k_{i}}=\prod\limits_{j=1}^{m}E_{l_{j}}
\]
has no solutions except for those given by known relationships between \(E_{4},E_{6},E_{8},E_{10},E_{14}.\)
Reviewer: Ahmet Tekcan (Bursa)Taylor coefficients of the Jacobi \(\theta_3(q)\) function.https://zbmath.org/1456.110662021-04-16T16:22:00+00:00"Wakhare, Tanay"https://zbmath.org/authors/?q=ai:wakhare.tanay-v"Vignat, Christophe"https://zbmath.org/authors/?q=ai:vignat.christopheSummary: We extend some results recently obtained by \textit{D. Romik} [Ramanujan J. 52, No. 2, 275--290 (2020; Zbl 07202103)] about the Taylor coefficients of the theta function \(\theta_3 (e^{- \pi})\) to the case \(\theta_3 (q)\) of a real valued variable \(0 < q < 1\). These results are obtained by carefully studying the properties of the cumulants associated to a Jacobi \(\theta_3\) (or discrete normal) distributed random variable. This article also states some integrality conjectures about rational sequences that generalize the one studied by Romik.Shintani functions for the holomorphic discrete series representation of \(\mathrm{GSp}_4(\mathbb R)\).https://zbmath.org/1456.110892021-04-16T16:22:00+00:00"Gejima, Kohta"https://zbmath.org/authors/?q=ai:gejima.kohtaThis paper is concerned with Shintani functions for the real reductive symmetric pair \((\mathrm{GSP}_4 (\mathbb R), (\mathrm{GL}_2 \times_{\mathrm{GL}_1} \mathrm{GL}_2) (\mathbb R))\). The author obtains an explicit formula for the Shintani functions for the holomorphic discrete series representation of \(\mathrm{GSP}_4 (\mathbb R)\) and proves their uniqueness. He also formulates an archimedean zeta integral of the type studied by \textit{A. Murase} and \textit{T. Sugano} [Math. Ann. 299, No. 1, 17--56 (1994; Zbl 0813.11032)] for the above mentioned symmetric pair and proves that the local zeta integral represents the local \(L\)-factor associated to the holomorphic discrete representations of \(\mathrm{GSP}_4 (\mathbb R)\).
Reviewer: Min Ho Lee (Cedar Falls)Singularities of ordinary deformation rings.https://zbmath.org/1456.110962021-04-16T16:22:00+00:00"Snowden, Andrew"https://zbmath.org/authors/?q=ai:snowden.andrew-wLet \(F\) be a finite extension of \({\mathbb Q}_p\), \(k\) a finite field of characteristic \(p\) and \(V_0\) a finite dimensional
\(k\)-vector space with a continuous representation of the absolute Galois group. Let \(\mathcal{O}\) be a finite totally ramified extension of \(W(k).\) In this situation there exists a universal ring \(R^{\mathrm{univ}}\) that parametizes framed deformations of \(V_0\) to Artinian \(\mathcal{O}\)-algebras. Let \(X\subset {\mathrm{MaxSpec}}(R^{\mathrm{univ}}[1/p])\) be the locus corresponding to ordinary representations, i.e., those that fulfill the following condition:
\[ V|_{I_F}=\begin{pmatrix}
\chi & *\\
0 &1
\end{pmatrix},
\]
where \(\chi\) is the cyclotomic character. Then it is known that \(X\) is Zariski closed and therefore equal to
\( {\mathrm{MaxSpec}}(R[1/p])\) for a unique \(\mathcal O\)-flat reduced quotient of \(R^{\mathrm{univ}}.\)
Some work devoted to understanding \(R\) was done by \textit{M. Kisin} [Invent. Math. 178, No. 3, 587--634 (2009; Zbl 1304.11043)]. In the paper under review, the author presents an original method which allows him to describe the functor of points of \(R\) as well as obtain some new results about \(R.\) For example, he shows that a minor modification of \(R\) is normal and Cohen-Macaulay but usually not Gorenstein.
Reviewer: Piotr Krasoń (Szczecin)Betti numbers of Shimura curves and arithmetic three-orbifolds.https://zbmath.org/1456.111102021-04-16T16:22:00+00:00"Frączyk, Mikołaj"https://zbmath.org/authors/?q=ai:fraczyk.mikolaj"Raimbault, Jean"https://zbmath.org/authors/?q=ai:raimbault.jeanSummary: We show that asymptotically the first Betti number \(b_1\) of a Shimura curve satisfies the Gauss-Bonnet equality \(2\pi(b_1-2)=\operatorname{vol}\) where \(\operatorname{vol}\) is hyperbolic volume; equivalently \(2g-2=(1+o(1))\operatorname{vol}\) where \(g\) is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3-orbifold asymptotically vanishes relatively to hyperbolic volume, that is \(b_1/\operatorname{vol}\to 0\). This generalizes previous results obtained by the first author [``Strong Limit Multiplicity for arithmetic hyperbolic surfaces and 3-manifolds'', Preprint, \url{arXiv:1612.05354}], on which we rely, and uses the same main tool, namely Benjamini-Schramm convergence.Hilbert modular forms and codes over \(\mathbb{F}_{p^2}\).https://zbmath.org/1456.110772021-04-16T16:22:00+00:00"Brown, Jim"https://zbmath.org/authors/?q=ai:brown.jim-l"Gunsolus, Beren"https://zbmath.org/authors/?q=ai:gunsolus.beren"Lilly, Jeremy"https://zbmath.org/authors/?q=ai:lilly.jeremy"Manganiello, Felice"https://zbmath.org/authors/?q=ai:manganiello.feliceSummary: Let \(p\) be an odd prime and consider the finite field \(\mathbb{F}_{p^2}\). Given a linear code \(\mathcal{C}\subset\mathbb{F}_{p^2}^n\), we use algebraic number theory to construct an associated lattice \(\Lambda_{\mathcal{C}}\subset\mathcal{O}_L^n\) for \(L\) an algebraic number field and \(\mathcal{O}_L\) the ring of integers of \(L\). We attach a theta series \(\theta_{\Lambda_{\mathcal{C}}}\) to the lattice \(\Lambda_{\mathcal{C}}\) and prove a relation between \(\theta_{\Lambda_{\mathcal{C}}}\) and the complete weight enumerator evaluated on weight one theta series.A sequence of modular forms associated with higher-order derivatives of Weierstrass-type functions.https://zbmath.org/1456.110502021-04-16T16:22:00+00:00"Aygunes, A. Ahmet"https://zbmath.org/authors/?q=ai:aygunes.aykut-ahmet"Simsek, Yılmaz"https://zbmath.org/authors/?q=ai:simsek.yilmaz"Srivastava, H. M."https://zbmath.org/authors/?q=ai:srivastava.hari-mohanSummary: In this article, we first determine a sequence \(\{f_n(\tau)\}_{n\in\mathbb N}\) of modular forms with weight
\[ 2^nk + 4(2^{n-1} - 1)\quad (n\in\mathbb N;\ k\in\mathbb N\backslash\{1\};\ \mathbb N:= \{1, 2, 3, \ldots\}). \]
