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Shimura defined a family of maps from the space of modular forms of half-integral weight to the space of modular forms of integral weight. Selberg in his unpublished work found explicitly this correspondence (the first Shimura map 𝒮 1 ) for the class of forms which are products of a Hecke eigenform of level one and a Jacobi theta function. Later, Cipra generalized the work of Selberg to the case where Jacobi theta functions are replaced by the theta functions associated to Dirichlet character of prime power moduli, and the level one Hecke eigenforms are replaced by newforms of arbitrary level. Hansen and Naqvi generalized Cipra's work (on the image of a class of modular forms under the first Shimura map 𝒮 1 ) to cover theta functions associated to Dirichlet characters of arbitrary moduli. In this paper, we show that the earlier results can be modified to get similar results for the t th Shimura lifts 𝒮 t , for any positive square-free integer t. [ABSTRACT FROM AUTHOR]

We discuss mock automorphic forms from the point of view of representation theory, that is, mock automorphic forms obtained from weak harmonic Maaß forms giving rise to nontrivial -cohomology. We consider the possibility of replacing the 'holomorphic' condition with a "cohomological" one when generalizing to general reductive groups. Such a candidate for replacement allows for growing Fourier coefficients, in contrast to automorphic forms under the Miatello-Wallach conjecture. In the second part, we provide an overview of the connection with BPS black hole counts as a physical motivation for studying mock automorphic forms. [ABSTRACT FROM AUTHOR]

In the present paper, we prove the 2-part of Birch and Swinnerton-Dyer conjecture for an explicit infinite family of rank 0 quadratic twists of the modular elliptic curve X 0 (14) , using an explicit form of the Waldspurger formula. We also give an explicit infinite family of rank 1 quadratic twists of X 0 (14) whose Tate–Shafarevich group is of odd cardinality. [ABSTRACT FROM AUTHOR]

We prove a vector-valued version of Muić's integral non-vanishing criterion for Poincaré series on the upper half-plane H. Moreover, we give an accompanying result on the construction of vector-valued modular forms in the form of Poincaré series. As an application of these results, we construct and study the non-vanishing of the classical and elliptic vector-valued Poincaré series. [ABSTRACT FROM AUTHOR]

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose graded traces are weight 3 2 weakly holomorphic modular forms satisfying certain special properties. We then use these modules to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups of quadratic twists of certain elliptic curves. [ABSTRACT FROM AUTHOR]

Let r Q (n) be the representation number of a nonnegative integer n by an integral positive definite quadratic form Q in 2 k variables. Let N be the level of Q. For a fixed prime p ∤ N , we assume that r Q (n) satisfies the relation r Q (1) r Q (p 2 n) = r Q (p 2) r Q (n) for all positive integers n with p ∤ n. We actually show that this relation holds for any prime p ∤ N when (− 1) k N is a fundamental discriminant. [ABSTRACT FROM AUTHOR]

We compare the homology of a congruence subgroup Γ of GL 2 (Z) with coefficients in the Steinberg modules over Q and over E , where E is a real quadratic field. If R is any commutative base ring, the last connecting homomorphism ψ Γ , E in the long exact sequence of homology stemming from this comparison has image in H 0 (Γ , St (Q 2 ; R)) generated by classes z β indexed by β ∈ E ∖ Q. We investigate this image. When R = C , H 0 (Γ , St (Q 2 ; C)) is isomorphic to a space of classical modular forms of weight 2, and the image lies inside the cuspidal part. In this case, z β is closely related to periods of modular forms over the geodesic in the upper half plane from β to its conjugate β ′. Assuming GRH we prove that the image of ψ Γ , E equals the entire cuspidal part. When R = Z , we have an integral version of the situation. We define the cuspidal part of the Steinberg homology, H 0 cusp (Γ , St (Q 2 ; Z)). Assuming GRH we prove that for any congruence subgroup, ψ Γ , E always has finite index in H 0 cusp (Γ , St (Q 2 ; Z)) , and if Γ = Γ 1 (N) ± or Γ 1 (N) , then the image is all of H 0 cusp (Γ , St (Q 2 ; Z)). If Γ = Γ 0 (N) ± or Γ 0 (N) , we prove (still assuming GRH) an upper bound for the size of H 0 cusp (Γ , St (Q 2 ; Z)) / Im (ψ Γ , E). We conjecture that the results in this paragraph are true unconditionally. We also report on extensive computations of the image of ψ Γ , E that we made for Γ = Γ 0 (N) ± and Γ = Γ 0 (N). Based on these computations, we believe that the image of ψ Γ , E is not all of H 0 cusp (Γ , St (Q 2 ; Z)) for these groups, for general N. [ABSTRACT FROM AUTHOR]

