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The classifying spaces of handlebody groups form a modular operad. Algebras over the handlebody operad yield systems of representations of handlebody groups that are compatible with gluing. We prove that algebras over the modular operad of handlebodies with values in an arbitrary symmetric monoidal bicategory |$\mathcal{M}$| (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in |$\mathcal{M}$|⁠. After specialization to a linear framework, this proves that consistent systems of handlebody group representations on finite-dimensional vector spaces are equivalent to ribbon Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the vector space assigned to an arbitrary handlebody in terms of a generalization of Lyubashenko's coend. Our main result can be used to obtain an ansular functor from vertex operator algebras subject to mild finiteness conditions. This includes examples of vertex operator algebras whose representation category has a non-exact monoidal product. [ABSTRACT FROM AUTHOR]

We study the cone of moving divisors on the moduli space |${\mathcal{A}}_{g}$| of principally polarized abelian varieties. Partly motivated by the generalized Rankin–Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on |${\mathcal{A}}_{g}$| for |$g\leq 4$|⁠ , and gives an explicit upper bound for the moving slope of |${\mathcal{A}}_{5}$| and a conjectural upper bound for the moving slope of |${\mathcal{A}}_{6}$|⁠. [ABSTRACT FROM AUTHOR]

We study the cone of moving divisors on the moduli space |${\mathcal{A}}_{g}$| of principally polarized abelian varieties. Partly motivated by the generalized Rankin–Cohen bracket, we construct a non-linear holomorphic differential operator that sends Siegel modular forms to Siegel modular forms, and we apply it to produce new modular forms. Our construction recovers the known divisors of minimal moving slope on |${\mathcal{A}}_{g}$| for |$g\leq 4$|⁠ , and gives an explicit upper bound for the moving slope of |${\mathcal{A}}_{5}$| and a conjectural upper bound for the moving slope of |${\mathcal{A}}_{6}$|⁠. [ABSTRACT FROM AUTHOR]

In this paper, we show how quantum modular forms naturally arise in the ergodic theory of circle rotations. Working with the classical Birkhoff sum |$S_{N}(\alpha)=\sum _{n=1}^{N} (\{ n \alpha \}-1/2)$|⁠ , we prove that the maximum and the minimum as well as certain exponential moments of |$S_{N}(r)$| as functions of |$r \in \mathbb{Q}$| satisfy a direct analogue of Zagier's continuity conjecture, originally stated for a quantum invariant of the figure-eight knot. As a corollary, we find the limit distribution of |$\max _{0 \le N

Harris and Venkatesh made a conjecture relating the derived Hecke operators and the adjoint motivic cohomology in the setting of weight one modular forms. This conjecture was proved under some conditions in the dihedral case by Darmon–Harris–Rotger–Venkatesh. We use a new approach to prove more general cases of the conjecture (up to sign). Our approach relies on Waldspurger's formula for the central value of Rankin L-series and Ichino's formula for the triple product L-function. [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 introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let |$\Gamma $| be a finite index subgroup of |$\textrm {SL}(2,\mathbb Z)$| with an action on a complex simple Lie algebra |$\mathfrak g$|⁠ , which can be extended to |$\textrm {SL}(2,{\mathbb {C}})$|⁠. We show that the Lie algebra of the corresponding |$\mathfrak {g}$| -valued modular forms is isomorphic to the extension of |$\mathfrak {g}$| over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups |$\Gamma (N), \, N\leq 6$|⁠ , is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras. [ABSTRACT FROM AUTHOR]

For any congruence subgroup |$\Gamma $|⁠ , we study the vertex operator algebra |$\Omega ^{ch}(\mathbb H,\Gamma)$| constructed from the |$\Gamma $| -invariant global sections of the chiral de Rham complex on the upper half plane, which are holomorphic at all the cusps. We construct a basis of |$\Omega ^{ch}(\mathbb H,\Gamma)$| in terms of modular forms of |$\Gamma $| and compute its character. We show that the vertex operations on |$\Omega ^{ch}(\mathbb H,\Gamma)$| are determined by a modification of the Rankin–Cohen brackets of modular forms. [ABSTRACT FROM AUTHOR]