We then present some applications of this sequence which are related to the Eisenstein series and the cusp forms. We also prove that higher-order derivatives of the Weierstrass type \(\wp_{2n}\)-functions are related to the above-mentioned sequence \(\{f_n(\tau)\}_{n\in\mathbb N}\) of modular forms.On certain degenerate Whittaker models for cuspidal representations of \({\mathrm{GL}_{k \cdot n}(\mathbb {F}_q)}\).https://zbmath.org/1456.110902021-04-16T16:22:00+00:00"Gorodetsky, Ofir"https://zbmath.org/authors/?q=ai:gorodetsky.ofir"Hazan, Zahi"https://zbmath.org/authors/?q=ai:hazan.zahiSummary: Let $\pi $ be an irreducible cuspidal representation of $\mathrm{GL}_{kn}(\mathbb {F}_q)$. Assume that $\pi = \pi _{\theta }$, corresponds to a regular character $\theta $ of $\mathbb {F}_{q^{kn}}^{*}$. We consider the twisted Jacquet module of $\pi $ with respect to a non-degenerate character of the unipotent radical corresponding to the partition $(n,n,\ldots ,n)$ of $kn$. We show that, as a $\mathrm{GL}_{n}(\mathbb {F}_q)$-representation, this Jacquet module is isomorphic to $\pi _{\theta \upharpoonright _{\mathbb {F}_n^*}} \otimes \text{St}^{\otimes (k-1)}$, where $\text{St}$ is the Steinberg representation of $\text{GL}_{n}(\mathbb {F}_q)$. This generalizes a theorem of \textit{D. Prasad} [Int. Math. Res. Not. 2000, No. 11, 579--595 (2000; Zbl 0983.22014)], who considered the case $k=2$. We prove and rely heavily on a formidable identity involving $q$-hypergeometric series and linear algebra.Identities about level 2 Eisenstein series.https://zbmath.org/1456.110482021-04-16T16:22:00+00:00"Xu, Ce"https://zbmath.org/authors/?q=ai:xu.ceSummary: In this paper we consider certain classes of generalized level 2 Eisenstein series by simple differential calculations of trigonometric functions. In particular, we give four new transformation formulas for some level 2 Eisenstein series. We can find that these level 2 Eisenstein series are reducible to infinite series involving hyperbolic functions. Moreover, some interesting new examples are given.Derived Langlands. Monomial resolutions of admissible representations.https://zbmath.org/1456.110032021-04-16T16:22:00+00:00"Snaith, Victor"https://zbmath.org/authors/?q=ai:snaith.victor-pIn this monograph, the author fits the theory of monomial resolutions in the subjects of Langlands programme and closely related topics. The fundamentals of the theory are developed in the first chapter, although the reader might want to consult [\textit{R. Boltje}, J. Algebra 246, No. 2, 811--848 (2001; Zbl 1006.20005)] to get a broader picture. This paper describes a category whose derived category is suitable environment for monomial resolutions when \(G\) is a finite group.
For a locally profinite group, the author of this monograph constructs monomial resolutions of its admissible \(k\)-representations which are recognized by the Langlands programme as objects related to questions arising in number theory. Monomial resolutions for \(\mathrm{GL}_2\) are described in both local and adélic case. They are motivated by the fact that certain subspace of automorphic representations appearing in its monomial resolution includes the classical spaces of modular forms. Additionally, the author poses certain conditions on Hecke operators under which they extend to the monomial resolution and he gives an example of a classical Hecke operator for which they are satisfied. Also, the monomial resolutions are constructed for \(\mathrm{GL}_n\) over a local field \(K\) gradually going from \(\mathrm{GL}_2 (K)\) to \(\mathrm{GL}_3(K)\) and eventually to a general case. The monograph also engages with Deligne representations, \(\varepsilon\)-factors, \(L\)-functions, Kondo-style invariants and Galois base change. Indications showing the utility of connecting these topics with monomial resolutions are given through examples. They suggest that one may be able to construct \(\varepsilon\)-factors and \(L\)-functions of [\textit{R. Godement} and \textit{H. Jacquet}, Zeta functions of simple algebras. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0244.12011)] in a simpler manner. Also, he illustrates the possibility of functoriality of Galois base change in context of finite general linear groups.
Throughout the monograph, the author explains the essential claims in detail and gives enough instructions for a reader to prove the other ones. Overviews of studied topics from the Langlands programme could be convenient for a reader interested in the results given in this book. On the other hand, a reader interested in a connection of monomial resolutions with topics of the Langlands programme has a motivation for further research.
Reviewer: Barbara Bošnjak (Zagreb)Counting and equidistribution for quaternion algebras.https://zbmath.org/1456.111862021-04-16T16:22:00+00:00"Lesesvre, Didier"https://zbmath.org/authors/?q=ai:lesesvre.didierSummary: We aim at studying automorphic forms of bounded analytic conductor in the totally definite quaternion algebra setting. We prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure. It leads to answering the Sato-Tate conjectures in this case, and contains the counting law of the universal family, with a power savings error term.Averaging over Narain moduli space.https://zbmath.org/1456.830672021-04-16T16:22:00+00:00"Maloney, Alexander"https://zbmath.org/authors/?q=ai:maloney.alexander"Witten, Edward"https://zbmath.org/authors/?q=ai:witten.edwardSummary: Recent developments involving JT gravity in two dimensions indicate that under some conditions, a gravitational path integral is dual to an average over an ensemble of boundary theories, rather than to a specific boundary theory. For an example in one dimension more, one would like to compare a random ensemble of two-dimensional CFT's to Einstein gravity in three dimensions. But this is difficult. For a simpler problem, here we average over Narain's family of two-dimensional CFT's obtained by toroidal compactification. These theories are believed to be the most general ones with their central charges and abelian current algebra symmetries, so averaging over them means picking a random CFT with those properties. The average can be computed using the Siegel-Weil formula of number theory and has some properties suggestive of a bulk dual theory that would be an exotic theory of gravity in three dimensions. The bulk dual theory would be more like \(\mathrm{U}(1)^{2D}\) Chern-Simons theory than like Einstein gravity.Kernels for products of Hilbert \(L\)-functions.https://zbmath.org/1456.110872021-04-16T16:22:00+00:00"Choie, YoungJu"https://zbmath.org/authors/?q=ai:choie.youngju"Zhang, Yichao"https://zbmath.org/authors/?q=ai:zhang.yichaoSummary: We study kernel functions of \(L\)-functions and products of \(L\)-functions of Hilbert cusp forms over real quadratic fields. This extends the results on elliptic modular forms in [\textit{N. Diamantis} and \textit{C. O'Sullivan}, Math. Ann. 346, No. 4, 897--929 (2010; Zbl 1257.11050)].Functoriality and the trace formula.https://zbmath.org/1456.110882021-04-16T16:22:00+00:00"Arthur, James"https://zbmath.org/authors/?q=ai:arthur.james-k|arthur.james-d|arthur.james-gSummary: We shall summarize two different lectures that were presented on beyond endoscopy, the proposal of Langlands to apply the trace formula to the principle of functoriality. We also include an elementary description of functoriality, and in the last section, some general reflections on where the study of beyond endoscopy might be leading.
