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We count the number of critical points of a modular form with real Fourier coefficients in a γ-translate of the standard fundamental domain ℱ (with γ ∈SL2(ℤ)). Whereas by the valence formula the (weighted) number of zeros of this modular form in γℱ is a constant only depending on its weight, we give a closed formula for this number of critical points in terms of those zeros of the modular form lying on the boundary of ℱ, the value of γ−1(∞) and the weight. More generally, we indicate what can be said about the number of zeros of a quasimodular form. [ABSTRACT FROM AUTHOR]

In this paper, we study mixed Hodge structures on the cohomology of locally symmetric varieties and give an application to modular forms. After proving vanishing of some Hodge numbers, we focus on the weight filtration on the last Hodge subspace of the middle degree cohomology. We prove that the weight filtration coincides with the corank filtration on the space of modular forms of canonical weight defined by the Siegel operators, and calculate the graded quotients. As an application, we deduce surjectivity of the total Siegel operators in many cases, and identify an obstruction space in the remaining case. [ABSTRACT FROM AUTHOR]

The degeneracies of 1/4 BPS states with unit torsion in heterotic string theory compactified on a six torus are given in terms of the Fourier coefficients of the reciprocal of the Igusa cusp Siegel modular form Φ 10 of weight 10. We use the symplectic symmetries of the latter to construct a fine-grained Rademacher-type expansion which expresses these BPS degeneracies as a regularized sum over residues of the poles of 1 / Φ 10 . The construction uses two distinct SL (2 , Z) subgroups of Sp (2 , Z) which encode multiplier systems, Kloosterman sums and Eichler integrals appearing therein. Additionally, it shows how the polar data are explicitly built from the Fourier coefficients of 1 / η 24 by means of a continued fraction structure. [ABSTRACT FROM AUTHOR]

For a prime p larger than 7, the Eisenstein series of weight p − 1 has some remarkable congruence properties modulo p. Those imply, for example, that the j -invariants of its zeros (which are known to be real algebraic numbers in the interval [ 0 , 1728 ]), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce "theta modular forms" of weight k ≥ 4 for the full modular group as the modular forms for which the first dim ⁡ (M k) Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j -invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions. [ABSTRACT FROM AUTHOR]

Let N ≥ 3 and r ≥ 1 be integers and p ≥ 2 be a prime such that p ∤ N. One can consider two different integral structures on the space of modular forms over Q , one coming from arithmetic via q -expansions, the other coming from geometry via integral models of modular curves. Both structures are stable under the Hecke operators; furthermore, their quotient is finite torsion. Our goal is to investigate the exponent of the annihilator of the quotient. We will apply methods due to Brian Conrad to the situation of modular forms of even weight and level Γ (N p r) over Q p (ζ N p r ) to obtain an upper bound for the exponent. We also use Klein forms to construct explicit modular forms of level p r whenever p r > 3 , allowing us to compute a lower bound which agrees with the upper bound. Hence we compute the exponent precisely. [ABSTRACT FROM AUTHOR]

The values of the Dedekind-Rademacher cocycle at certain real quadratic arguments are shown to be global p-units in the narrow Hilbert class field of the associated real quadratic field, as predicted by the conjectures of Darmon-Dasgupta (2006) and Darmon-Vonk (2021). The strategy for proving this result combines the approach of prior work of the authors (2021) with one crucial extra ingredient: the study of infinitesimal deformations of irregular Hilbert Eisenstein series of weight 1 in the anti-parallel direction. [ABSTRACT FROM AUTHOR]

We introduce a sequence of families of lattice polarized K3 surfaces. This sequence is closely related to complex reflection groups of exceptional type. Namely, we obtain modular forms coming from the inverse correspondences of the period mappings attached to our sequence. We study a non-trivial relation between our modular forms and invariants of complex reflection groups. Especially, we consider a family concerned with the Shephard-Todd group No.34 based on arithmetic properties of lattices and algebro-geometric properties of the period mappings. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytic ranks of the Hasse–Weil L-function. We consider the products for the completed Hasse–Weil L-function in the framework of the theory of modular forms. We propose the new conjectures for the deformed Fourier sine and cosine integral formulas related to the completed Hasse–Weil L-functions and Mittag-Leffler function. We show the entire-function solutions for the fractional diffusion equations in mathematical physics. We also discuss the trivial zeros of the Hasse–Weil L-function. [ABSTRACT FROM AUTHOR]