A general Voronoï summation formula for the (metaplectic) double cover of |$\operatorname{GL}_2$| is derived via the representation theoretic framework à la Ichino–Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms. [ABSTRACT FROM AUTHOR]

We establish new results on equations and bilinear forms with modular square roots. The main motivation and application of these results is our new bound on the fourth moment of the error term in the asymptotic formula for the twisted second moment of half integral weight Dirichlet series on average over moduli. [ABSTRACT FROM AUTHOR]

In this paper, we derive topological and number theoretical consequences of the rigidity of elliptic genera, which are special modular forms associated to compact almost complex manifolds. On the geometry side, we prove that rigidity implies relations between the Betti numbers and the index of a compact symplectic manifold admitting a Hamiltonian action of a circle with isolated fixed points. We investigate the case of maximal index and toric actions. On the number theoretical side, we prove that from each compact almost complex manifold of index greater than one, that can be endowed with the action of a circle with isolated fixed points, one can derive non-trivial relations among Eisenstein series. We give explicit formulas coming from the standard action on |${{\mathbb{C}}} P^n$|⁠. [ABSTRACT FROM AUTHOR]

Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of |$p$| -adic and |$\textrm{mod}\; p$| modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: (1) the analytic continuation at unramified primes |$p$| to the whole Shimura variety of the |$\textrm{mod}\; p$| reduction of |$p$| -adic Maass–Shimura operators a priori defined only over the |$\mu $| -ordinary locus, and (2) the construction of new |$\textrm{mod}\; p$| theta operators that do not arise as the |$\textrm{mod}\; p$| reduction of Maass–Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree. [ABSTRACT FROM AUTHOR]

We study the meromorphy of various Euler products of degree two attached to holomorphic Hecke eigen cusp forms for the elliptic modular group, including Ramanujan's |$\Delta $| -function. [ABSTRACT FROM AUTHOR]

We consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo–Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted variant of the Holowinsky–Soundararajan theorem involving restrictions of normalized Hilbert modular forms arising via base change. [ABSTRACT FROM AUTHOR]

We describe the ring of modular forms of degree |$2$| in characteristic |$2$| using its relation with curves of genus |$2$|⁠. [ABSTRACT FROM AUTHOR]

We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry–Ganguly–Kowalski–Michel and Kowalski–Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution that comes from the asymptotic mixed distribution of Salié sums. [ABSTRACT FROM AUTHOR]

We study rationality properties of geodesic cycle integrals of meromorphic modular forms associated to positive definite binary quadratic forms. In particular, we obtain finite rational formulas for the cycle integrals of suitable linear combinations of these meromorphic modular forms. [ABSTRACT FROM AUTHOR]

This article is a continuation of our previous work [ 7 ] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field |$K$|⁠ , where the modular form in question was assumed to be ordinary at a fixed odd prime |$p$|⁠. We formulate integral Iwasawa main conjectures at non-ordinary primes |$p$| for suitable twists of the base change of a newform |$f$| to an imaginary quadratic field |$K$| where |$p$| splits, over the cyclotomic |${\mathbb{Z}}_p$| -extension, the anticyclotomic |${\mathbb{Z}}_p$| -extensions (in both the definite and the indefinite cases) as well as the |${\mathbb{Z}}_p^2$| -extension of |$K$|⁠. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed |$p$| -adic |$L$| -functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances). [ABSTRACT FROM AUTHOR]

We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic four-fold is a non-singular curve of genus |$631$|⁠. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely |$3,780$| elliptic curves of minimal degree with fixed (general) |$j$| -invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-Kähler varieties with fixed |$j$| -invariant in terms of Gromov–Witten invariants. In |$K3^{[2]}$| -type this leads to explicit formulas of these counts in terms of modular forms. [ABSTRACT FROM AUTHOR]

