Abstract: We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the “building complex” associated to level 3 structures at the prime 2. Finally, we note the existence of a number of connective models of the spectrum TMF(Γ0(3)).

Abstract: 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 Γ 0 (N) to modular forms for the Weil representation. We also show that the elements of the Mathieu group M 23 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: Denote by L(a, b) the free complex Lie algebra on the two generators a and b. For each integer m ≥ 0 there is a derivation 2m on L(a, b) that satisfies 2m([a, b]) = 0 and 2m(a) = ad(a) (b). In this paper we study the derivation subalgebra u generated by the 2m. In particular, we study the relations between the 2m and find that these relations are related to the period polynomials of modular forms.

Abstract: Let {FN} and {GM} be families of primitive automorphic L-functions for GLn(AQ) and GLm(AQ), respectively, such that, as N,M → ∞, the statistical behavior (1-level density) of the low-lying zeros of L-functions in FN (resp., GM ) agrees with that of the eigenvalues near 1 of matrices in G1 (resp., G2) as the size of the matrices tend to infinity, where each Gi is one of the classical compact groups (unitary U, symplectic Sp, or orthogonal O, SO(even), SO(odd)). Assuming that the convolved families of L-functions FN × GM are automorphic, we study their 1-level density. (We also study convolved families of the form f ×GM for a fixed f .) Under natural assumptions on the families (which hold in many cases) we can associate to each family L of L-functions a symmetry constant cL equal to 0 (resp., 1 or −1) if the corresponding low-lying zero statistics agree with those of the unitary (resp., symplectic or orthogonal) group. Our main result is that cF×G = cF · cG : the symmetry type of the convolved family is the product of the symmetry types of the two families. A similar statement holds for the convolved families f × GM . We provide examples built from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An interesting special case is to convolve two families of elliptic curves with positive rank. In this case the symmetry group of the convolution is independent of the ranks, in accordance with the general principle of multiplicativity of the symmetry constants (but the ranks persist, before taking the limit N, M →∞, as lower-order terms).

Abstract: For the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.

Abstract: In four dimensional string theories with N=4 and N=8 supersymmetries one can often define twisted index in a subspace of the moduli space which captures additional information on the partition function than the ones contained in the usual helicity trace index. We compute several such indices in type IIB string theory on K3 x T^2 and T^6, and find that they share many properties with the usual helicity trace index that captures the spectrum of quarter BPS states in N=4 supersymmetric string theories. In particular the partition function is a modular form of a subgroup of Sp(2,Z) and the jumps across the walls of marginal stability are controlled by the residues at the poles of the partition function. However for large charges the logarithm of this index grows as 1/n times the entropy of a black hole carrying the same charges where n is the order of the symmetry generator that is used to define the twisted index. We provide a macroscopic explanation of this phenomenon using quantum entropy function formalism. The leading saddle point corresponding to the attractor geometry fails to contribute to the twisted index, but a Z_n orbifold of the attractor geometry produces the desired contribution.

Abstract: In this paper we continue the program pioneered by D’Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ansatze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots.

Abstract: For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f = sum(a_n q^n) be a normalized new modular eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f, we define two Coleman maps with values in the Iwasawa algebra of Zp^* (after extending scalars to some extension of Qp). Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case a_p=0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture.

Abstract: We classify newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. Under the extended Riemann hypothesis for odd real Dirichlet characters, these newforms are finite in number. We produce tables for weights 3 and 4, where finiteness holds unconditionally.

Abstract: We provide evidence for the existence of a family of generalized Kac-Moody (GKM) superalgebras, N, whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Δk/2(Z), for (N, k) = (1, 10), (2, 6), (3, 4), and possibly (5, 2). The square of the automorphic form is the modular transform of the generating function of the degeneracy of CHL dyons in asymmetric N-orbifolds of the heterotic string compactified on T6. The new generalized Kac-Moody superalgebras all arise as different `automorphic corrections' of the same Lie algebra and are closely related to a generalized Kac-Moody superalgebra constructed by Gritsenko and Nikulin. The automorphic forms, Δk/2(Z), arise as additive lifts of Jacobi forms of (integral) weight k/2 and index 1/2. We note that the orbifolding acts on the imaginary simple roots of the unorbifolded GKM superalgebra, 1, leaving the real simple roots untouched. We anticipate that these superalgebras will play a role in understanding the `algebra of BPS states' in CHL compactifications.

Abstract: We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these L-functions. We also establish an equidistribution result for the Hecke eigenvalues weighted by these L-values

Abstract: We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of even and odd Coleman maps for normalised new forms of arbitrary weights and relate Pollack's $p$-adic $L$-functions to the even and odd Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the "main conjecture" for CM modular forms.

Abstract: In this paper a modular version of the classical Korovkin theorem in multivariate modular function spaces is obtained and applications to some multivariate discrete and integral operators, acting in Orlicz spaces, are given.

