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In a landmark paper published in 1957, Kneser (Arch. Math. 8, 241–250 (1957)) introduced a method for enumerating classes in the genus of a definite, integral quadratic form. This method has been deeply influential, on account of its theoretical importance as well as its practicality. In this survey, we exhibit Kneser's method of neighbors and indicate some of its applications in number theory. [ABSTRACT FROM AUTHOR]

Viazovska proved that the E8 lattice sphere packing is the densest sphere packing in 8 dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations. [ABSTRACT FROM AUTHOR]

In this paper, we have realized the left–right symmetric model (LRSM) with modular symmetry. Most of the previous works on LRSM have been done considering discrete flavor symmetry, but this work has been carried out using Γ (3) modular group which is isomorphic to nonabelian discrete symmetry group A 4 . The advantage of using modular symmetry is the nonrequirement for the use of extra particles called "flavons". In this model, the Yukawa couplings are expressed in terms of modular forms (Y 1 , Y 2 , Y 3). In this work, we have studied minimal LRSM for both type-I and type-II dominances. Here, we have calculated the values for the Yukawa couplings and then plotted it against the sum of the neutrino masses. The results obtained are well within the experimental limits for the desired values of sum of neutrino masses. We have also briefly analyzed the effects of the implications of modular symmetry on neutrinoless double beta decay with the new physics contributions and the correlation of lepton flavor violation and lightest neutrino mass within the framework of modular symmetric LRSM. [ABSTRACT FROM AUTHOR]

A partition of a positive integer n is called $\ell $ -regular if none of its parts is divisible by $\ell $. Denote by $b_{\ell }(n)$ the number of $\ell $ -regular partitions of n. We give a complete characterisation of the arithmetic of $b_{23}(n)$ modulo $11$ for all n not divisible by $11$ in terms of binary quadratic forms. Our result is obtained by establishing a relation between the generating function for these values of $b_{23}(n)$ and certain modular forms having complex multiplication by ${\mathbb Q}(\sqrt {-69})$. [ABSTRACT FROM AUTHOR]

For a positive integer N, let X0(N)$X_0(N)$ be the modular curve over Q$\mathbf {Q}$ and J0(N)$J_0(N)$ its Jacobian variety. We prove that the rational cuspidal subgroup of J0(N)$J_0(N)$ is equal to the rational cuspidal divisor class group of X0(N)$X_0(N)$ when N=p2M$N=p^2M$ for any prime p and any squarefree integer M. To achieve this, we show that all modular units on X0(N)$X_0(N)$ can be written as products of certain functions Fm,h$F_{m, h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on X0(N)$X_0(N)$ under a mild assumption. [ABSTRACT FROM AUTHOR]

We obtain an explicit zero-free region for a weighted sum of L -functions over the orthogonal basis of Hecke eigen cusp forms of a large integral weight for the full modular group. We also estimate the number of such forms whose L value does not vanish at a given point inside this zero-free region. [ABSTRACT FROM AUTHOR]

The goal of this note is to provide a general lower bound on the number of even values of the Fourier coefficients of an arbitrary eta-quotient F , over any arithmetic progression. Namely, if g a , b (x) denotes the number of even coefficients of F in degrees n ≡ b (mod a) such that n ≤ x , then we show that g a , b (x) / x is unbounded for x large. Note that our result is very close to the best bound currently known even in the special case of the partition function p (n) (namely, x log log x , proven by Bellaïche and Nicolas in 2016). Our argument substantially relies upon, and generalizes, Serre's classical theorem on the number of even values of p (n) , combined with a recent modular-form result by Cotron et al. on the lacunarity modulo 2 of certain eta-quotients. Interestingly, even in the case of p (n) first shown by Serre, no elementary proof is known of this bound. At the end, we propose an elegant problem on quadratic representations, whose solution would finally yield a modular form-free proof of Serre's theorem. [ABSTRACT FROM AUTHOR]

We provide comparison results for anticyclotomic p -adic L -functions attached to Hida families of modular forms. The main result is a comparison between the anticyclotomic restriction of the three variable p -adic L -function introduced by Skinner and Urban, and the anticyclotomic L -function constructed by means of p -adic families of Gross points in the setting of definite quaternion algebras. Several auxiliary results involving anticyclotomic p -adic L -functions introduced by Chida-Hsieh and Büyükboduk–Lei are also proved. [ABSTRACT FROM AUTHOR]