Kings, Lei, Loeffler and Zerbes constructed in [LLZ], [KLZ1] a three-variable Euler system -.g; h/of Beilinson-Flach elements associated to a pair of Hida families .g; h/and exploited it to obtain applications to the arithmetic of elliptic curves, extending the earlier work [BDR]. The aim of this article is to show that this Euler system also encodes arithmetic information concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when g and h specialize in weight 1 to p-stabilizations of eigenforms such that one is dual to the other. We encounter an exceptional zero phenomenon which forces the specialization of -.g; h/to vanish and we are led to study the derivative of this class. The main result we obtain is the proof of the main conjecture of [DLR4] on iterated integrals and the main conjecture of [DR1] for Beilinson-Flach elements in the adjoint setting. The main point of this paper is that the methods of [DLR1], [DLR4] and [CH], where the above conjectures are proved when the weight 1 eigenforms have CM, do not apply to our setting and new ideas are required. In the previous works, a crucial ingredient is a factorization of p-adic L-functions, which in our scenario is not available due to the lack of critical points. Instead we resort to the principle of improved Euler systems and p-adic L-functions to reduce our problems to questions which can be resolved using Galois deformation theory. We expect this approach may be adapted to prove other cases of the Elliptic Stark Conjecture and of its generalizations that appear in the literature. [ABSTRACT FROM AUTHOR]

In the theory of harmonic Maaß forms and mock modular forms, mock theta functions are distinguished examples which arose from q-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular forms. Here, we introduce distinguished examples of these forms, which we call higher depth mock theta functions, and develop q-hypergeometric expressions for them. We provide three examples of mock theta functions of depth two, each arising by multiplying a classical mock theta function with a certain specialization of a universal mock theta function. In addition, we give their modular completions, and relate each to a q-hypergeometric series. [ABSTRACT FROM AUTHOR]

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let p be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L‐function of the base‐change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch–Kato conjecture predicts that this should force the existence of non‐trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Salazar and the second author to construct certain p‐adic Abel–Jacobi maps, which we use to propose a construction of such classes via Stark–Heegner cycles. This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms. [ABSTRACT FROM AUTHOR]

The purpose of this article is proving the equality of two natural L‐invariants attached to the adjoint representation of a weight one cusp form, each defined by purely analytic, respectively, algebraic means. The proof departs from Greenberg's definition of the algebraic L‐invariant as a universal norm of a canonical Zp‐extension of Qp associated to the representation. We relate it to a certain 2×2 regulator of p‐adic logarithms of global units by means of class field theory, which we then show to be equal to the analytic L‐invariant computed in Rivero and Rotger [J. Eur. Math. Soc., to appear]. [ABSTRACT FROM AUTHOR]

The sup‐norm problem in analytic number theory asks for the largest value taken by a given automorphic form. We observe that the function‐field version of this problem can be reduced to the geometric problem of finding the largest dimension of the ith stalk cohomology group of a given Hecke eigensheaf at any point. This problem, in turn, can be reduced to the intersection‐theoretic problem of bounding the 'polar multiplicities' of the characteristic cycle of the Hecke eigensheaf, which in known cases is the nilpotent cone of the moduli space of Higgs bundles. We solve this problem for newforms on GL2(AFq(T)) of squarefree level, leading to bounds on the sup‐norm that are stronger than what is known in the analogous problem for newforms on GL2(AQ) (that is, classical holomorphic and Maaß modular forms.) [ABSTRACT FROM AUTHOR]

Recent advances in ecosystem theory prescribe that companies need to develop offers that are modular and form unique or supermodular complementarities with other offers. However, both academic and managerial knowledge of the strategies that especially small and medium sized enterprises (SMEs) can use to engage in existing ecosystems for value creation remains scattered and predominantly vague. This article thus aims to explore applicable ecosystem engagement strategies from the perspective of SMEs, as discussed in previous scholarly literature. In so doing, the article puts forward and elaborates three distinct strategies that SMEs can apply to become part of value-creating ecosystems. In this way, the findings contribute to the literature on ecosystems. [ABSTRACT FROM AUTHOR]