For the entire collection see [Zbl 1401.11005].The power of 2: small primes in number theory.https://zbmath.org/1456.110972021-04-16T16:22:00+00:00"Thorne, Jack A."https://zbmath.org/authors/?q=ai:thorne.jack-aFrom the text: The first proposition in Euclid's \textit{Elements} gives the construction, with ruler and compass, of the equilateral triangle. Later, Euclid shows how to construct a regular \(n\)-gon for \(n=5\) and \(n=15\), and how to pass from a construction of the regular \(n\)-gon to a construction of the regular \(2n\)-gon. For which other values of \(n\) does a construction of the regular \(n\)-gon using rule rand compass exist?
The answer to this ancient question was given roughly 2000 years later, by Gauss.
My research is in algebraic number theory, and in particular the Langlands program, which aims to give the ultimate non-abelian generalisation of class field theory, a topic which has its roots in topics treated in \textit{Disquisitiones} (consider quadratic reciprocity, the ideal class group and the reduction theory of binary quadratic forms, to name but a few).A magnetic modular form.https://zbmath.org/1456.110582021-04-16T16:22:00+00:00"Li, Yingkun"https://zbmath.org/authors/?q=ai:li.yingkun"Neururer, Michael"https://zbmath.org/authors/?q=ai:neururer.michael-oOn the numbers of the form \(x^2 + 11y^2\).https://zbmath.org/1456.110422021-04-16T16:22:00+00:00"Kreuzer, Martin"https://zbmath.org/authors/?q=ai:kreuzer.martin"Rosenberger, Gerhard"https://zbmath.org/authors/?q=ai:rosenberger.gerhardPrimes expressible as \(x^2+ny^2\) for a positive integer \(n\) were
studied by Euler. For number like \(n=1,2,3\) etc., the quadratic
reciprocity law already gives characterization of these primes as
precisely those belonging to certain congruence classes. However,
there are only finitely many \(n\) for which such a characterization
can be given. Euler wrote down a list of 65 `Idoneus Numerus'
(roughly translated as `convenient numbers'), the largest of which
is 1848; this list is expected to be complete but this is not yet
proved. Here, \(n\) is said to be `convenient' if each positive
integer that is expressible as \(x^2+ny^2\) with \(\gcd(x^2,ny^2)=1\), is
uniquely so expressible if, and only if, \(n\) is the power of a prime
or twice such a number. The smallest ``inconvenient'' number is \(11\).
Ring class field theory enables us to characterize prime numbers
that are expressible as \(x^2+ny^2\). One considers the order
\(\mathbb{Z}[\sqrt{-n}]\) in the corresponding imaginary quadratic
field (so, this is the full ring of integers when \(n \equiv 1,2\) mod
\(4\)) and its ring class field. If \(p\) is an odd prime not dividing
\(n\), and also, not dividing the discriminant of a monic integer
polynomial \(f_n\) of degree \(h(-4n)\) whose root is an algebraic
integer that generates the ring class field, the necessary and
sufficient condition for \(p\) to be expressible as \(x^2+ny^2\) is that
\(-n\) is a quadratic residue mod \(p\) and \(f_n(a) \equiv 0\) mod \(p\)
has a solution \(a \in \mathbb{Z}\). For \(n \equiv 1\) or \(2\) mod \(4\),
this condition is in terms of the class field of
\(\mathbb{Q}(\sqrt{-n})\).
The first inconvenient number is \(11\) and, in particular, the
above-mentioned theorem characterizes primes expressible as
\(x^2+11y^2\); the polynomial \(f_{11}(x) = x^3-2x^2+2x-2\) in this
case. However, the authors of the paper address the more difficult
task of characterizing positive integers (not just primes) that are
expressible as \(x^2+ 11y^2\) with gcd\((x,11y)=1\). They use the
detailed structure of the class group \(G_{11}\) of level \(11\) to
obtain and phrase their results. The group \(G_{11}\) (is the image in
\(PSL_2(\mathbb{R})\) of the set of matrices of the form \(\begin{pmatrix} a
\sqrt{11} & b \\ c & d \sqrt{11}\end{pmatrix}\) or \(\begin{pmatrix} a & b \sqrt{11} \cr
c \sqrt{11} & d\end{pmatrix}\) where \(a,b,c,d\) are integers and the matrix has
determinant \(1\).
In particular, they show that \(G_{11}\) has four conjugacy classes of
elliptic elements of order \(2\) represented by \(t_1,t_2,t_3,t_4\) say,
where \(t_1 = \begin{pmatrix} 0 & 1 \cr -1 & 0\end{pmatrix}\). A conjugate of \(t_1\) in
\(G_{11}\) gives a matrix in \(G_{11}\) whose \((1,2)\)-th entry is
expressible as \(x^2+11y^2\) with gcd\((x,11y)=1\). Conversely,
corresponding to any positive integer \(n\) for which \(-11\) is a
quadratic residue modulo \(n\), the authors construct an elliptic
element \(A_n\) of order \(2\) in \(G_{11}\) which is conjugate to exactly
one of the \(t_i\)'s, and this is \(t_1\) if and only if \(n\) is
expressible as \(x^2+11y^2\). Thus, there are four sets \(S_i\)
consisting of those \(n\) for which the \((1,2)\)-th entry of \(A_n\) is
\(n\) and \(A_n\) is conjugate to \(t_i\) (\(1 \leq i \leq 4\)). The authors
prove quite easily that \(S_2=S_3\) and \(S_4=2S_2\) consists precisely
the positive integers \(\equiv 2\) modulo \(4\). The set of interest is
\(S_1\) and the difficult task they accomplish is to distinguish
between \(S_1\) and \(S_2\).
For the entire collection see [Zbl 1435.20002].
Reviewer: Balasubramanian Sury (Bangalore)Quantitative non-vanishing results on \(L\)-functions.https://zbmath.org/1456.111692021-04-16T16:22:00+00:00"Jiang, Yujiao"https://zbmath.org/authors/?q=ai:jiang.yujiao"Lü, Guangshi"https://zbmath.org/authors/?q=ai:lu.guangshiSummary: In this paper, we establish a quantitative non-vanishing result on a class of twisted \(L\)-functions \(\mathcal{A}(s,\chi)\) of degree \(k\), which satisfy some mild and standard assumptions. As a corollary, we show that for a positive proportion of characters in a specific set, the special values \(\mathcal{A}(\beta,\chi)\) are non-vanishing for \(\Re\beta>1-\frac{1}{k}\). In particular, our argument holds for Rankin-Selberg \(L\)-functions and symmetric square \(L\)-functions on certain higher rank groups, and leads to some new non-trivial results for the first time.Profinite invariants of arithmetic groups.https://zbmath.org/1456.200232021-04-16T16:22:00+00:00"Kammeyer, Holger"https://zbmath.org/authors/?q=ai:kammeyer.holger"Kionke, Steffen"https://zbmath.org/authors/?q=ai:kionke.steffen"Raimbault, Jean"https://zbmath.org/authors/?q=ai:raimbault.jean"Sauer, Roman"https://zbmath.org/authors/?q=ai:sauer.romanOne says that an arithmetic group has the conguence subgroup property if the conguence kernel of it is finite.