In this paper, we construct generalized L-functions associated to meromorphic modular forms of weight 1 2 for the theta group with a single simple pole in the fundamental domain. We then consider their behavior toward i∞ and relate this to the Riemann zeta function. [ABSTRACT FROM AUTHOR]

Let 풮k(Γ0(N),χ) denote the space of holomorphic cuspforms with Dirichlet character χ and modular subgroup Γ0(N). We will characterize the space of newforms 풮knew(Γ 0(N),χ) as the intersection of eigenspaces of a particular family of Hecke operators, generalizing previous work of Baruch–Purkait to forms with nontrivial character. We achieve this by obtaining representation theoretic results in the p-adic case which we then de-adelize into relations of classical Hecke operators. [ABSTRACT FROM AUTHOR]

We examine hypergeometric functions in the finite field, p-adic and classical settings. In each setting, we prove a formula which splits the hypergeometric function into a sum of lower order functions whose arguments differ by roots of unity. We provide multiple applications of these results, including new reduction and summation formulas for finite field hypergeometric functions, along with classical analogues; evaluations of special values of these functions which apply in both the finite field and p-adic settings; and new relations to Fourier coefficients of modular forms. [ABSTRACT FROM AUTHOR]

We prove that weights of two Siegel modular forms of nonquadratic nebentypus should satisfy some congruence relations if these modular forms are congruent to each other. Applying this result, we prove that there are no mod p singular forms of nonquadratic nebentypus. Here we consider the case where the Fourier coefficients of the modular forms are algebraic integers, and we emphasize that p is a rational prime. Moreover, we construct some examples of mod p singular forms of nonquadratic nebentypus using the Eisenstein series studied by Takemori. [ABSTRACT FROM AUTHOR]

We investigate the subspace of the homology of a congruence subgroup Γ of SL 3 (ℤ) with coefficients in the Steinberg module St (ℚ 3) which is spanned by certain modular symbols formed using the units of a totally real cubic field E. By Borel–Serre duality, H 0 (Γ , St (ℚ 3)) is isomorphic to H 3 (Γ , ℚ). The Borel–Serre duals of the modular symbols in question necessarily lie in the cuspidal cohomology H cusp 3 (Γ , ℚ). Their span is a naturally defined subspace C (Γ , E) of H cusp 3 (Γ , ℚ). Using a computer, we study where C (Γ , E) sits between 0 and H cusp 3 (Γ , ℚ). On the basis of our computations, we conjecture that ∑ E C (Γ , E) = H cusp 3 (Γ , ℚ) , and we raise the question as to whether for each E individually it might always be true that C (Γ , E) = H cusp 3 (Γ , ℚ). [ABSTRACT FROM AUTHOR]

This article is concerned with the Fourier coefficients of cusp forms (not necessarily eigen-forms) of half-integer weight lying in the plus space. We give a soft proof that there are infinitely many fundamental discriminants D such that the Fourier coefficients evaluated at |D| are non-zero. By adapting the resonance method, we also demonstrate that such Fourier coefficients must take quite large values. [ABSTRACT FROM AUTHOR]

This is the second and final part of ‘Topological twists of massive SQCD’. Part I is available at Lett. Math. Phys. 114 (2024) 3, 62. In this second part, we evaluate the contribution of the Coulomb branch to topological path integrals for N = 2 supersymmetric QCD with N f ≤ 3 massive hypermultiplets on compact four-manifolds. Our analysis includes the decoupling of hypermultiplets, the massless limit and the merging of mutually non-local singularities at the Argyres–Douglas points. We give explicit mass expansions for the four-manifolds P 2 and K3. For P 2 , we find that the correlation functions are polynomial as function of the masses, while infinite series and (potential) singularities occur for K3. The mass dependence corresponds mathematically to the integration of the equivariant Chern class of the matter bundle over the moduli space of Q-fixed equations. We demonstrate that the physical partition functions agree with mathematical results on Segre numbers of instanton moduli spaces. [ABSTRACT FROM AUTHOR]