Trypanosoma cruzi is the causal agent of Chagas Disease and is a unicellular parasite that infects a wide variety of mammalian hosts. The parasite exhibits auxotrophy by L-Met; consequently, it must be acquired from the extracellular environment of the host, either mammalian or invertebrate. Methionine (Met) oxidation produces a racemic mixture (R and S forms) of methionine sulfoxide (MetSO). Reduction of L-MetSO (free or protein-bound) to L-Met is catalyzed by methionine sulfoxide reductases (MSRs). Bioinformatics analyses identified the coding sequence for a free-R -MSR (fR MSR) enzyme in the genome of T. cruzi Dm 28c. Structurally, this enzyme is a modular protein with a putative N-terminal GAF domain linked to a C-terminal TIP41 motif. We performed detailed biochemical and kinetic characterization of the GAF domain of fR MSR in combination with mutant versions of specific cysteine residues, namely, Cys12, Cys98, Cys108, and Cys132. The isolated recombinant GAF domain and full-length fR MSR exhibited specific catalytic activity for the reduction of free L-Met(R)SO (non-protein bound), using tryparedoxins as reducing partners. We demonstrated that this process involves two Cys residues, Cys98 and Cys132. Cys132 is the essential catalytic residue on which a sulfenic acid intermediate is formed. Cys98 is the resolutive Cys, which forms a disulfide bond with Cys132 as a catalytic step. Overall, our results provide new insights into redox metabolism in T. cruzi , contributing to previous knowledge of L-Met metabolism in this parasite. [Display omitted] • TcfR MSR is a modular protein formed by GAF and TIP41 domains. • The GAF domain of TcfR MSR, but not the TIP41 domain, has MSR activity. • TcfR MSR GAF catalyzes the specific reduction of L-Met(R)SO using TXNs or TRX. • Cys132 and Cys98 are catalytic and resolutive residues, respectively. • The TcfR MSR protein has cytoplasmic localization in epimastigote cells. [ABSTRACT FROM AUTHOR]

We continue our study of the density of the odd values of eta-quotients, here focusing on the m -regular partition functions b m for m even. Based on extensive computational evidence, we propose an elegant conjecture which, in particular, completely classifies such densities: Let m = 2 j m 0 with m 0 odd. If 2 j < m 0 , then the odd density of b m is 1/2; moreover, such density is equal to 1/2 on every (nonconstant) subprogression A n + B. If 2 j > m 0 , then b m , which is already known to have density zero, is identically even on infinitely many non-nested subprogressions. This and all other conjectures of this paper are consistent with our "master conjecture" on eta-quotients presented in the previous work. In general, our results on b m for m even determine behaviors considerably different from the case of m odd. Also interesting, it frequently happens that on subprogressions A n + B , b m matches the parity of the multipartition functions p t , for certain values of t. We make a suitable use of Ramanujan-Kolberg identities to deduce a large class of such results; as an example, b 28 (49 n + 12) ≡ p 3 (7 n + 2) (mod 2). Additional consequences are several "almost always congruences" for various b m , as well as new parity results specifically for b 11. We wrap up our work with a much simpler proof of the main result of a recent paper by Cherubini-Mercuri, which fully characterized the parity of b 8. [ABSTRACT FROM AUTHOR]

A generalization of Serre's p -adic Eisenstein series in the case of Siegel modular forms is studied and a coincidence between a p -adic Siegel Eisenstein series and a genus theta series associated with a quaternary quadratic form is proved. [ABSTRACT FROM AUTHOR]

The arithmetic properties of the ordinary partition function p(n) have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form p(ℓn+β) = 0 (mod ℓ) for the primes ℓ = 5,7, 11, and it is known that there are no others of this form. On the other hand, for every prime ℓ = 5 there are infinitely many examples of congruences of the form p(ℓQmn+β) = 0 (mod ℓ) where Q = 5 is prime and m= 3. This leaves open the question of the existence of such congruences when m= 1 or m= 2 (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve X(ℓQ), Galois representations and the arithmetic large sieve. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the algebraic nature of the value of a higher Green function on an orthogonal Shimura variety at a single CM point. This is motivated by a conjecture of Gross and Zagier in the setting of higher Green functions on the product of two modular curves. In the process, we will study analogue of harmonic Maass forms in the setting of Hilbert modular forms, and obtain results concerning the arithmetic of their holomorphic part Fourier coefficients. As a consequence, we answer a question of Zagier in his 1986 ICM proceeding. [ABSTRACT FROM AUTHOR]