We establish an effective version of the André–Oort conjecture for linear subspaces of |$Y(1)^n_{\mathbb{C}} \approx \mathbb{A}_{\mathbb{C}}^n$|⁠. This gives the first effective nontrivial results of André–Oort type for higher-dimensional varieties in products of modular curves. [ABSTRACT FROM AUTHOR]

We define an algebraic set in |$23$| -dimensional projective space whose |${{\mathbb{Q}}}$| -rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree 2 paramodular groups. Weight |$3$| examples provide antisymmetric canonical differential forms on Siegel modular three-folds. Weight |$2$| is the minimal weight and these examples, via the paramodular conjecture, give evidence for the modularity of some rank 1 abelian surfaces defined over |$\mathbb{Q}$|⁠. [ABSTRACT FROM AUTHOR]

We describe a novel method for bounding the dimension |$d$| of the largest simple Hecke submodule of |$S_{2}(\Gamma _{0}(N);\mathbb{Q})$| from below. Such bounds are of interest because of their relevance to the structure of |$J_{0}(N)$|⁠ , for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. We prove conditional bounds, the strongest of which is |$d\gg _{\epsilon } N^{1/2-\epsilon }$| over a large set of primes |$N$|⁠ , contingent on Soundararajan's heuristics for the class number problem and Artin's conjecture on primitive roots. For prime levels |$N\equiv 7\mod 8,$| our method yields an unconditional bound of |$d\geq \log _{2}\log _{2}(\frac{N}{8})$|⁠ , which is larger than the known bound of |$d\gg \sqrt{\log \log N}$| due to Murty–Sinha and Royer. A stronger unconditional bound of |$d\gg \log N$| can be obtained in more specialized (but infinitely many) cases. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of |$S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$|⁠. [ABSTRACT FROM AUTHOR]

In this short article, we investigate the topology of the moduli space of two-convex embedded tori |$S^{n-1}\times S^1\subset \mathbb{R}^{n+1}$|⁠. We prove that for |$n\geq 3$| this moduli space is path connected, and that for |$n=2$| the connected components of the moduli space are in bijective correspondence with the knot classes associated to the embeddings. Our proof uses a variant of mean curvature flow with surgery developed in our earlier article [ 3 ] where neck regions are deformed to tiny strings instead of being cut out completely, an approach which preserves the global topology, embeddedness, as well as two-convexity. [ABSTRACT FROM AUTHOR]

We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke $$L$$ -functions that were recently proved by the authors. Our result produces an improvement for the best unconditional bounds toward two conjectures of Lang and Trotter regarding the distribution of traces of Frobenius for elliptic curves and holomorphic cuspidal modular forms. We also obtain new results on the distribution of primes represented by positive-definite integral binary quadratic forms. [ABSTRACT FROM AUTHOR]

In this note, we prove that there exists a conjugate pair of classical Hilbert modular cuspidal eigenforms over Q(√ 5) of partial weight 1 which do not arise from the induction of a Grössencharacter from a complex multiplication extension of Q( √ 5). [ABSTRACT FROM AUTHOR]

In this paper, we study certain partial and false theta functions in connection to vertex operator algebras and conformal field theory. We prove a variety of results concerning the asymptotics of modified characters of irreducible modules of certain W-algebras of singlet type, which allows us in particular to determine their (analytic) quantum dimensions. Our results are fully consistent with the previous conjectures on fusion rings for these vertex algebras. More importantly, we prove quantum modularity( à la Zagier) of the numerator part of irreducible characters of singlet algebra modules, thus demonstrating that quantum modular forms naturally appear in many "sufficiently nice" irrational vertex algebras. It is interesting that quantum modularity persists on the whole set of rationals as in the original Zagier's example coming from Kontsevich's "strange series". In the last part, slightly independent of all this, we also discuss Nahm-type q-hypergometric series in connection to tails of colored Jones polynomials of certain torus knots and characters of modules for the (1, p)-singlet algebra. [ABSTRACT FROM AUTHOR]