The main result of the paper states that the sign of the Euler characteristic of an arithmetic group \(\Gamma\) with the congruence subgroup property is determined by its profinite completion (or equivalently by the family of its finite quotients). More precisely, Theorem 1.1 states that two arithmetic groups \(\Gamma_1,\Gamma_2\) with the conguence subgroup property and commensurable profinite completions have the same sign of their Euler characteristic.
Note that by the result of \textit{M. Aka} [J. Algebra 352, No. 1, 322--340 (2012; Zbl 1254.20026)] an arithmetic group with the conguence subgroup property is determined by its profinite completion up to finitely many isomorphism calsses among arithmetic groups.
It is also shown in the paper that two natural generalizations of Theorem 1.1 do not hold. Namely, the Euler charateristic of \(\Gamma\) is not determined by the profinite completion. Also the profinite completion does not determine the sign of the Euler characteristic of finitely generated residually finite group of type \(F\).
Reviewer: Pavel Zalesskij (Brasília)Generalizations of Alder's conjecture via a conjecture of Kang and Park.https://zbmath.org/1456.050122021-04-16T16:22:00+00:00"Duncan, Adriana L."https://zbmath.org/authors/?q=ai:duncan.adriana-l"Khunger, Simran"https://zbmath.org/authors/?q=ai:khunger.simran"Swisher, Holly"https://zbmath.org/authors/?q=ai:swisher.holly"Tamura, Ryan"https://zbmath.org/authors/?q=ai:tamura.ryanSummary: Let \(\varDelta_d^{(a,b)}(n) = q_d^{(a)}(n) - Q_d^{(b)}(n)\) where \(q_d^{(a)}(n)\) counts the number of partitions of \(n\) into parts with difference at least \(d\) and size at least \(a\), and \(Q_d^{(b)}(n)\) counts the number of partitions into parts \(\equiv \pm b \pmod{d + 3}\). In 1956, Alder conjectured that \(\varDelta_d^{(1,1)}(n) \ge 0\) for all positive \(n\) and \(d\). This conjecture was proved partially by \textit{G. E. Andrews} [Pac. J. Math. 36, 279--284 (1971; Zbl 0195.31201)], by
\textit{A. J. Yee} [J. Reine Angew. Math. 616, 67--88 (2008; Zbl 1221.11205)], and was fully resolved by \textit{C. Alfes} et al. [Proc. Am. Math. Soc. 139, No. 1, 63--78 (2011; Zbl 1242.11075)]. Alder's conjecture generalizes several well-known partition identities, including Euler's theorem that the number of partitions of \(n\) into odd parts equals the number of those into distinct parts, as well as the first of the famous Rogers-Ramanujan identities. \textit{S.-Y. Kang} and \textit{E. Y. Park} [Discrete Math. 343, No. 7, Article ID 111882, 5 p. (2020; Zbl 1440.05041)] constructed an extension of Alder's conjecture which relates to the second Rogers-Ramanujan identity by considering the difference \(\varDelta_d^{(a,b,-)}(n) = q_d^{(a)}(n) - Q_d^{(b,-)}(n)\), where \(Q_d^{(b,-)}(n)\) counts the number of partitions into parts \(\equiv \pm b \pmod{d + 3}\) excluding the part \(d+3-b\). Kang and Park [loc. cit.] conjectured that \(\varDelta_d^{(2,2,-)}(n)\ge 0\) for all \(d\ge 1\) and \(n\ge 0\), and proved this when \(d = 2^r - 2\) and \(n\) is even. Here, we prove Kang and Park's conjecture for all \(d\ge 62\). Toward proving the remaining cases, we adapt work of Alfes et al. [loc. cit.] to generate asymptotics for the related functions. Additionally, we present a more generalized conjecture for higher \(a=b\) and prove it for infinite classes of \(n\) and \(d\).Angular changes of Fourier coefficients at primes.https://zbmath.org/1456.110552021-04-16T16:22:00+00:00"Kumar, Balesh"https://zbmath.org/authors/?q=ai:kumar.balesh"Viswanadham, G. K."https://zbmath.org/authors/?q=ai:viswanadham.g-kSummary: We study the angle changes of Fourier coefficients of cusp forms and \(q\)-exponents of generalized modular functions at primes. More precisely, we prove that both these subsequences, under certain conditions, fall infinitely often outside any given wedge \(\mathcal{W}(\theta _1, \theta _2):=\{re^{i\theta }: r>0, \theta \in [\theta _1,\theta _2]\}\) with \(0\le \theta _2-\theta _1< \pi \).Algebraic results for the values \(\vartheta_3(m\tau)\) and \(\vartheta_3(n\tau)\) of the Jacobi theta-constant.https://zbmath.org/1456.111312021-04-16T16:22:00+00:00"Elsner, Carsten"https://zbmath.org/authors/?q=ai:elsner.carsten"Luca, Florian"https://zbmath.org/authors/?q=ai:luca.florian"Tachiya, Yohei"https://zbmath.org/authors/?q=ai:tachiya.yoheiSet \(\mathbb H= \{ \tau\in\mathbb C ; \Im(\tau)>0\}\). Let \(m\) and \(n\) be distinct integers and \(\tau\in\mathbb H\). Then the authors prove that at least two of numbers \(e^{\pi i\tau}\), \(1+2\sum_{j=1}^\infty e^{\pi in\tau j^2}\) and \(1+2\sum_{j=1}^\infty e^{\pi im\tau j^2}\) are algebraically independent over \(\mathbb Q\). In particular, the two numbers \(1+2\sum_{j=1}^\infty e^{\pi in\tau j^2}\) and \(1+2\sum_{j=1}^\infty e^{\pi im\tau j^2}\) are algebraically independent over \(\mathbb Q\) for any \(\tau\in\mathbb H\) such that \(e^{\pi i\tau}\) is an algebraic number.