In this article, we construct zeta morphisms for the universal deformations of odd absolutely irreducible two dimensional mod p Galois representations satisfying some mild assumptions, and prove that our zeta morphisms interpolate Kato's zeta morphisms for Galois representations associated to Hecke eigen cusp newforms. The existence of such morphisms was predicted by Kato's generalized Iwasawa main conjecture. Based on Kato's original construction, we construct our zeta morphisms using many deep results in the theory of p -adic (local and global) Langlands correspondence for GL 2 / Q . As an application of our zeta morphisms and the recent article (Kim et al. in On the Iwasawa invariants of Kato's zeta elements for modular forms, 2019, arXiv:1909.01764v2), we prove a theorem which roughly states that, under some μ = 0 assumption, Iwasawa main conjecture without p -adic L -function for f holds if this conjecture holds for one g which is congruent to f . [ABSTRACT FROM AUTHOR]

Urban social movements, alliances and organisations use a range of strategies to contest injustice and inequality by making urban authority accountable to the 'power of the people'. While there is irrepressible diversity in the strategies used to build and enact people power across global urban contexts, movement participants rarely start from scratch in developing these strategies. There are patterns of contestation that resonate and circulate between urban movements. This article outlines an approach to analysing the patterning, diffusion and translation of commonly used urban people power strategies in and between cities. To track these strategies, we mapped hundreds of examples of urban people power, conducted field research across a range of cities, and co-designed findings and frameworks at a two week gathering of urban activists from a diverse group of cities. We identified five strategies of people power that are commonly repeated across different urban contexts—playing by the rules, mobilising, organising, prefiguring, and running for office. We outline the key characteristics of these different strategies, and discuss the different channels and forms of diffusion that facilitate their exchange across diverse urban contexts. All forms of diffusion involve the hard work of translation and learning, but this work takes different forms for different strategies. We find that where people power takes on a more visible and modular form it is more able to spread through weak ties and digital channels. However, where people power practices are more practice-based and harder to see, strong ties and personal relationships are often needed to spread the strategy. This is illustrated with a case study of the diffusion and translation of an organising strategy between housing activists in Cape Town and Barcelona. Our research and modelling of people power strategies and diffusion is offered as a contribution to on-going work of movement translation. [ABSTRACT FROM AUTHOR]

We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed φ-congruence subgroups, are obtained by reducing homomorphisms φ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi-unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasiunipotent case, we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve y² = x³ - 1728 defined by the commutator subgroup of the modular group. [ABSTRACT FROM AUTHOR]

Given a primitive, non-CM, holomorphic cusp form f with normalized Fourier coefficients a(n) and given an interval I ⊂ [ - 2 , 2 ] , we study the least prime p such that a (p) ∈ 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. [ABSTRACT FROM AUTHOR]

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].

Modular data is a significant invariant of a modular tensor category. We pursue an approach to the classification of modular data of modular tensor categories by building the modular S and T matrices directly from irreducible representations of SL 2 (Z / n Z) . We discover and collect many conditions on the SL 2 (Z / n Z) representations to identify those that correspond to some modular data. To arrive at concrete matrices from representations, we also develop methods that allow us to select the proper basis of the SL 2 (Z / n Z) representations so that they have the form of modular data. We apply this technique to the classification of rank-6 modular tensor categories, obtaining a classification of modular data, up to Galois conjugation and changing spherical structure. Most of the calculations can be automated using a computer algebraic system, which can be employed to classify modular data of higher rank modular tensor categories. Our classification employs a hybrid of automated computational methods and by-hand calculations. [ABSTRACT FROM AUTHOR]

We reformulate the question of the absence of global anomalies of heterotic string theory mathematically in terms of a certain natural transformation TMF ∙ → (I Z Ω string) ∙ - 20 , from topological modular forms to the Anderson dual of string bordism groups, using the Segal–Stolz–Teichner conjecture. We will show that this natural transformation vanishes, implying that heterotic global anomalies are always absent. The fact that TMF 21 (pt) = 0 plays an important role in the process. Along the way, we also discuss how the twists of TMF can be described under the Segal–Stolz–Teichner conjecture, by using the result of Freed and Hopkins concerning anomalies of quantum field theories. The paper contains separate introductions for mathematicians and for string theorists, in the hope of making the content more accessible to a larger audience. The sections are also demarcated cleanly into mathematically rigorous parts and those which are not. [ABSTRACT FROM AUTHOR]