Abstract: We show that a cuspidal normalized Hecke eigenform g of level one and even weight is uniquely determined by the central values of the family of Rankin– Selberg L-functions \({L(s, f\otimes g)}\) , where f runs over the Hecke basis of the space of cusp forms of level one and weight k with k varying over an infinite set of even integers.

Abstract: We characterize the 2‐line of the p ‐local Adams‐Novikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p 5. We give a similar characterization of the 1‐line, reinterpreting some earlier work of A Baker and G Laures. These results are then used to deduce that, for ‘ a prime which generates Z p , the spectrum Q.‘/ detects the and families in the stable stems. 55Q45; 55Q51, 55N34, 11F33

Abstract: Theory of Modular Forms of One Variable: Group Action of the Modular Group The Gamma and Zeta Functions Zeta Functions of Modular Forms Dimension Formulae Bernoulli Identities and Applications Euler Sums and Recent Development Theory of Modular Forms of Several Variables: Theory of Modular Forms of Several Variables The Full Modular Group The Fourier Coefficients of Eisenstein Series Theory of Jacobi Forms Hecke Operators and Jacobi Forms Singular Modular Forms on the Exceptional Domain.

Abstract: We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.

Abstract: A variety of interesting connections with modular forms, mock theta functions and Rogers-Ramanujan type identities arise in consideration of partitions in which the smaller integers are repeated as summands more often than the larger summands. In particular, this concept leads to new interpretations of the Rogers-Selberg identities and Bailey’s modulus 9 identities.

Abstract: We study the possible weights of an irreducible 2-dimensional modular mod p representation of the absolute Galois group of F, where F is a totally real field which is totally ramified at p, and the representation is tamely ramified at the prime above p. In most cases we determine the precise list of possible weights; in the remaining cases we determine the possible weights up to a short and explicit list of exceptions.

Abstract: Despite the presence of many famous examples, the precise interplay between basic hypergeometric series and modular forms remains a mystery. We consider this problem for canonical spaces of weight 3/2 harmonic Maass forms. Using recent work of Zwegers, we exhibit forms that have the property that their holomorphic parts arise from Lerch-type series, which in turn may be formulated in terms of the Rogers{Fine basic hypergeometric series.

Abstract: We study the SU(2) Witten–Reshetikhin–Turaev invariants for Seifert manifolds associated with the Arnold 14 unimodal singularities. We show that the invariants are decomposed based on a value of linking pairings. Discussed also is a relationship with modular forms with weight-3/2.

Abstract: In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Particular care is taken in order to exclude nontrivial inner twists.

Abstract: We prove that the theta correspondence for the dual pair \(({\widetilde{\textup{SL}_2}}, \textit{PB}^\times)\), for B an indefinite quaternion algebra over ℚ, acting on modular forms of odd square-free level, preserves rationality and p-integrality in both directions. As a consequence, we deduce the rationality of certain period ratios of modular forms and even p-integrality of these ratios under the assumption that p does not divide a certain L-value. The rationality is applied to give a direct construction of isogenies between new quotients of Jacobians of Shimura curves, completely independent of Faltings’ isogeny theorem.

Abstract: We give the algebra of quasimodular forms a collection of Rankin-Cohen operators. These operators extend those defined by Cohen on modular forms and, as for modular forms, the first of them provide a Lie structure on quasimodular forms. They also satisfy a ``Leibniz rule'' for the usual derivation. Rankin-Cohen operators are useful for proving arithmetic identities. In particular we give an interpretation of the Chazy equation and explain why such an equation has to exist.

Abstract: We present solutions of the holomorphic anomaly equations for compact two-parameter Calabi-Yau manifolds which are hypersurfaces in weighted projective space. In particular we focus on K3-fibrations where due to heterotic type II duality the topological invariants in the fibre direction are encoded in certain modular forms. The formalism employed provides holomorphic expansions of topological string amplitudes everywhere in moduli space.

Abstract: Let L >= 3. Using the moduli interpretation, we define certain elliptic modular forms of level Gamma(L) over any field k where 6L is invertible and k contains the Lth roots of unity. These forms generate a graded algebra R_L, which, over C, is generated by the Eisenstein series of weight 1 on Gamma(L). The main result of this article is that, when k=C, the ring R_L contains all modular forms on Gamma(L) in weights >= 2. The proof combines algebraic and analytic techniques, including the action of Hecke operators and nonvanishing of L-functions. Our results give a systematic method to produce models for the modular curve X(L) defined over the Lth cyclotomic field, using only exact arithmetic in the L-torsion field of a single Q-rational elliptic curve E^0.

Abstract: We solve the diophantine equations x 4 + dy 2 = z p for d = 2 and d = 3 and any prime p > 349 and p > 131 respectively. The method consists in generalizing the ideas applied by Frey, Ribet and Wiles in the solution of Fermat’s Last Theorem, and by Ellenberg in the solution of the equation x 4 + y 2 = z p , and we use Q-curves, modular forms and inner twists. In principle our method can be applied to solve this type of equations for other values of d.