In this article, we consider the problem of determining formulas for the number of representations of a natural number n by a sum of figurate numbers with certain positive integer coefficients. To achieve this, we prove that the associated generating function gives rise to a modular form of integral weight under certain conditions on the coefficients when even number of higher figurate numbers are considered. In particular, we obtain the modular property of the generating function corresponding to a sum of even number of triangular numbers with coefficients. We also obtain the modularity property of the generating function of mixed forms involving figurate numbers (including the squares and triangular numbers) with coefficients and forms of the type m 2 + m n + n 2 with coefficients. In particular, we show the modularity of the generating function of odd number of squares and odd number of triangular numbers (with coefficients). As a consequence, explicit formulas for the number of representations of these mixed forms are obtained using a basis of the corresponding space of modular forms of integral weight. We also obtain several applications concerning the triangular numbers with coefficients similar to the ones obtained in Ono et al. (Aequat Math 50:73–94, 1995). In 2016, Xia et al. (Int J Number Theory 12:945–954) considered some special cases of mixed forms and obtained the number of representations of these 21 mixed forms using the (p, k) parametrization method. We also derive these 21 formulas using our method and further obtain as a consequence, the (p, k) parametrization of the Eisenstein series E 4 (τ) and its duplications. It is to be noted that the (p, k) parametrization of E 4 and its duplications were derived by a different method in [3, 8]. We illustrate our method with several examples. [ABSTRACT FROM AUTHOR]

Interaction networks are a widely used tool to understand the dynamics of plant–pollinator ecological communities. However, while most mutualistic networks have been defined at the species level, ecological processes such as pollination take place at different scales, including the individual or patch levels. Yet, current approaches studying fine‐grain sub‐specific plant–pollinator networks only account for interactions among nodes belonging to a single plant species due to the conceptual and mathematical limitations of modeling simultaneously several plant species each composed of several nodes. Here, we introduce a multilayer diffusion network framework that allows modeling simple diffusion processes between nodes pertaining to the same or different layers (i.e. species). It is designed to depict from the network structure the potential conspecific and heterospecific pollen flows among plant individuals or patches. This potential pollen flow is modeled as a transport‐like system, in which pollen grain movements are represented as random‐walkers that diffuse on an ensemble of bipartite layers of conspecific plants and their shared pollinators. We exemplify this physical conceptualization using a dataset of nine fine‐grain sub‐specific plant–pollinator networks from a Mediterranean grassland of annual plants, where plant nodes represent groups of conspecifics within patches of 1 m2. The diffusion networks show pollinators effectively connecting sets of patches of the same and different plant species, forming a modular structure. Interestingly, different properties of the network structure, such as the conspecific pollen arrival probability and the number of conspecific subgraphs in which plants are embedded, are critical for the seed production of different plant species. We provide a simple but robust set of metrics to calculate potential pollen flow and scale down network ecology to functioning properties at the individual or patch level, where most ecological processes take place, hence moving forward the description and interpretation of species‐rich communities across scales. [ABSTRACT FROM AUTHOR]

Serre showed that for any integer m , a (n) ≡ 0 (mod m) for almost all n , where a (n) is the n th Fourier coefficient of any modular form with rational coefficients. In this paper, we consider a certain class of cuspforms and study # { a (n) (mod m) } n ≤ x over the set of integers with O (1) many prime factors. Moreover, we show that any residue class a ∈ ℤ / m ℤ can be written as the sum of at most 13 Fourier coefficients, which are polynomially bounded as a function of m. [ABSTRACT FROM AUTHOR]

We prove the best possible upper bounds of the gaps between non-vanishing Fourier coefficients of half-integral weight cuspforms. This improves the works of Balog–Ono and Thorner. We also show an asymptotic formula of central modular L -values for short intervals. [ABSTRACT FROM AUTHOR]

We determine the asymptotic distribution of Manin's iterated integrals of length at most 2. For all lengths, we compute all the asymptotic moments. We show that if the length is at least 3, these moments do in general not determine a unique distribution. [ABSTRACT FROM AUTHOR]

We prove that the Cohn–Elkies linear programming bound for sphere packing is not sharp in dimension 6. The proof uses duality and optimization over a space of modular forms, generalizing a construction of Cohn–Triantafillou [Math. Comp. 91 (2021), pp. 491–508] to the case of odd weight and non-trivial character. [ABSTRACT FROM AUTHOR]

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev. [ABSTRACT FROM AUTHOR]

By using modular functions on the upper complex half-plane, we study a class of strain energies for crystalline materials whose global invariance originates from the full symmetry group of the underlying lattice. This follows Ericksen's suggestion which aimed at extending the Landau-type theories to encompass the behavior of crystals undergoing structural phase transformation, with twinning, microstructure formation, and possibly associated plasticity effects. Here we investigate such Ericksen-Landau strain energies for the modelling of reconstructive transformations, focusing on the prototypical case of the square-hexagonal phase change in 2D crystals. We study the bifurcation and valley-floor network of these potentials, and use one in the simulation of a quasi-static shearing test. We observe typical effects associated with the micro-mechanics of phase transformation in crystals, in particular, the bursty progress of the structural phase change, characterized by intermittent stress-relaxation through microstructure formation, mediated, in this reconstructive case, by defect nucleation and movement in the lattice. [ABSTRACT FROM AUTHOR]