The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series. [ABSTRACT FROM AUTHOR]

The u-plane integral is the contribution of the Coulomb branch to correlation functions of N = 2 gauge theory on a compact four-manifold. We consider the u-plane integral for correlators of point and surface observables of topologically twisted theories with gauge group SU (2) , for an arbitrary four-manifold with (b 1 , b 2 +) = (0 , 1) . The u-plane contribution equals the full correlator in the absence of Seiberg–Witten contributions at strong coupling, and coincides with the mathematically defined Donaldson invariants in such cases. We demonstrate that the u-plane correlators are efficiently determined using mock modular forms for point observables, and Appell–Lerch sums for surface observables. We use these results to discuss the asymptotic behavior of correlators as function of the number of observables. Our findings suggest that the vev of exponentiated point and surface observables is an entire function of the fugacities. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be an integer. Let q be a prime power such that q ≡ 1 (mod k) if q is even, or, q ≡ 1 (mod 2 k) if q is odd. The generalized Paley graph of order q, G k (q) , is the graph with vertex set F q where ab is an edge if and only if a - b is a kth power residue. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in G k (q) , K 4 (G k (q)) , which holds for all k. This generalizes the results of Evans, Pulham and Sheehan on the original ( k = 2 ) Paley graph. We also provide a formula, in terms of Jacobi sums, for the number of complete subgraphs of order three contained in G k (q) , K 3 (G k (q)) . In both cases, we give explicit determinations of these formulae for small k. We show that zero values of K 4 (G k (q)) (resp. K 3 (G k (q)) ) yield lower bounds for the multicolor diagonal Ramsey numbers R k (4) = R (4 , 4 , ... , 4) (resp. R k (3) ). We state explicitly these lower bounds for small k and compare to known bounds. We also examine the relationship between both K 4 (G k (q)) and K 3 (G k (q)) , when q is prime, and Fourier coefficients of modular forms. [ABSTRACT FROM AUTHOR]

The algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra T consisting of symmetric polynomials in the part sizes and multiplicities. [ABSTRACT FROM AUTHOR]

In this article, we construct a differential modular form of non-zero order and integral weight for compact Shimura curves over totally real fields bigger than Q . The construction uses the theory of lifting ordinary mod p Hilbert modular forms to characteristic 0 as well as the theory of Igusa curve. This is the analogue of the construction of Buium in the case of modular curves parametrizing elliptic curves with level structures. [ABSTRACT FROM AUTHOR]

There are various reasons why a naive analog of the Maeda conjecture has to fail for Drinfeld cusp forms. Focussing on double cusp forms and using the link found by Teitelbaum between Drinfeld cusp forms and certain harmonic cochains, we observed a while ago that all obvious counterexamples disappear for certain Hecke-invariant subquotients of spaces of Drinfeld cusp forms of fixed weight, which can be defined naturally via representation theory. The present work extends Teitelbaum's isomorphism to an adelic setting and to arbitrary levels, it makes precise the impact of representation theory, it relates certain intertwining maps to hyperderivatives of Bosser-Pellarin, and it begins an investigation into dimension formulas for the subquotients mentioned above. We end with some numerical data for A = F 3 [ t ] that displays a new obstruction to an analog of a Maeda conjecture by discovering a conjecturally infinite supply of F 3 (t) -rational eigenforms with combinatorially given (conjectural) Hecke eigenvalues at the prime t. [ABSTRACT FROM AUTHOR]

We prove (under certain assumptions) the irreducibility of the limit σ 2 of a sequence of irreducible essentially self-dual Galois representations σ k : G Q → GL 4 (Q ¯ p) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to 1 ⊕ ρ ⊕ χ with ρ irreducible, two-dimensional of determinant χ , where χ is the mod p cyclotomic character. More precisely, we assume that σ k are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as k → 2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to ρ ) which we assume are non-zero. [ABSTRACT FROM AUTHOR]

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard L -function attached to vector-valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted L -function beyond the usual range of absolute convergence. Our results include also the case of metaplectic Siegel modular forms. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series. [ABSTRACT FROM AUTHOR]