Given two eigenforms, it is a natural question to ask if the product of the eigenforms is again an eigenform. In the case of elliptic modular forms, this was answered in the full level case by Duke and Ghate and in the general level case by Johnson. In this paper, we consider the case of genus two Siegel modular forms with general congruence level. [ABSTRACT FROM AUTHOR]

False theta functions are functions that are closely related to classical theta functions and mock theta functions. In this paper, we study their modular properties at all ranks by forming modular completions analogous to modular completions of indefinite theta functions of any signature and thereby develop a structure parallel to the recently developed theory of higher depth mock modular forms. We then demonstrate this theoretical base on a number of examples up to depth three coming from characters of modules for the vertex algebra W 0 (p) A n , 1 ≤ n ≤ 3 , and from Ẑ -invariants of three-manifolds associated with gauge group SU(3). [ABSTRACT FROM AUTHOR]

We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the epigraph of the quadratic can be obtained from inequalities for the underlying supermodular set function by lifting them into nonlinear inequalities in the original space of variables. Explicit forms of the convex-hull description are given, both in the original space of variables and in an extended formulation via conic quadratic-representable inequalities, along with a polynomial separation algorithm. Computational experiments indicate that the lifted supermodular inequalities in conic quadratic form are quite effective in reducing the integrality gap for quadratic optimization with indicators. [ABSTRACT FROM AUTHOR]

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In 2015, Lovejoy and Osburn discovered 12 q -hypergeometric series and proved that their Fourier coefficients can be understood as counting functions of ideals in certain quadratic fields. In this paper, we study their modular and quantum modular properties and show that they yield three vector-valued quantum modular forms on the group |$\Gamma_0 (2)$|⁠. [ABSTRACT FROM AUTHOR]

This paper develops a new chain model for the commutative graph complex GC 2 which takes Lie graph homology as an input. Our main technical result is the identification of a large contractible complex of (certain) tadpoles and higher genus vertices of the Feynman transform of Lie graph homology. Using this result we identify the anti-invariants of Lie graph homology in genus 2 with relations between bracketings of conjectural generators of grt 1 in depth 2 modulo depth 3, unifying two a priori disparate appearances of the space of modular cusp forms in the study of graph homology. [ABSTRACT FROM AUTHOR]

By the modularity theorem, every rigid Calabi–Yau threefold X has associated modular form f such that the equality of L‐functions L(X,s)=L(f,s)$L(X,s)=L(f,s)$ holds. In this case, period integrals of X are expected to be expressible in terms of the special values L(f,1)$L(f,1)$ and L(f,2)$L(f,2)$. We propose a similar interpretation of period integrals of a nodal model of X. It is given in terms of certain variants of a Mellin transform of f. We provide numerical evidence toward this interpretation based on a case of double octics. [ABSTRACT FROM AUTHOR]

For | q | < 1 , define f i = ∏ n = 1 ∞ (1 − q i n) , and let (A (q) , B (q)) be any of the pairs (f 1 4 , f 1 8 f 2 2 ) , (f 1 4 , f 1 1 0 f 3 2 ) , (f 1 6 , f 2 4 f 1 2 ) , (f 1 6 , f 1 1 4 f 2 4 ) , (f 1 1 0 , f 2 6 f 1 2 ) , (f 1 1 4 , f 3 5 f 1 ) , (f 1 1 4 , f 2 8 f 1 2 ). For any such pair (A (q) , B (q)) , define the sequences { a (n) } and { b (n) } to be the coefficients of q n of A (q) and B (q) , respectively. Then for each pair it is shown that a (n) vanishes if and only if b (n) vanishes. In each case, a criterion is given which states precisely when a (n) = b (n) = 0. Moreover, for the pairs (f 1 2 6 , f 3 9 f 1 ) , (f 1 2 6 , f 2 1 6 f 1 6 ) it is shown that a (n) = b (n) = 0 if 1 2 n + 1 3 satisfies a criteria of Serre for a (n) = 0. [ABSTRACT FROM AUTHOR]

We prove that Rankin–Cohen brackets of Eisenstein series generate the whole space of cuspforms in several cases. This generalizes the classical result that products of two Eisenstein series generate the whole space of modular forms. [ABSTRACT FROM AUTHOR]