Ramanujan's partition congruences modulo ℓ ∈ { 5 , 7 , 11 } assert that p (ℓ n + δ ℓ) ≡ 0 (mod ℓ) , where 0 < δ ℓ < ℓ satisfies 24 δ ℓ ≡ 1 (mod ℓ). By proving Subbarao's conjecture, Radu showed that there are no such congruences when it comes to parity. There are infinitely many odd (resp. even) partition numbers in every arithmetic progression. For primes ℓ ≥ 5 , we give a new proof of the conclusion that there are infinitely many m for which p (ℓ m + δ ℓ) is odd. This proof uses a generalization, due to the second author and Ramsey, of a result of Mazur in his classic paper on the Eisenstein ideal. We also refine a classical criterion of Sturm for modular form congruences, which allows us to show that the smallest such m satisfies m < (ℓ 2 - 1) / 24 , representing a significant improvement to the previous bound. [ABSTRACT FROM AUTHOR]

Let L be a lattice. We exhibit algorithms for calculating Tits buildings and orbits of vectors in L for certain subgroups of the orthogonal group O (L). We discuss how these algorithms can be applied to determine the configuration of boundary components in the Baily-Borel compactification of orthogonal modular varieties and to improve the performance of computer arithmetic of orthogonal modular forms. [ABSTRACT FROM AUTHOR]

In this paper, we show that the p-adic limit of a family of Eisenstein series on the exceptional domain where the exceptional group of type E_{7,3} acts is an ordinary modular form for a congruence subgroup. [ABSTRACT FROM AUTHOR]

We study minimal and toroidal compactifications of p -integral models of Hilbert modular varieties. We review the theory in the setting of Iwahori level at primes over p , and extend it to certain finer level structures. We also prove extensions to compactifications of recent results on Iwahori-level Kodaira–Spencer isomorphisms and cohomological vanishing for degeneracy maps. Finally we apply the theory to study q -expansions of Hilbert modular forms, especially the effect of Hecke operators at primes over p over general base rings. [ABSTRACT FROM AUTHOR]

For an isotropic subgroup H of a discriminant form D there exists a lift from modular forms for the Weil representation of the discriminant form H ⊥ / H to modular forms for the Weil representation of D. We determine a set of discriminant forms such that all modular forms for any discriminant form are induced from the discriminant forms in this set. Furthermore for any discriminant form in this set there exist modular forms that are not induced from smaller discriminant forms. [ABSTRACT FROM AUTHOR]

Let p ∈ { 3 , 5 , 7 } and let Δ denote the weight twelve modular form arising from Ramanujan's tau function. We show that Δ is congruent to an Eisenstein series E k , χ , ψ modulo p for explicit choices of k and Dirichlet characters χ and ψ . We then prove formulae describing the Iwasawa invariants of the Mazur–Tate elements attached to Δ , confirming numerical data gathered by the authors in a previous work. [ABSTRACT FROM AUTHOR]

We consider the fourteen families W of Calabi–Yau threefolds with one complex structure parameter and Picard–Fuchs equation of hypergeometric type, like the mirror of the quintic in P 4 . Mirror symmetry identifies the masses of even-dimensional D-branes of the mirror Calabi–Yau M with four periods of the holomorphic (3, 0)-form over a symplectic basis of H 3 (W , Z) . It was discovered by Chad Schoen that the singular fiber at the conifold of the quintic gives rise to a Hecke eigenform of weight four under Γ 0 (25) , whose Hecke eigenvalues are determined by the Hasse–Weil zeta function which can be obtained by counting points of that fiber over finite fields. Similar features are known for the thirteen other cases. In two cases we further find special regular points, so called rank two attractor points, where the Hasse–Weil zeta function gives rise to modular forms of weight four and two. We numerically identify entries of the period matrix at these special fibers as periods and quasiperiods of the associated modular forms. In one case we prove this by constructing a correspondence between the conifold fiber and a Kuga–Sato variety. We also comment on simpler applications to local Calabi–Yau threefolds. [ABSTRACT FROM AUTHOR]