Reviewer: Jaroslav Hančl (Ostrava)Arithmetic diagonal cycles on unitary Shimura varieties.https://zbmath.org/1456.140312021-04-16T16:22:00+00:00"Rapoport, M."https://zbmath.org/authors/?q=ai:rapoport.michael"Smithling, B."https://zbmath.org/authors/?q=ai:smithling.brian-d"Zhang, W."https://zbmath.org/authors/?q=ai:zhang.wei.1The paper under review has three main objectives, clearly and explicitly stated by the authors in its introduction. The first one of these, achieved in Section 6, is to give a detailed and explicit variant of the arithmetic Gan-Gross-Prasad conjecture for Shimura varieties of PEL type associated to certain unitary groups. This conjecture concerns the order of vanishing at the point \(s = 1/2\) of the \(L\)-functions \(L(s,\pi,R)\) associated to certain automorphic representations \(\pi\) of the adelic points of an algebraic group \(\widetilde{HG}\) defined over \(\mathbb{Q}\), which is introduced in Section 2.1. More precisely, Conjecture 6.10 of the paper under review relates the aforementioned order of vanishing to the dimension of the module of Hecke-equivariant homomorphisms between the \(K\)-invariant finite part \(\pi_f^K\) of the automorphic representation \(\pi\), where \(K \subseteq \widetilde{HG}(\mathbb{A}_f)\) is some open and compact subgroup of the finite-adelic points of \(\widetilde{HG}\), and the cyclic Hecke module \(\mathcal{Z}_{K,0} \subseteq \mathrm{CH}^{n - 1}(M_K(\widetilde{HG}))_{\mathbb{Q},0}\) generated by a suitable cohomologically trivial algebraic cycle constructed on the canonical model \(M_K(\widetilde{HG})\) of the Shimura variety \(\mathrm{Sh}_K(\widetilde{HG})\) associated to \(K\) and \(\widetilde{HG}\). Note that \(\dim(M_K(\widetilde{HG})) = 2 n - 3\), so that the module \(\mathcal{Z}_{K,0}\) is defined by an algebraic cycle of degree just above the middle dimension of the Shimura variety \(M_K(\widetilde{HG})\). Moreover, Conjecture 6.12 predicts that \(L(s,\pi,R)\) vanishes at \(s = 1/2\) with order exactly one if and only if a suitable linear form \(\ell_K\), defined on the Chow group \(\mathrm{CH}^{n - 1}(M_K(\widetilde{HG}))_{\mathbb{Q},0}\), is non-trivial when restricted to the \(\pi_f^K\)-isotypic component of \(\mathcal{Z}_{K,0}\). Note that the linear form \(\ell_K\) is defined using the Beilinson-Bloch height pairing for algebraic cycles, whose construction is recalled in Section 6.1 of the paper under review, and is conditional on certain conjectures concerning the existence of suitable integral models (Conjecture 6.1) and the possibility to lift algebraic cycles from the generic fibre of these integral models (Conjecture 6.2). However, as the authors observe in Remark 6.18, the Beilinson-Bloch height pairing is unconditionally defined when \(n = 2\), and it coincides with the Néron-Tate height pairing. In particular, the arithmetic Gan-Gross-Prasad conjecture should be closely related in this case to the \textit{Gross-Zagier formula on Shimura curves}, studied in the foundational book [\textit{X. Yuan} et al., The Gross-Zagier formula on Shimura curves. Princeton, NJ: Princeton University Press (2013; Zbl 1272.11082)].
The second goal of the paper under review, achieved in Section 8, is to formulate a global analogue of the \textit{arithmetic fundamental lemma} and \textit{arithmetic transfer} conjectures, which are local conjectures proposed respectively by the third author of the paper under review [\textit{W. Zhang}, Invent. Math. 188, No. 1, 197--252 (2012; Zbl 1247.14031)] and by the same authors of the paper under review [\textit{M. Rapoport} et al., Math. Ann. 370, No. 3--4, 1079--1175 (2018; Zbl 1408.14143)]. The advantage of these new conjectures over the arithmetic Gan-Gross-Prasad conjecture is that they bypass the use of the Beilinson-Bloch height pairing, which is only conjecturally defined, by working with specific integral models of the Shimura varieties in question, and using the arithmetic intersection pairing defined in [\textit{H. Gillet} and \textit{C. Soulé}, Publ. Math., Inst. Hautes Étud. Sci. 72, 93--174 (1990; Zbl 0741.14012)]. We note that Theorem 8.14 of the paper under review reduces the global Conjectures 8.2 and 8.8 to the semi-global Conjecture 8.13. Moreover, Theorem 8.15 of the paper under review proves that half of Conjecture 8.13 holds true if \(n \leq 3\), where again \(n := (\dim(M_K(\widetilde{HG})) + 3)/2\).
The third main goal of the paper under review, achieved in Sections 4 and 5, is precisely to define the integral model of the Shimura variety \(M_K(\widetilde{HG})\) which appears in the aforementioned global and semi-global conjectures. Note that \(M_K(\widetilde{HG})\) is associated to a suitable moduli problem for abelian varieties with level structure, whose definition is spelled out in Section 3.2. Moreover, \(M_K(\widetilde{HG})\) is closely related to the Shimura varieties considered in [\textit{R. E. Kottwitz}, J. Am. Math. Soc. 5, No. 2, 373--444 (1992; Zbl 0796.14014)] and [\textit{M. Harris} and \textit{R. Taylor}, The geometry and cohomology of some simple Shimura varieties. With an appendix by Vladimir G. Berkovich. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)].
To conclude, the paper under review deepens considerably our understanding of the arithmetic Gan-Gross-Prasad conjectures for Shimura varieties associated to unitary groups, and makes clear the links between the local and global pictures. The notation and terminologies employed in the paper under review can easily seem overwhelmingly tecnical, as it is often the case in this research area. However, the authors make a great effort of clarity, and the most important notations are summarized when needed.
Reviewer: Riccardo Pengo (Lyon)Aspects of the \(s\) transformation bootstrap.https://zbmath.org/1456.813552021-04-16T16:22:00+00:00"Brehm, Enrico M."https://zbmath.org/authors/?q=ai:brehm.enrico-m"Das, Diptarka"https://zbmath.org/authors/?q=ai:das.diptarkaA hypergeometric version of the modularity of rigid Calabi-Yau manifolds.https://zbmath.org/1456.110732021-04-16T16:22:00+00:00"Zudilin, Wadim"https://zbmath.org/authors/?q=ai:zudilin.wadimThis paper considers the fourteen one-parameter families of Calabi-Yau
threefolds whose periods are expressed in terms of hypergeometric functions.
For these fourteen families, periods are solutions of hypergeometric equations
with parameter \((r, 1-r, t, 1-t)\), where
\begin{multline*}
(r,t)=\Big(\frac{1}{2},\frac{1}{2}\Big),\Big(\frac{1}{2},\frac{1}{3}\Big),\Big(\frac{1}{2},\frac{1}{4}\Big),
\Big(\frac{1}{2},\frac{1}{6}\Big),\Big(\frac{1}{3}\Big),\Big(\frac{1}{3},\frac{1}{4}\Big),\Big(\frac{1}{3},\frac{1}{6}\Big),\\
\Big(\frac{1}{4},\frac{1}{4}\Big),\Big(\frac{1}{4},\frac{1}{6}\Big),\Big(\frac{1}{6},\frac{1}{6}\Big),\Big(\frac{1}{5},\frac{2}{5}\Big),
\Big(\frac{1}{8},\frac{3}{8}\Big),\Big(\frac{1}{10},\frac{3}{10}\Big),\Big(\frac{1}{12},\frac{5}{12}\Big).
\end{multline*}
At a conifold point, any of these Calabi-Yau threefolds becomes rigid,
and the \(p\)-th coefficient \(a(p)\) of the corresponding modular form of weight \(4\)
can be recovered from the truncated partial sums of the corresponding
hypergeometric series modulo a higher power of \(p\), where \(p\) is any good prime \(>5\).
This paper discusses relationships between the critical values of the \(L\)-series of the modular form
and the values of a related basis of solutions to the hypergeometric differential equation.
It is numerically observed that the critical \(L\)-values are \(\mathbb{Q}\)-proportional to the
hypergeometric values \(F_1(1), F_2(1), F_3(1)\), where \(F_j(z)\) are solutions of the hypergeometric
equation for the hypergeometric function \(F_0(z)=_4F_3(z)\) with parameters \((r, 1-r, t, 1-t)\).