Let (𝕋 f , 𝔪 f) denote the mod p local Hecke algebra attached to a normalized Hecke eigenform f , which is a commutative algebra over some finite field 𝔽 q of characteristic p and with residue field 𝔽 q . By a result of Carayol we know that, if the residual Galois representation ρ ¯ f : G ℚ → GL 2 (𝔽 q) is absolutely irreducible, then one can attach to this algebra a Galois representation ρ f : G ℚ → GL 2 (𝕋 f) that is a lift of ρ ¯ f . We will show how one can determine the image of ρ f under the assumptions that (i) the image of the residual representation contains SL 2 (𝔽 q) , (ii) 𝔪 f 2 = 0 and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain p -elementary abelian extensions of big non-solvable number fields. [ABSTRACT FROM AUTHOR]

For a positive integer N , we say that ∞ is a Weierstrass point on the modular curve X 0 (N) if there is a non-zero cusp form of weight 2 on Γ 0 (N) which vanishes at ∞ to order greater than the genus of X 0 (N). If p is a prime with p ∤ N , Ogg proved that ∞ is not a Weierstrass point on X 0 (p N) if the genus of X 0 (N) is 0. We prove a similar result for even weights k ≥ 4. We also study the space of weight k cusp forms on Γ 0 (N) vanishing to order greater than the dimension. [ABSTRACT FROM AUTHOR]

In this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulae for them. This implies an elementary "double copy" formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Néron–Tate heights on curves, single-valued multiple zeta values and polylogarithms. The results of the present paper are used in [F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, preprint 2019, https://arxiv.org/abs/1910.01107] to prove a recent conjecture of Stieberger which relates the coefficients in a Laurent expansion of two different kinds of periods of twisted cohomology on the moduli spaces of curves ℳ0,n of genus zero with n marked points. We also study a morphism between certain rings of "motivic" periods, called the de Rham projection, which provides a bridge between complex periods and single-valued periods in many situations of interest. [ABSTRACT FROM AUTHOR]

We give a lower bound of the number of zeros of the modular forms E k + E ℓ + − E k + ℓ + for large enough k and ℓ on each of the arc boundary and on the vertical boundary of the fundamental domain for the Fricke group Γ 0 + (2) , where E k + is the Eisenstein series of weight k for Γ 0 + (2). [ABSTRACT FROM AUTHOR]

The article discusses the construction of a one-variable p-adic L-function for Rankin-Selberg families using the imaginary quadratic field K and the eigenform of weight k. The p-adic L-function is a continuous function on a space of p-adic Hecke characters. Also cited is the use of Theorems 5.48 and 6.21 to allow the p-adic L-function to satisfy an interpolation property equal to central L-values.

The article discusses the supersingular Rankin-Selberg p-adic L-functions by constructing a 1-variable anticyclotomic p-adic L-function linked to a given normalized eigenform like a newform or normalized Eisenstein series. Also cited are the use of an imaginary quadratic field and the fundamental discriminant of K to create a Heegner hypothesis for N, and the equation for a complex-valued Hecke character.

The article discusses the p-adic Maass-Shimura operator and its role in the construction of p-adic L-functions over imaginary quadratic fields K. Other topics include the horizontal lifting of the Hodge-Tate filtration, the equation of the Hodge-Tate sequence, the characteristics of a relative Hodge-Tate decomposition, and the satisfactory properties of the operator like algebraicity.

The article discusses the construction of p-adic L-functions over imaginary quadratic fields K based on the works of N. Katz, M. Bertolini, H. Darmon and K. Prasanna. Topics include Katz's theory of p-adic modular forms, the application of p-adic differential operators in modular forms, and the differential operator Atkin-Serre operator.

Inspired by work done for systems of polynomial exponential equations, we study systems of equations involving the modular j function. We show general cases in which these systems have solutions, and then we look at certain situations in which the modular Schanuel conjecture implies that these systems have generic solutions. An unconditional result in this direction is proven for certain polynomial equations on j with algebraic coefficients. [ABSTRACT FROM AUTHOR]

Soient F un corps de fonctions sur Fq, A l'anneau des fonctions régulières hors d'une place ∞ et p un idéal premier de A. En premier lieu, nous développons la théorie de Hida pour les formes modulaires de Drinfeld de rang r qui sont de pente nulle pour l'opérateur de Hecke Uπ convenablement défini. En second lieu, nous montrons en pente finie l'existence de familles de formes modulaires de Drinfeld variant continûment selon le poids. En troisième lieu, nous prouvons un résultat de classicité: une forme modulaire de Drinfeld surconvergente de pente suffisamment petite par rapport au poids est une forme modulaire de Drinfeld classique. [ABSTRACT FROM AUTHOR]