Don Zagier, former director at the Bonn Max Planck Institute for Mathematics, has been awarded the Gumin Prize for Mathematics of the Year 2024. The Carl Friedrich von Siemens Foundation honors his groundbreaking research in number theory and the theory of modular forms. The Gumin Prize is awarded every three to four years to outstanding mathematicians in Germany, Austria, or Switzerland. Zagier speaks in an interview about his work, his collaboration with other mathematicians, and his former doctoral student Maryna Viazovska, who received the Fields Medal. He also comments on current developments in mathematics and mentions the Taniyama-Weil conjecture as a problem he would like to see solved. Zagier also mentions his personal story as a prodigy. [Extracted from the article]

Deligne proved in [Extensions centrales non résiduellement finies de groupes arithmetiques, C. R. Acad. Sci. Paris 287 (1978) 203–208] (see also 7.1 in [R. Hill, Fractional weights and non-congruence subgroups, in Automorphic Forms and Representations of Algebraic Groups Over Local Fields, eds. H. Saito and T. Takahashi, Surikenkoukyuroku Series, Vol. 1338 (2003), pp. 71–80]) that the weights of Siegel modular forms on any congruence subgroup of the Siegel modular group of genus g > 1 must be integral or half integral. Actually he proved that for a system v (M) of complex numbers of absolute value 1 v (M) det (C Z + D) r (r ∈ ℝ) (0. 1) can be an automorphy factor only if 2 r is integral. We give a different proof for this. It uses Mennicke's result [Zur Theorie der Siegelschen Modulgruppe, Math. Ann. 159 (1965) 115–129] that subgroups of finite index of the Siegel modular group are congruence subgroups and some techniques from [Solution of the congruence subgroup problem for SL n (n ≥ 3) and Sp 2 n (n ≥ 2) , Publ. Math. Inst. Hautes Études Sci. 33 (1967) 59–137] of Bass–Milnor–Serre. [ABSTRACT FROM AUTHOR]

To prove that a modular variety is of general type, there are three types of obstructions: reflective, cusp and elliptic obstructions. In this paper, we give a quantitative estimate of the reflective obstructions for the unitary case. This shows in particular that the reflective obstructions are small enough in higher dimension, say greater than 138. Our result reduces the study of the Kodaira dimension of unitary modular varieties to the construction of a cusp form of small weight in a quantitative manner. As a byproduct, we formulate and partially prove the finiteness of Hermitian lattices admitting reflective modular forms, which is a unitary analog of the conjecture by Gritsenko-Nikulin in the orthogonal case. Our estimate of the reflective obstructions uses Prasad's volume formula. [ABSTRACT FROM AUTHOR]

In this paper, we consider universality in short intervals for the zeta-function attached to a normalized Hecke-eigen cusp form with respect to the modular group. For this, we apply a conjecture for the mean square in short interval on the critical strip for that zeta-function. The proof of the obtained universality theorem is based on a probabilistic limit theorem in the space of analytic functions. [ABSTRACT FROM AUTHOR]

We begin with the observation that the p -adic completion of any affine elliptic curve with ordinary reduction possesses Frobenius lifts ϕ that are "Lie invariant mod p " in the sense that the "normalized" action of ϕ on 1-forms preserves mod p the space of invariant 1-forms. Our main result is that, after removing the 2-torsion sections, the above situation can be "infinitesimally deformed" in the sense that the above mod p result has a mod p 2 analogue. We end by showing that, in contrast with the case of elliptic curves, the following holds: if G is a linear algebraic group over a number field and if G is not a torus then for all but finitely many primes p the p -adic completion of G does not possess a Frobenius lift that is Lie invariant mod p. [ABSTRACT FROM AUTHOR]

In this paper, we study universal sums of triangular numbers and squares. Specifically, we prove that a sum of triangular numbers and squares is universal if and only if it represents 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 10 , 13 , 14 , 15 , 18 , 19 , 20 , 23 , 27 , 28 , 34 , 41 , 47 , and 48. [ABSTRACT FROM AUTHOR]

We provide for primes p ≥ 5 a method to compute valuations appearing in the "formal" Katz expansion of the family E κ ⁎ V (E κ ⁎) derived from the family of Eisenstein series E κ ⁎. We will describe two algorithms: the first one to compute the Katz expansion of an overconvergent modular form and the second one, which uses the first algorithm, to compute valuations appearing in the "formal" Katz expansion. Based on data obtained using these algorithms we make a precise conjecture about a constant appearing in the overconvergence rates related to the classical Eisenstein series at level p. The study of these overconvergence rates of the members of this family goes back to a conjecture of Coleman. [ABSTRACT FROM AUTHOR]