Let ℓ ≥ 5 be prime. For the partition function p (n) and 5 ≤ ℓ ≤ 31 , Atkin found a number of examples of primes Q ≥ 5 such that there exist congruences of the form p (ℓ Q 3 n + β) ≡ 0 (mod ℓ). Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every ℓ. In this paper, for a wide range of c ∈ F ℓ , we prove congruences of the form p (ℓ Q 3 n + β 0) ≡ c ⋅ p (ℓ Q n + β 1) (mod ℓ) for infinitely many primes Q. For a positive integer r , let p r (n) be the r -colored partition function. Our methods yield similar congruences for p r (n). In particular, if r is an odd positive integer for which ℓ > 5 r + 19 and 2 r + 2 ≢ 2 ± 1 (mod ℓ) , then we show that there are infinitely many congruences of the form p r (ℓ Q 3 n + β) ≡ 0 (mod ℓ). Our methods involve the theory of modular Galois representations. [ABSTRACT FROM AUTHOR]

In the present paper, we prove that the first homology group of Bianchi 3-fold is generated by the special Bianchi modular symbols. Also, we construct the integral valued p -adic L -function of p -ordinary Bianchi Hecke eigenforms by taking the Lefschetz-Poincaré Pairing of the cohomology class attached to Bianchi modular forms and Bianchi modular symbols on Bianchi 3-folds. By using the homology generation result, we prove that the μ -invariant of some isotopic components of the p -adic L -function of certain Bianchi Hecke eigenforms vanishes for a positive proportion of ordinary prime ideals. The main ingredient of the proof of the generation result is analyzing the fast convergent series expression of the integration of the Bianchi modular forms along the special modular symbols. [ABSTRACT FROM AUTHOR]

A manifestation of the black hole information loss problem is that the two-point function of probe operators in a large Anti-de Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity. We call operators that cluster in time "mixing" and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double state (KMS state), if mixing operators form an algebra (close under multiplication), the resulting algebra must be a von Neumann type III1 factor. In other words, the physically intuitive requirement that all nonconserved operators should exponentially mix is so strong that it fixes the observable algebra to be an exotic algebra called a type III1 factor. More generally, for an arbitrary out-of-equilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra, it is a type III1 von Neumann factor. In a theory of Generalized Free Fields (GFF), we show that if the two-point function clusters in time, all operators are mixing, and the algebra is a type III1 factor. For example, in 풩 = 4 SYM, above the Hawking-Page phase transition, clustering of the single trace operators implies that the algebra is a type III1 factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C∗-algebra and von Neumann subalgebras of GFF associated with time bands and, more generally, open sets of the bulk spacetime using the HKLL reconstruction map. [ABSTRACT FROM AUTHOR]

We study the possibility to generate the quark mass hierarchies as well as the CKM quark mixing and CP violation without fine-tuning in a quark flavour model with modular A4 symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point τ = i∞, τ being the vacuum expectation value of the modulus. We consider first a model in which the up-type and down-type quark mass matrices Mu and Md involve modular forms of level 3 and weights 6, 4 and 2 and each depends on four constant parameters. Two ratios of these parameters, gu and gd, can be sources of the CP violation. If Mu and Md depend on the same τ, it is possible to reproduce the up-type and down-type quark mass hierarchies in the considered model for |gu| ~ O (10) with all other constants being in magnitude of the same order. However, reproducing the CP violation in the quark sector is problematic. A correct description of the quark mass hierarchies, the quark mixing and CP violation is possible close to τ = i∞ with all constant being in magnitude of the same order and complex gu and gd, if there are two different moduli τu and τd in the up-type and down-type quark sectors. We also consider the case of Mu and Md depending on the same τ and involving modular forms of weights 8, 4, 2 and 6, 4, 2, respectively, with Mu receiving a tiny SUSY breaking or higher dimensional operator contribution. Both the mass hierarchies of up-type and down-type quarks as well and the CKM mixing angles and CP violating phase are reproduced successfully with one complex parameter and all parameters being in magnitude of the same order. The relatively large value of Im τ, needed for describing the down-type quark mass hierarchies, is crucial for obtaining the correct up-type quark mass hierarchies. [ABSTRACT FROM AUTHOR]