The De Concini-Procesi wonderful models of the braid arrangement of type An+1 are equipped with a natural Sn action, but only the minimal model admits an "hidden" symmetry, that is, an action of Sn+1 that comes from its moduli space interpretation. In this paper, we explain why the non minimal models do not admit this extended action: they are "too small". In particular, we construct a supermaximal model which is the smallest model that can be projected on to the maximal model and again admits an extended Sn+1 action. We give an explicit description of a basis for the integer cohomology of this supermaximal model. Furthermore, we deal with another hidden extended action of the symmetric group: we observe that the symmetric group Sn+k acts by permutation on the set of k-codimensional strata of the minimal model. Even if this happens at a purely combinatorial level, it gives rise to an interesting permutation action on the elements of a basis of the integer cohomology. [ABSTRACT FROM AUTHOR]

Given P,an Apollonian Circle Packing, and a circle C0 = ∂B(z0,r0) in P, color the set of disks in P tangent to C0 red.What proportionofthe concentric circle C∊ = ∂B(z0,r0 + ∊) is red, and what is the behavior of this quantity as ∊→ 0? Using equidistribution of closed horocycles on the modular surface H2/SL(2,Z), we show that the answer is = 0.9549 .... We also describe an observation due to Alex Kontorovich connecting the 3/π rate of this convergence in the Farey-Ford packing to the Riemann hypothesis. For the analogous problem for Soddy sphere packings, we find that the limiting radial density is √3/2V3T = 0.853 ..., where VT denotes the volume of an ideal hyperbolic tetrahedron with dihedral angles π/3. [ABSTRACT FROM AUTHOR]

We construct some examples of origamis (square-tiled surfaces) such that the Hodge bundles over the corresponding SL(2,R)-orbits on the moduli space admit nontrivial isotropic SL(2,R)-invariant subbundles. This answers a question posed to the authors by Eskin and Forni. [ABSTRACT FROM AUTHOR]

We construct "generalized Heegner cycles" on a variety fibered over a Shimura curve, defined over a number field. We show that their images under the p-adic Abel-Jacobi map coincide with the values (outside the range of interpolation) of a p-adic L-function L p( f, X) which interpolates special values of the Rankin-Selberg convolution of a fixed newform f and a theta-series θχ attached to an unramified Hecke character of an imaginary quadratic field K. This generalizes previous work of Bertolini, Darmon, and Prasanna, which demonstrated a similar result in the case of modular curves. Our main tool is the theory of Serre-Tate coordinates, which yields p-adic expansions of modular forms at CM points, replacing the role of q-expansions in computations on modular curves. [ABSTRACT FROM AUTHOR]

We describe a practical method for finding an L-function without first finding an associated underlying object. The procedure involves using the Euler product, the Ramanujan bound, and the approximate functional equation in a new way. No use is made of the functional equation of twists of the L-function. The method is used to find a large number of Maass forms on and to give the first examples of Maass forms of higher level on GL(3), and on GL(4) and Sp(4). [ABSTRACT FROM PUBLISHER]

We construct a Poincaré section for the horocycle flow on the modular surface , and study the associated first return map, which coincides with a transformation [the Boca-Cobeli-Zaharescu (BCZ) map] defined by Boca et al. [7]. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors. [ABSTRACT FROM AUTHOR]

Previous work constructed a generalized truncated Brown–Peterson spectrum of chromatic height 2 at the prime 2 as an -ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an -ring map from the spectrum of topological modular forms to this truncated Brown–Peterson spectrum, and that this orientation fits into a diagram of -ring spectra lifting a classical diagram of modules over the mod-2 Steenrod algebra. In an appendix, we document how to organize Morava's forms of K-theory into a sheaf of -ring spectra. [ABSTRACT FROM AUTHOR]

We give an asymptotic for the number of strongly unimodal sequences of weight n, denoted u*(n) with an error of size O(n1+ε). The result relies on an identity expressing the generating function for u*(n) as a mixed mock modular form. [ABSTRACT FROM PUBLISHER]