This confirms the prediction of Golyshev concerning gamma structures [\textit{V. Golyshev} and \textit{A. Mellit}, J. Geom. Phys. 78, 12--18 (2014; Zbl 1284.33001)].
Reviewer: Noriko Yui (Kingston)Globally analytic \(p\)-adic representations of the pro-\(p\) Iwahori subgroup of \(\mathrm{GL}(2)\) and base change. II: A Steinberg tensor product theorem.https://zbmath.org/1456.112102021-04-16T16:22:00+00:00"Clozel, Laurent"https://zbmath.org/authors/?q=ai:clozel.laurentThe current work is the second part of author's paper, the first part of which is [Bull. Iran. Math. Soc. 43, No. 4, 55--76 (2017; Zbl 1423.11187)]. The first part of this paper is devoted to the study of Iwasawa algebra of the pro-\(p\) Iwahori subgroup of GL\((2, L)\) for an unramified extension \(L\) of degree \(r\) of \(\mathbb{Q}_p\) and gave a presentation of it by generators and relations, imitating [Doc. Math. 16, 545--559 (2011; Zbl 1263.22011)]. A natural base change map then appears that, however, is well defined only for the globally analytic distributions on the groups, seen as rigid-analytic spaces. In Section 1 of [Bull. Iran. Math. Soc. 43, No. 4, 55--76 (2017; Zbl 1423.11187)], the author stated that this should be related to a construction of base change for representations of these groups, similar to \textit{R. Steinberg}'s tensor product theorem [Nagoya Math. J. 22, 33--56 (1963; Zbl 0271.20019)] for algebraic groups over finite fields.
In this paper under review, the author gives such a construction, and show that it is compatible with the (\(p\)-adic) Langlands correspondence in the case of the principal series for GL\((2)\). He exploits the base change map for globally analytic distributions constructed there, relating distributions on the pro-\(p\) Iwahori subgroup of GL\((2)\) over \(\mathbb{Q}_p\) and those on the pro-\(p\) Iwahori subgroup of GL\((2, L)\) where \(L\) is an unramified extension of \(\mathbb{Q}_p\). This is used to obtain a functor, the `Steinberg tensor product', relating globally analytic \(p\)-adic representations of these two groups. We are led to extend the theory, sketched by \textit{M. Emerton} [Locally analytic vectors in representations of locally \(p\)-adic analytic groups. Providence, RI: American Mathematical Society (AMS) (2017; Zbl 1430.22020)], of these globally analytic representations. In the last section, the author showed that this functor exhibits, for principal series, Langlands' base change (at least for the restrictions of these representations to the pro-\(p\) Iwahori subgroups.)
For the entire collection see [Zbl 1401.20003].
Reviewer: Wei Feng (Beijing)Dedekind sums arising from newform Eisenstein series.https://zbmath.org/1456.110622021-04-16T16:22:00+00:00"Stucker, T."https://zbmath.org/authors/?q=ai:stucker.t"Vennos, A."https://zbmath.org/authors/?q=ai:vennos.a"Young, M. P."https://zbmath.org/authors/?q=ai:young.matthew-p|young.malcolm-pThe Maaß space for paramodular groups.https://zbmath.org/1456.110812021-04-16T16:22:00+00:00"Heim, Bernhard"https://zbmath.org/authors/?q=ai:heim.bernhard-ernst"Krieg, Aloys"https://zbmath.org/authors/?q=ai:krieg.aloysSummary: We describe a characterization for the Maaß space associated with the paramodular group of degree \(2\) and square-free level \(N\). As an application we show that the Maaß space is invariant under all Hecke operators. As a consequence, we conclude that the associated Siegel-Eisenstein series belongs to the Maaß space.Equations of hyperelliptic Shimura curves.https://zbmath.org/1456.110452021-04-16T16:22:00+00:00"Guo, Jia-Wei"https://zbmath.org/authors/?q=ai:guo.jiawei"Yang, Yifan"https://zbmath.org/authors/?q=ai:yang.yifanSummary: By constructing suitable Borcherds forms on Shimura curves and using Schofer's formula for norms of values of Borcherds forms at CM points, we determine all of the equations of hyperelliptic Shimura curves \(X_{0}^{D}(N)\). As a byproduct, we also address the problem of whether a modular form on Shimura curves \(X_{0}^{D}(N)/W_{D,N}\) with a divisor supported on CM divisors can be realized as a Borcherds form, where \(X_{0}^{D}(N)/W_{D,N}\) denotes the quotient of \(X_{0}^{D}(N)\) by all of the Atkin-Lehner involutions. The construction of Borcherds forms is done by solving certain integer programming problems.Arithmetic topology in Ihara theory. II: Milnor invariants, dilogarithmic Heisenberg coverings and triple power residue symbols.https://zbmath.org/1456.112162021-04-16T16:22:00+00:00"Hirano, Hikaru"https://zbmath.org/authors/?q=ai:hirano.hikaru"Morishita, Masanori"https://zbmath.org/authors/?q=ai:morishita.masanoriIn this paper, a wide class of triple quadratic (resp., cubic) residue symbols \([p_1,p_2,p_3]\) of primes \(p_i\) (\(i=1,2,3\)
in \(\mathbb{Q}\) (resp., \(\mathbb{Q}(\sqrt{-3})\)) is connected to the mod \(\ell\) Milnor invariants introduced in previous work [\textit{H. Kodani} et al., Publ. Res. Inst. Math. Sci. 53, No. 4, 629--688 (2017; Zbl 1430.11082)] as certain coefficients of Magnus series of Frobenius elements arising from Ihara theory on Galois representations in the pro-\(\ell\) fundamental groups of punctured projective lines. Dilogarithmic mod \(\ell\) Heisenberg ramified covering \(D(\ell)\) of \(\mathbb{P}^1\) plays a central role ``as a higher analog of the dilogarithmic function for the gerbe associated to the mod \(\ell\) Heisenberg group''.
The monodromy transformations of certain functions on \(D(\ell)\) along the pro-\(\ell\) longitudes of Frobenius elements
turn out to capture the aimed power residue symbols via Wojtkowiak's work on the \(\ell\)-adic Galois polylogarithms.