In the last letter to Hardy, Ramanujan [ Collected Papers , Cambridge Univ. Press, 1927; Reprinted, Chelsea, New York, 1962] introduced seventeen functions defined by q-series convergent for |q|<1 with a complex variable q, and called these functions "mock theta functions". Subsequently, mock theta functions were widely studied in the literature. In the survey of B. Gordon and R. J. McIntosh [ A survey of classical mock theta functions , Partitions, q-series, and modular forms, Dev. Math., vol. 23, Springer, New York, 2012, pp. 95–144], they showed that the odd (resp. even) order mock theta functions are related to the function g_3(x,q) (resp. g_2(x,q)). These two functions are usually called "universal mock theta functions". D. R. Hickerson and E. T. Mortenson [Proc. Lond. Math. Soc. (3) 109 (2014), pp. 382–422] expressed all the classical mock theta functions and the two universal mock theta functions in terms of Appell–Lerch sums. In this paper, based on some q-series identities, we find four functions, and express them in terms of Appell–Lerch sums. For example, \begin{equation*} 1+(xq^{-1}-x^{-1}q)\sum _{n=0}^{\infty }\frac {(-1;q)_{2n}q^{n}}{(xq^{-1},x^{-1}q;q^2)_{n+1}}=2m(x,q^2,q). \end{equation*} Then we establish some identities related to these functions and the universal mock theta function g_2(x,q). These relations imply that all the classical mock theta functions can be expressed in terms of these four functions. Furthermore, by means of q-series identities and some properties of Appell–Lerch sums, we derive four radial limit results related to these functions. [ABSTRACT FROM AUTHOR]

We prove one inclusion in the Iwasawa main conjecture, and the Bloch--Kato conjecture in analytic rank 0, for the symmetric cube of a level 1 modular form. [ABSTRACT FROM AUTHOR]

Let f and g be two different holomorphic cusp froms or Maass cusp forms for the full modular group S L (2 , Z) . We are interested in coefficients of Rankin–Selberg L-functions, and establish some bounds for ∑ n ≤ x λ sym i f × sym j g (n) , ∑ n ≤ x λ f (n i) λ g (n j) , ∑ n ≤ x | λ sym i f × sym j g (n) | , ∑ n ≤ x | λ f (n i) λ g (n j) | , and ∑ n ≤ x max { | λ sym i f × sym j g (n) | 2 φ , | λ sym i f × sym j g (n + h) | 2 φ } , where φ > 0 and h is a fixed positive integer. [ABSTRACT FROM AUTHOR]

We construct a family of q-series with rational coefficients satisfying a variant of the extended double shuffle equations, which are a lift of a given Q -valued solution of the extended double shuffle equations. We call these q-series combinatorial (bi-)multiple Eisenstein series, and in depth one they coincide with (classical) Eisenstein series. Combinatorial multiple Eisenstein series can be seen as an interpolation between the given Q -valued solution of the extended double shuffle equations (as q → 0 ) and multiple zeta values (as q → 1 ). In particular, they are q-analogues of multiple zeta values closely related to modular forms. Their definition is inspired by the Fourier expansion of multiple Eisenstein series introduced by Gangl-Kaneko-Zagier. Our explicit construction is done on the level of their generating series, which we show to be a so-called symmetril and swap invariant bimould. [ABSTRACT FROM AUTHOR]

Consider a family of modular forms of weight 2, all of whose residual (mod p) Galois representations are isomorphic. It is well known that their corresponding Iwasawa λ -invariants may vary. In this paper, we study this variation from a quantitative perspective, providing lower bounds on the frequency with which these λ -invariants grow or remain stable. [ABSTRACT FROM AUTHOR]

We use the theory of topological modular forms to constrain bosonic holomorphic CFTs, which can be viewed as (0, 1) SCFTs with trivial right-moving supersymmetric sector. A conjecture by Segal, Stolz and Teichner requires the constant term of the partition function to be divisible by specific integers determined by the central charge. We verify this constraint in large classes of physical examples, and rule out the existence of an infinite set of extremal CFTs, including those with central charges c = 48 , 72 , 96 and 120. [ABSTRACT FROM AUTHOR]

We prove that all mod |$p$| Hilbert modular forms arise via multiplication by generalized partial Hasse invariants from forms whose weight falls within a certain minimal cone. This answers a question posed by Andreatta and Goren and generalizes our previous results that treated the case where |$p$| is unramified in the totally real field. Whereas our previous work made use of deep Jacquet–Langlands type results on the Goren–Oort stratification (not yet available when |$p$| is ramified), here we instead use properties of the stratification at Iwahori level, which are more readily generalizable to other Shimura varieties. [ABSTRACT FROM AUTHOR]