We prove the exceptional zero conjecture for the symmetric powers of complex multiplication cuspidal eigenforms at ordinary primes. In other words, we determine the trivial zeros of the associated p-adic L-functions, compute the L-invariants, and show that they agree with Greenberg’s L-invariants. In the appendix, we prove a functional equation for some of the p-adic L-functions we construct. [ABSTRACT FROM AUTHOR]

We show that, under suitable assumptions, the Hecke eigensystems arising from (mod p) modular forms of PEL type associated to an algebraic group of type A or C, coincide with the Hecke eigensystems arising from (mod p) algebraic modular forms associated to an inner form I of G. The form I has to be compact modulo center at infinity and satisfies: for . In order to realize the Hecke correspondence, we first adapt a result of Rapoport and Zink to obtain an explicit uniformization of the superspecial locus of PEL Shimura varieties over (such a uniformization in the general context of PEL Shimura varieties was first described by C.-F. Yu). We then compare (mod p) modular forms with superspecial modular forms by exploiting the Hermitian -structure of the cotangent spaces of the p-divisible groups of the superspecial points. We study in detail the case in which G is of type A, and we prove that the number of (mod p) Hecke eigensystems arising from unitary modular forms for a fixed quadratic imaginary field, having signature (r,s), prime-to-p level N≥3 and variable weight satisfies the estimate for p large, where g=r+s. The paper generalizes results of J-P. Serre for GL(2) and Ghitza for GSp(2g) (g>1). [ABSTRACT FROM PUBLISHER]

We show how the Langlands–Kottwitz method can be used to determine the local factors of the Hasse–Weil zeta-function of the modular curve at places of bad reduction. On the way, we prove a conjecture of Haines and Kottwitz in this special case. [ABSTRACT FROM PUBLISHER]

We bound the j-invariant of S-integral points on arbitrary modular curves over arbitrary number fields, in terms of the congruence group defining the curve, assuming a certain Runge condition is satisfied by our objects. We then apply our bounds to prove that for sufficiently large prime p, the points of with r < 1 are either cusps or complex multiplication points. This can be interpreted as the non-existence of quadratic elliptic -curves with higher prime-power degree. [ABSTRACT FROM PUBLISHER]

Let f be a self-dual Hecke–Maass cusp form for GL(3). We show that f is uniquely determined by central values of GL(2) twists of its L-function. More precisely, if g is another self-dual GL(3) Hecke–Maass cusp form such that for all holomorphic Hecke eigenforms for all k ≡ 0 (mod 4), then f = g. [ABSTRACT FROM PUBLISHER]

We examine the Fourier coefficients of modular forms in a canonical basis for the space of weakly holomorphic modular forms of weights 4, 6, 8, 10, and 14, and show that these coefficients are often highly divisible by the primes 2, 3, and 5. [ABSTRACT FROM AUTHOR]

The theta function of a positive definite even lattice of even rank generates a representation of on the group algebra of the discriminant form of the lattice. This representation goes back to Jacobi and is called Weil representation. We derive an explicit formula for the action in terms of the genus of the lattice. This generalizes classical results of Schoeneberg and Weil. We use the formula to calculate the lift from scalar-valued modular forms on Γ0(N) to modular forms for the Weil representation. We also show that the elements of the Mathieu group M23 correspond naturally to reflective automorphic products of singular weight, and we construct three generalized Kac–Moody superalgebras representing supersymmetric superstrings in dimensions 10, 6, and 4. [ABSTRACT FROM PUBLISHER]

Let K be a totally real field. We present an asymptotic formula for the number of Hilbert modular cusp forms f with given ramification at every place v of K. When v is an infinite place, this means specifying the weight of f at k, and when v is finite, this means specifying the restriction to inertia of the local Weil–Deligne representation attached to f at v. Our formula shows that with essentially finitely many exceptions, the cusp forms of K exhibit every possible sort of ramification behavior. [ABSTRACT FROM PUBLISHER]