In [9] , Gun, Murty and Rath studied non-vanishing and transcendental nature of special values of a varying class of L -functions and their derivatives. This led to a number of works by several authors in different set-ups including studying higher derivatives. However, all these works were focused around the central point of the critical strip. In this article, we extend the study to arbitrary points in the critical strip. [ABSTRACT FROM AUTHOR]

We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real K$K$‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of TMF$\mathrm{TMF}$ is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of TMF$\mathrm{TMF}$ and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group. [ABSTRACT FROM AUTHOR]

We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4, 12, 12, 14, 18, 20 and 30. We also give a complete dictionary between the Dixmier-Ohno invariants of ternary quartics and the above generators. [ABSTRACT FROM AUTHOR]

Let Fg,n be the moduli space of n-pointed K3 surfaces of genus g with at worst rational double points. We establish an isomorphism between the ring of pluricanonical forms on Fg,n and the ring of certain orthogonal modular forms, and give applications to the birational type of Fg,n. We prove that the Kodaira dimension of Fg,n stabilizes to 19 when n is sufficiently large. Then we use explicit Borcherds products to find a lower bound of n where Fg,n has nonnegative Kodaira dimension, and compare this with an upper bound where Fg,n is unirational or uniruled using Mukai models of K3 surfaces in g ≤ 20. This reveals the exact transition point of Kodaira dimension in some g. [ABSTRACT FROM AUTHOR]

For a positive integer t ≥ 2 , let b t (n) denote the number of t-regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for b 9 (n) and b 19 (n) . We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of b 9 (n) and b 19 (n) modulo 2. For t ∈ { 6 , 10 , 14 , 15 , 18 , 20 , 22 , 26 , 27 , 28 } , Keith and Zanello conjectured that there are no integers A > 0 and B ≥ 0 for which b t (A n + B) ≡ 0 (mod 2) for all n ≥ 0 . We prove that, for any t ≥ 2 and prime ℓ , there are infinitely many arithmetic progressions A n + B for which ∑ n = 0 ∞ b t (A n + B) q n ≢ 0 (mod ℓ) . Next, we obtain quantitative estimates for the distributions of b 6 (n) , b 10 (n) and b 14 (n) modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and 13-regular partition functions. [ABSTRACT FROM AUTHOR]

We survey the connections between modular forms and representations of Galois groups that are predicted by the Langlands programme. We focus in particular on the applications of congruences between modular forms (through automorphy lifting theorems) to an improved understanding of these connections, including the author's recent joint work with James Newton on the existence of the symmetric power liftings of Hilbert modular forms. [ABSTRACT FROM AUTHOR]

We give a general conjecture concerning the existence of Eisenstein congruences between weight k ≥ 3 newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character χ of conductor N. The special cases M = 1 and M = p prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof. [ABSTRACT FROM AUTHOR]

Our results investigate mock theta functions and quantum modular forms via quantum q-series identities. After Lovejoy, quantum q-series identities are such that they do not hold as an equality between power series inside the unit disc in the classical sense, but do hold at dense sets of roots of unity on the boundary. We establish several general (multivariable) quantum q-series identities and apply them to various settings involving (universal) mock theta functions. As a consequence, we surprisingly show that limiting, finite, universal mock theta functions at roots of unity for which their infinite counterparts do not converge are quantum modular. Moreover, we show that these finite limiting universal mock theta functions play key roles in (generalized) Ramanujan radial limits. A further corollary of our work reveals that the finite Kontsevich–Zagier series is a kind of "universal quantum mock theta function," in that it may be used to evaluate odd-order Ramanujan mock theta functions at roots of unity. (We also offer a similar result for even-order mock theta functions.) Finally, to complement the notion of a quantum q-series identity and the results of this paper, we also define what we call an "antiquantum q-series identity' and offer motivating general results with applications to third-order mock theta functions. [ABSTRACT FROM AUTHOR]

We give explicit expressions for MacMahon's generalized sums-of-divisors q-series A r and C r by relating them to (odd) multiple Eisenstein series. Recently, these sums-of-divisors have been studied in the context of quasimodular forms, vertex algebras, N = 4 SU(N) Super–Yang–Mills theory, and the study of congruences of partitions. We relate them to a broader mathematical framework and give explicit expressions for both q-series in terms of Eisenstein series and their odd variants. [ABSTRACT FROM AUTHOR]

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke's work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions. [ABSTRACT FROM AUTHOR]