Reviewer: Hiroaki Nakamura (Osaka)Algebraic cycles and residues of degree \(8 L\)-functions of GSp(4) \(\times\) GL(2).https://zbmath.org/1456.110842021-04-16T16:22:00+00:00"Lemma, Francesco"https://zbmath.org/authors/?q=ai:lemma.francescoSummary: We prove a cohomological formula for noncritical residues of degree 8 automorphic \(L\)-functions of \(\text{GSp}(4) \times \text{GL}(2)\) in the spirit of Beilinson conjecture. We rely on the cohomological interpretation of an automorphic period integral and on the study of Novodvorsky's integral representation of the \(L\)-functions.Identities of cycle integrals of weak Maass forms.https://zbmath.org/1456.110492021-04-16T16:22:00+00:00"Alfes-Neumann, Claudia"https://zbmath.org/authors/?q=ai:alfes-neumann.claudia"Schwagenscheidt, Markus"https://zbmath.org/authors/?q=ai:schwagenscheidt.markusSummary: We prove identities between cycle integrals of non-holomorphic modular forms arising from applications of various differential operators to weak Maass formsIrreducibility of automorphic Galois representations of low dimensions.https://zbmath.org/1456.110982021-04-16T16:22:00+00:00"Xia, Yuhou"https://zbmath.org/authors/?q=ai:xia.yuhouSummary: Let \(\pi \) be a polarizable, regular algebraic, cuspidal automorphic representation of \(\mathrm{ GL }_n(\mathbb{A}_F)\), where \(F\) is an imaginary CM field and \(n \le 6\). We show that there is a Dirichlet density 1 set \(\mathfrak{L}\) of rational primes, such that for all \(l\in \mathfrak{L}\), the \(l\)-adic Galois representations associated to \(\pi \) are irreducible.Further improvement on bounds for \(L\)-functions related to \(\mathrm{GL}(3)\).https://zbmath.org/1456.110852021-04-16T16:22:00+00:00"Sun, Haiwei"https://zbmath.org/authors/?q=ai:sun.haiwei"Ye, Yangbo"https://zbmath.org/authors/?q=ai:ye.yangboFrom the monster to Thompson to O'Nan.https://zbmath.org/1456.110632021-04-16T16:22:00+00:00"Duncan, John F. R."https://zbmath.org/authors/?q=ai:duncan.john-f-rSummary: The commencement of monstrous moonshine is a connection between the largest sporadic simple group-the monster-and complex elliptic curves. Here we explain how a closer look at this connection leads, via the Thompson group, to recently observed relationships between the non-monstrous sporadic simple group of O'Nan and certain families of elliptic curves defined over the rationals. We also describe umbral moonshine from this perspective.
For the entire collection see [Zbl 1452.17002].An exact formula for \(\mathbf{U}(3)\) Vafa-Witten invariants on \(\mathbb{P}^2\).https://zbmath.org/1456.110752021-04-16T16:22:00+00:00"Bringmann, Kathrin"https://zbmath.org/authors/?q=ai:bringmann.kathrin"Nazaroglu, Caner"https://zbmath.org/authors/?q=ai:nazaroglu.canerSummary: Topologically twisted \(\mathcal{N} = 4\) super Yang-Mills theory has a partition function that counts Euler numbers of instanton moduli spaces. On the manifold \(\mathbb{P}^2\) and with gauge group \(\mathrm{U}(3)\) this partition function has a holomorphic anomaly which makes it a mock modular form of depth two. We employ the circle method to find a Rademacher expansion for the Fourier coefficients of this partition function. This is the first example of the use of circle method for a mock modular form of a higher depth.``cases? for \(\varphi\) also'' a question raised by Ramanujan.https://zbmath.org/1456.330152021-04-16T16:22:00+00:00"Vasuki, K. R."https://zbmath.org/authors/?q=ai:vasuki.kaliyur-ranganna"Bhuvan, E. N."https://zbmath.org/authors/?q=ai:bhuvan.e-nAnalytical approach to Ramanujan type and Ramanujan's modular equations of degree 7.https://zbmath.org/1456.110612021-04-16T16:22:00+00:00"Mahadevaswamy"https://zbmath.org/authors/?q=ai:mahadevaswamy.Determining cuspforms from critical values of convolution \(L\)-functions and Rankin-Cohen brackets.https://zbmath.org/1456.110562021-04-16T16:22:00+00:00"Lanphier, Dominic"https://zbmath.org/authors/?q=ai:lanphier.dominicSupercuspidal ramifications and traces of adjoint lifts.https://zbmath.org/1456.110522021-04-16T16:22:00+00:00"Banerjee, Debargha"https://zbmath.org/authors/?q=ai:banerjee.debargha"Mandal, Tathagata"https://zbmath.org/authors/?q=ai:mandal.tathagataSummary: In this paper, we write down the local Brauer classes of the endomorphism algebras of motives attached to non-CM primitive Hecke eigenforms for the supercuspidal prime \(p = 2\). The same for odd supercuspidal primes are determined by \textit{S. Bhattacharya} and \textit{E. Ghate} [Proc. Am. Math. Soc. 143, No. 11, 4669--4684 (2015; Zbl 1378.11053)]. We also treat the case of odd unramified supercuspidal primes of level zero also removing a mild hypothesis of them. As an intermediate step, we write down a description of the inertial Galois representation even for \(p = 2\) generalizing the construction of \textit{E. Ghate} and \textit{A. Mézard} [Trans. Am. Math. Soc. 361, No. 5, 2243--2261 (2009; Zbl 1251.11044)]. Some numerical examples using age and MFDB are provided supporting some of our theorems.Computing special \(L\)-values of certain modular forms with complex multiplication.https://zbmath.org/1456.110572021-04-16T16:22:00+00:00"Li, Wen-Ching Winnie"https://zbmath.org/authors/?q=ai:li.wen-ching-winnie"Long, Ling"https://zbmath.org/authors/?q=ai:long.ling"Tu, Fang-Ting"https://zbmath.org/authors/?q=ai:tu.fang-tingThis is an expository article in which the authors present two explicit examples
of computing special \(L\)-values of modular forms admitting complex multiplication.
Two methods for this task are discussed. One uses hypergeometric functions,
and the other one Eisenstein series, and approaches are rather computational.
Two main examples are given in the following theorems. Here \(\eta(\tau)\)
denotes the Dedekind eta-function.
Theorem 1: Let \(\psi\) be the idéle class character of \({\mathbb{Q}}(\sqrt{-1})\)
such that \(L(\psi, s-\frac{1}{2})\) is the Hasse-Weil \(L\)-function of the CM elliptic
curve \(E_1:y^4+x^2=1\) of conductor \(32\). Then
\[2L\Big(\psi,\frac{1}{2}\Big)^2=L(\psi^2,1).\]
In terms of cusp forms with CM by \({\mathbb{Q}}(\sqrt{-1})\), the above identity is reformulated as
\[2L(\eta(4\tau)^2\eta(8\tau)^2,1)^2=L(\eta(4\tau)^6,2),\]
where \(\eta(4\tau)^2\eta(8\tau)^2\) is the weight \(2\) level \(32\) cuspidal eigenform corresponding
to \(\psi\), and \(\eta(4\tau)^6\) is the weight \(3\) level \(16\) cuspidal eigenform corresponding
to \(\psi^2\).
Theorem 2: Let \(\chi\) be the idéle class character of \({\mathbb{Q}}(\sqrt{-3})\) such
that \(L(\chi,s-\frac{1}{2})\) is the Hasse-Weil \(L\)-function of the CM elliptic curve
\(E_2: x^3+y^3=\frac{1}{4}\) of conductor \(36\). Then
\[\frac{3}{2}L\Big(\chi,\frac{1}{2}\Big)^2=L(\chi^2,1)\quad\text{and}\quad
\frac{8}{3}L\Big(\chi,\frac{1}{2}\Big)^3=L\Big(\chi^3,\frac{3}{2}\Big).\]
In terms of cusp forms with CM by \({\mathbb{Q}}(\sqrt{-3})\), these identities are reformulated
respectively as follows:
\[\frac{3}{2}L(\eta(6\tau)^4,1)^2=L(\eta(2\tau)^3\eta(6\tau)^3,2)\]
and
\[\frac{8}{3}L(\eta(6\tau)^4,1)^3=L(\eta(3\tau)^8,3).\]
Here \(\eta(6\tau)^4\) is the level \(36\) weight \(2\) cuspidal Hecke eigenform corresponding
to \(\chi\), \(\eta(2\tau)^3\eta(6\tau)^3\) is the level \(12\) weight \(3\) Hecke eigenform
corresponding to \(\chi^2\), and \(\eta(3\tau)^8\) is the level \(9\) weight \(4\) Hecke eigenform
corresponding to \(\chi^3\).