Motivated by navigation and control problems in robotics, Ghrist and Peterson introduced a class of non-positively curved (NPC) cubical complexes arising as configuration spaces of reconfigurable systems, best regarded as discretized state space representations of embodied agents such as a multi-jointed robotic arm. In current real world applications, agents are increasingly required to respond autonomously to sensory input in order for them to contend with a priori unknown obstacles to navigation. In particular, the configuration spaces in question may not be known in advance. This motivates the following problem formulation: Given a NPC cubical complex C and a point-separating collection Σ of Boolean queries on its 0-skeleton, C (0) , find an efficient algorithm for learning C from the outputs provided by Σ along an appropriately chosen path in C . In this note, we tackle the problem of identifying C when it is known that C is CAT(0). We show that the collection of canonical hyperplanes of C is the unique solution of a sub-modular minmax problem over the space of point-separating systems of Boolean queries on C (0) , which may also be formulated in terms of the quadratic form associated with the graph Laplacian of C (1) . [ABSTRACT FROM AUTHOR]

This paper is a continuation of our previous works where we study maps from X 0 (N) , N ≥ 1 , into P 2 constructed via modular forms of the same weight and criteria that such a map is birational (see Muić in Monatsh Math 180(3):607–629, 2016). In the present paper, our approach is based on the theory of primitive elements in finite separable field extensions. We prove that in most of the cases the constructed maps are birational. We consider particular cases and their equations in P 2 . [ABSTRACT FROM AUTHOR]

In 2012, Chan posed six conjectures on congruences for Appell–Lerch sums and five of them were proved by mathematicians. In this paper, we confirm the remaining conjectural congruence of Chan by utilizing an identity due to Hickerson and Mortenson and the theory of modular forms. [ABSTRACT FROM AUTHOR]

In this paper we study the analytic properties of the standard L -function attached to vector valued quaternionic modular forms using the Rankin-Selberg method. This involves the construction of vector valued theta series, which we obtain by applying some differential operators on Jacobi-theta series studied by Krieg. Such differential operators are obtained from the Howe-Weyl duality for the pair S p n (C) × G L 2 n (C). [ABSTRACT FROM AUTHOR]

This paper continues work of the earlier articles with the same title. For two classes of modular forms f : • para-Eisenstein series α k and • coefficient forms ℓ k a , where k ∈ N and a is a non-constant element of F q [ T ] , the growth behavior on the fundamental domain and the zero loci Ω (f) as well as their images BT (f) in the Bruhat-Tits building BT are studied. We obtain a complete description for f = α k and for those of the forms ℓ k a where k ≤ deg ⁡ a. It turns out that in these cases, α k and ℓ k a are strongly related, e.g., BT (ℓ k a ) = BT (α k) , and that BT (α k) is the set of Q -points of a full subcomplex of BT with nice properties. As a case study, we present in detail the outcome for the forms α 2 in rank 3. [ABSTRACT FROM AUTHOR]

Klein forms are used to construct generators for the graded algebra of modular forms of level 7. Dissection formulas for the series imply Ramanujan type congruences modulo powers of 7 for a family of generating functions that subsume the counting function for 7-core partitions. The broad class of arithmetic functions considered here enumerate colored partitions by weights determined by parts modulo 7. The method is a prototype for similar analysis of modular forms of level 7 and at other prime levels. As an example of the utility of the dissection method, the paper concludes with a derivation of novel congruences for the number of representations by x 2 + x y + 2 y 2 in exactly k ways. [ABSTRACT FROM AUTHOR]

In this note, we confirm a conjectural formula on the number of representations of the square of an odd prime by a sum of an odd number of squares, i.e., # { (x 1 , ... , x 2 k + 1) ∈ Z 2 k + 1 : x 1 2 + ⋯ + x 2 k + 1 2 = p 2 } , proposed by Cooper. [ABSTRACT FROM AUTHOR]

We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1, 0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0, 1) supersymmetry and, possibly, a residual flavor symmetry. The equivariant topological Witten genus of this 2d theory then produces a new invariant of the 4-manifold equipped with a principle bundle, valued in the ring of equivariant weakly holomorphic (topological) modular forms. We describe basic properties of this map and present a few simple examples. As a byproduct, we obtain some new results on 't Hooft anomalies of 6d (1, 0) theories and a better understanding of the relation between 2d (0, 1) theories and TMF spectra. [ABSTRACT FROM AUTHOR]