We express recent double-sums studied by Wang, Yee, and Liu in terms of two types of Hecke-type double-sum building blocks. When possible we determine the (mock) modularity. We also express a recent q -hypergeometric function of Andrews as a mixed mock modular form. [ABSTRACT FROM AUTHOR]

The study of Andrews–Beck type congruences for partitions has its origin in the work by Andrews, who proved two congruences on the total number of parts in the partitions of n with the Dyson rank, conjectured by George Beck. Recently, Lin, Peng and Toh proved many Andrews–Beck type congruences for k -colored partitions. Moreover, they posed eight conjectural congruences. In this paper, we confirm two congruences modulo 1 1 by utilizing some q -series techniques and the theory of modular forms. [ABSTRACT FROM AUTHOR]

Suppose j N (τ) and j N ⁎ (τ) are the Hauptmoduln of the congruence subgroup Γ 0 (N) and the Fricke group Γ 0 ⁎ (N) , respectively. In [7] , the authors predicted that, like Klein's j -function, the Fourier coefficients of j N (τ) and j N ⁎ (τ) in some arithmetic progression are both even and odd with density 1 2. In this article, we can find some arithmetic progression of n where the Fourier coefficients of j 6 (τ) (resp. j 6 ⁎ (τ) and j 10 (τ)) are almost always even. Furthermore, using Hecke eigenforms and Rogers-Ramanujan continued fraction, we obtain infinite families of congruences for j 6 (τ) , j 6 ⁎ (τ) , j 10 (τ) , and j 10 ⁎ (τ). [ABSTRACT FROM AUTHOR]

For any given totally real number field K , we compute the Fourier developments of the Jacobi Eisenstein series over K at the cusp at infinity. As main application we prove, for any K with class number 1, that the L -series of the Jacobi Eisenstein series of weight k ≥ 3 for indices with rank and modified level 1 coincide with the L -series of the Eisenstein series of weight 2 k − 2 on the full Hilbert modular group of K. Moreover, under this correspondence the Fourier coefficients of the Jacobi Eisenstein series are related to the twisted L -series of the Hilbert Eisenstein series at the critical point by a Waldspurger type identity. This is a first step in the proof that Skoruppa's and Zagier's lifting from Jacobi forms over Q to elliptic modular forms holds true over arbitrary totally real number fields too. [ABSTRACT FROM AUTHOR]

In this paper, we derive the Saito-Kurokawa isomorphism on the space of newforms for Maaß spezialschar of degree 2, weight k , level M , where 32 | M and primitive character χ modulo M with χ (− 1) = (− 1) k and χ 2 is primitive modulo M / 2. We first develop the corresponding theory of newforms for respective spaces of half-integral weight modular forms, Jacobi forms and then for Maass forms and as a consequence we get the Saito-Kurokawa isomorphism. [ABSTRACT FROM AUTHOR]

We show that if the dimension of the space of cuspforms is greater than or equal to three, then any three even periods are linearly independent. We also prove an asymptotic result for an arbitrary number of even periods. These results are achieved by studying the Rankin-Cohen brackets of Eisenstein series. [ABSTRACT FROM AUTHOR]

We study A4× A4× A4 modular symmetric flavor models to realize quark mass hierarchies and mixing angles without fine-tuning. Mass matrices are written in terms of modular forms. At modular fixed points τ = i∞ and ω, A4 is broken to Z3 residual symmetry. When the modulus τ is deviated from the fixed points, modular forms show hierarchies depending on their residual charges. Thus, we obtain hierarchical structures in mass matrices. Since we begin with A4× A4× A4, the residual symmetry is Z3× Z3× Z3 which can generate sufficient hierarchies to realize quark mass ratios and absolute values of the CKM matrix |VCKM| without fine-tuning. Furthermore, CP violation is studied. We present necessary conditions for CP violation caused by the value of τ. We also show possibilities to realize observed values of the Jarlskog invariant JCP, quark mass ratios and CKM matrix |VCKM| simultaneously, if (10) adjustments in coefficients of Yukawa couplings are allowed or moduli values are non-universal. [ABSTRACT FROM AUTHOR]