Andrews recently introduced k-marked Durfee symbols, which are a generalization of partitions that are connected to moments of Dyson's rank statistic. He used these connections to find identities relating to their generating functions as well as to prove Ramanujan-type congruences for these objects and find relations between them. In this paper, we show that the hypergeometric generating functions for these objects are natural examples of quasimock theta functions, which are defined as the holomorphic parts of harmonic Maass forms and their derivatives. In particular, these generating functions may be viewed as analogs of Ramanujan's mock theta functions with arbitrarily high weight. We use the automorphic properties to prove the existence of infinitely many congruences for the Durfee symbols. Furthermore, we show that as k varies, the modularity of the k-marked Durfee symbols is precisely dictated by the case k = 2. Finally, we use this relation in order to prove the existence of general congruences for rank moments in terms of level one modular forms of bounded weight. [ABSTRACT FROM PUBLISHER]

For a quaternion algebra B over a totally real field F unramified at every finite place and ramified at most infinite places of F, we prove that the space of ℤ[1/E]-integral Hilbert modular forms of weight 2 and of level 1 is spanned over ℤ[1/E] by the theta series of the norm form of B. Here E=6d(F)∏ψ (the numerator of L(−1,ψ2)), where d(F) is the discriminant of F and ψ runs over all unramified characters of Gal(F¯/F). [ABSTRACT FROM AUTHOR]

We study the distribution of zeros of holomorphic modular forms. Assuming the generalized Riemann hypothesis, we show that the zeros of Hecke eigenforms for the modular group become equidistributed with respect to the hyperbolic measure on the modular domain as the weight grows. [ABSTRACT FROM AUTHOR]

Following Apéry's proof of the irrationality of ζ(3), Beukers found an elegant reinterpretation of Apéry's arguments using modular forms. We show how Beukers arguments can be adapted to a p-adic setting. In this context, certain functional equations arising from Eichler integrals are replaced by the notion of overconvergent p-adic modular forms, and the periods themselves arise not as coefficients of period polynomials, but as constant terms of p-adic Eisenstein series. We prove that the analogue of ζ(3) is irrational for p = 2 and 3, as well as the 2-adic analogue of Catalan's constant. [ABSTRACT FROM AUTHOR]

We associate p-adic modular forms of half-integral weight to nearly holomorphic modular forms of half-integral weight. A p-adic version of the Kohnen-Shimura map establishes a connection between p-adic modular forms of half-integral weight and p-adic modular forms of weight 2, discovered by Bruinier and Ono. It provides an answer, in the framework of p-adic modular forms, to a question by Borcherds about a relation between his isomorphism and the Kohnen-Shimura map. As applications of our construction, we prove interesting congruences for traces of singular moduli and for Borcherds exponents. In particular, we partially answer some questions posed by Ono. [ABSTRACT FROM AUTHOR]

We study the Witten-Reshetikhin-Turaev SU(2) invariant for the Seifert manifold with 4-singular fibers. We define the Eichler integrals of the modular forms with half-integral weight, and we show that the invariant is rewritten as a sum of the Eichler integrals. Using a nearly modular property of the Eichler integral, we give an exact asymptotic expansion of the WRT invariant in N → ∞. We reveal that the number of dominating terms, which is the number of the nonvanishing Eichler integrals in a limit τ→ N ∈ℤ, is related to that of lattice points inside 4-dimensional simplex, and we discuss a relationship with the irreducible representations of the fundamental group. [ABSTRACT FROM AUTHOR]

Matrix representations of Hecke operators on classical holomorphic cusp forms and corresponding period polynomials are well known. In this paper we define Hecke operators on period functions introduced recently by Lewis and Zagier and show how they are related to the Hecke operators on Maass cusp forms. Moreover, we give an explicit general compatibility criterion for formal sums of matrices to represent Hecke operators on period functions. An explicit example of such matrices with only nonnegative entries is constructed. [ABSTRACT FROM AUTHOR]

We derive an explicit upper bound for the number N(g,N,p) of systems of Hecke eigenvalues coming from Siegel modular forms mod p) of dimension g and level N relatively prime to p. In the special case of elliptic modular forms (g=1), our result agrees with recent work of G. Herrick. [ABSTRACT FROM AUTHOR]