Using the Gelfand-Kapranov-Zelevinskĭ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana integrals with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel Γ ̂ ‐ class evaluation in the ambient spaces of the mirror, while the imaginary part of the integral in this regime is determined by the Γ ̂ ‐ class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the integrals when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogeneous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the banana integral in very short time to very high numerical precision for all values of the physical parameters. Using modular properties of the periods we determine the value of the maximal cut equal mass four-loop integral at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins. [ABSTRACT FROM AUTHOR]

We extend the dictionary between the BPS spectrum of Heterotic strings and the one of F-/M-theory compactifications on K3 fibered Calabi-Yau 3-folds to cases with higher rank non-Abelian gauge groups and in particular to dual pairs between Heterotic CHL orbifolds and compactifications on Calabi-Yau 3-folds with a compatible genus one fibration. We show how to obtain the new supersymmetric index purely from the Calabi-Yau geometry by reconstructing the Noether-Lefschetz generators, which are vector-valued modular forms. There is an isomorphism between the latter objects and vector-valued lattice Jacobi forms, which relates them to the elliptic genera and twisted-twined elliptic genera of six- and five-dimensional Heterotic strings. The meromorphic Jacobi forms generate the dimensions of the refined cohomology of the Hilbert schemes of symmetric products of the fiber and allow us to refine the BPS indices in the fiber and therefore to obtain, conjecturally, actual state counts. Using the properties of the vector-valued lattice Jacobi forms we also provide a mathematical proof of the non-Abelian weak gravity conjecture for F-/M-theory compactified on this general class of fibered Calabi-Yau 3-folds. [ABSTRACT FROM AUTHOR]

We apply differential operators to modular forms on orthogonal groups O(2, ℓ) to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature (2, 2) and (2, 3), for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama. [ABSTRACT FROM AUTHOR]

Let K = Q (ζ 8) be the cyclotomic field over Q of the extension degree 4. We give an integral lattice construction on Q (ζ 8) induced from codes over the ring R = F 2 [ u ] / ⟨ u 4 ⟩ . We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a code over R . If C is a Type II code of length l, we find that the complete weight enumerator of C gives a Jacobi form of weight l and the index 2l over the maximal totally real subfield k = Q (ζ 8 + ζ 8 - 1) of K. Also, we see that Hilbert–Siegel modular form of weight n and genus g can be seen in terms of the complete joint weight enumerator of codes C j , for 1 ≤ j ≤ g over R . [ABSTRACT FROM AUTHOR]

In recent work, methods from the theory of modular forms were used to obtain Fourier uniqueness results in several key dimensions (d = 1, 8, 24), in which a function could be uniquely reconstructed from the values of it and its Fourier transform on a discrete set, with the striking application of resolving the sphere packing problem in dimensions d = 8 and d = 24. In this short note, we present an alternative approach to such results, viable in even dimensions, based instead on the uniqueness theory for the Klein- Gordon equation. Since the existing method for the Klein-Gordon uniqueness theory is based on the study of iterations of Gauss-type maps, this suggests a connection between the latter and methods involving modular forms. The derivation of Fourier uniqueness from the Klein-Gordon theory supplies conditions on the given test function for Fourier interpolation, which are hoped to be optimal or close to optimal. [ABSTRACT FROM AUTHOR]

We survey divisibility properties of the Fourier coefficients of modular functions inspired by Ramanujan. Then using recent results of the generalized Hecke operator on harmonic Maass functions and known divisibility of Fourier coefficients of modular functions, we establish congruence relations of the Fourier coefficients of certain modular functions and mock modular functions of various levels. [ABSTRACT FROM AUTHOR]

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) "small divisors" summatory functions σ ψ sm (n). More precisely, in terms of the weight 2 quasimodular Eisenstein series E 2 (τ) and a generic Shimura theta function 𝜃 ψ (τ) , we show that there is a constant α ψ for which ε ψ + (τ) : = α ψ ⋅ E 2 (τ) 𝜃 ψ (τ) + 1 𝜃 ψ (τ) ∑ n = 1 ∞ σ ψ sm (n) q n is a half integral weight (polar) mock modular form. These include generating functions for combinatorial objects such as the Andrews s p t -function and the "consecutive parts" partition function. Finally, in analogy with Serre's result that the weight 2 Eisenstein series is a p -adic modular form, we show that these forms possess canonical congruences with modular forms. [ABSTRACT FROM AUTHOR]