Geometrically, weight \(2, 3\) and \(4\) cusp forms come from elliptic curves \(E_1\) and \(E_2\),
K3 surfaces, and a Calabi-Yau threefold.
Reviewer: Noriko Yui (Kingston)Incongruences for modular forms and applications to partition functions.https://zbmath.org/1456.110712021-04-16T16:22:00+00:00"Garthwaite, Sharon Anne"https://zbmath.org/authors/?q=ai:garthwaite.sharon-anne"Jameson, Marie"https://zbmath.org/authors/?q=ai:jameson.marieThe study of arithmetic properties of coefficients of modular forms has a rich history, including deep results regarding congruences in arithmetic progressions. Following the work of \textit{S.~Ahlgren} and \textit{B.~Kim} [Math. Proc. Camb. Philos. Soc. 158, No. 1, 111--129 (2015; Zbl 1371.11086)], \textit{N.~Andersen} [Q. J. Math. 65, No. 3, 781--805 (2014; Zbl 1302.11025)], and \textit{S.~Löbrich} [Res. Number Theory 3, Paper No. 18, 8 p. (2017; Zbl 1405.11044)], the authors further investigate congruence properties for coefficients of certain modular forms.
Let \(B\in\mathbb{Z}\), \(k\in\frac{1}{2}\mathbb{Z}\), and \(N\in\mathbb{Z}^+\), and denote
\[
\mathcal{S}(B,k,N,\chi):=\left\{\eta^B(\tau)F(\tau): ~F(\tau)\in M_k^!(\Gamma_0(N),\chi)\right\},
\]
where \(M_k^!(\Gamma_0(N),\chi)\) is the space of weakly holomorphic modular forms of weight \(k\) and level \(N\) with character \(\chi\) and \(\eta(\tau)\) is the Dedekind eta function. The authors prove a general theorem concerning the non-existence of congruences about the coefficients of modular forms. More precisely, if \(f(\tau)=q^{B/24}\sum\limits_{n\geq n_0}a(n)q^n\in\mathcal{S}(B,k,N,\chi)\) have rational \(\ell\)-integral coefficients (\(\ell\) is prime), let \(m\in\mathbb{Z}^+\) and let \(t_0\in\mathbb{Z}\) such that \(a(t_0)\not\equiv0\pmod{\ell}\) and \(a(n)=0\) for all \(n<t_0\) with \(n\equiv t_0\pmod{m}\). The for all \(t\in\{0,\cdots,m-1\}\) such that
\[
t\equiv t_0d^2+B\dfrac{d^2-1}{24}\pmod{m}
\]
for some integer \(d\) with \(\gcd(d, 6Nm)=1\), then \(a(mn+t)\not\equiv0\pmod{\ell}\).
Next, the authors apply this method to several noteworthy examples, including \(k\)-colored generalized Frobenius partition function \(c\phi_k(n)\) and the two mock theta functions \(f(q)\) and \(\omega(q)\). For example, for any prime \(\ell\geq5\) such that \(k\not\equiv0\pmod{\ell}\), if \(\left(\frac{k(k-24)}{\ell}\right)=-1\), then
\[
c\phi_k(\ell n+t)\not\equiv0\pmod{\ell}
\]
for all \(t\) except possibly \(t\equiv k(1-\ell^2)/24\pmod{\ell}\).
Reviewer: Dazhao Tang (Chongqing)The universal family of semistable \(p\)-adic Galois representations.https://zbmath.org/1456.112232021-04-16T16:22:00+00:00"Hartl, Urs"https://zbmath.org/authors/?q=ai:hartl.urs-t"Hellmann, Eugen"https://zbmath.org/authors/?q=ai:hellmann.eugenLet \(K\) be a finite field extension of \(\mathbb{Q}_p\) and let \(\mathscr{G}_k\) be its absolute Galois group. We construct the universal family of filtered \((\varphi, N)\)-modules, or (more generally) the universal family of \((\varphi, N)\)-modules with a Hodge-Pink lattice, and study its geometric properties. Building on this, we construct the universal family of semistable \(\mathscr{G}_k\)-representations in \(p\)-algebras. All these universal families are parametrized by moduli spaces which are Artin stacks in schemes or in adic spaces locally of finite type over \(\mathbb{Q}_p\) in the sense of Huber. This has conjectural applications to the \(p\)-adic local Langlands program.
The study of such families was begun in [\textit{M. Kisin}, Prog. Math. 253, 457--496 (2006; Zbl 1184.11052); J. Am. Math. Soc. 21, No. 2, 513--546 (2008; Zbl 1205.11060); \textit{G. Pappas} and \textit{M. Rapoport}, Mosc. Math. J. 9, No. 3, 625--663 (2009; Zbl 1194.14032)] and in [\textit{E. Hellmann}, J. Inst. Math. Jussieu 12, No. 4, 677--726 (2013; Zbl 1355.11065)], where a universal family of filtered \(\varphi\)-modules was constructed and, building on this, a universal family of crystalline representations with Hodge-Tate weights in \({0, 1}\). The approach is based on Kisin's integral \(p\)-adic Hodge theory cf. [\textit{M. Kisin}, Prog. Math. 253, 457--496 (2006; Zbl 1184.11052)].
In this article, these results are generalized in two directions. First, the authors consider more general families of \(p\)-adic Hodge-structure, namely families of \((\varphi, N)\)-modules together with a so called Hodge-Pink lattice. The inspiration to work with Hodge-Pink lattices instead of filtrations is taken from the analogous theory over function fields; see [\textit{R. Pink}, ``Hodge structures over function fields'', Preprint, \url{http://www.math.ethz.ch/~pinkri/ftp/HS.pdf}; \textit{A. Genestier} and \textit{V. Lafforgue}, Ann. Sci. Éc. Norm. Supér. (4) 44, No. 2, 263--360 (2011; Zbl 1277.14036); \textit{U. Hartl}, Ann. Math. (2) 173, No. 3, 1241--1358 (2011; Zbl 1304.11050)]. It was already applied to Kisin's integral \(p\)-adic Hodge theory by \textit{A. Genestier} and \textit{V. Lafforgue} [Compos. Math. 148, No. 3, 751--789 (2012; Zbl 1328.11112)] in the absolute case for \(\varphi\)-modules over \(\mathbb{Q}_p\). Second, the authors generalize [\textit{E. Hellmann}, J. Inst. Math. Jussieu 12, No. 4, 677--726 (2013; Zbl 1355.11065)] to the case of semistable representations.
Reviewer: Mouad Moutaoukil (Fès)