Modular Forms Of Weight Research Articles

Modular $S_3$-invariant flavor model in SU(5) grand unified theory

Abstract We present a flavor model with $S_3$ modular invariance in the framework of SU(5) grand unified theory (GUT). The $S_3$ modular forms of weights $2$ and $4$ give the quark and lepton mass matrices with a common complex parameter, the modulus $\tau$. The GUT relation of down-type quarks and charged leptons is imposed by the vacuum expectation value (VEV) of the adjoint 24-dimensional Higgs multiplet in addition to the VEVs of $5$ and $\bar 5$ Higgs multiplets of SU(5). The observed Cabibbo–Kobayashi–Maskawa and Pontecorvo–Maki–Nakagawa–Sakata mixing parameters as well as the mass eigenvalues are reproduced properly. We discuss the leptonic charge–parity phase and the effective mass of the neutrinoless double beta decay with the sum of neutrino masses.

$$\Lambda $$-adic families of Jacobi forms

We show that Hida’s families of p-adic elliptic modular forms generalize to p-adic families of Jacobi forms. We also construct p-adic versions of theta lifts from elliptic modular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hida and Stevens on the related case of half-integral weight modular forms.

Romik's conjecture for the Jacobi theta function

Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point i. He then conjectured that the Taylor coefficients d(n) either vanish or are periodic modulo any prime p; this was proved by the combined efforts of Scherer and Guerzhoy-Mertens-Rolen, with the latter trio considering arbitrary half integral weight modular forms. We refine previous work for p≡1(mod4) by displaying a concise algebraic relation between d(n+p−12) and d(n) related to the p-adic factorial, from which we can deduce periodicity with an effective period.

Rankin-Eisenstein classes for modular forms

In this paper we make a systematic study of certain motivic cohomology classes ("Rankin-Eisenstein classes") attached to the Rankin--Selberg convolution of two modular forms of weight $\ge 2$. The main result is the computation of the $p$-adic syntomic regulators of these classes. As a consequence we prove many cases of the Perrin-Riou conjecture for Rankin--Selberg convolutions of cusp forms.

Bounds for twisted symmetric square L-functions via half-integral weight periods

Abstract We establish the first moment bound $$\begin{align*}\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_\varepsilon p^{5/4+\varepsilon} \end{align*}$$ for triple product L-functions, where $\Psi $ is a fixed Hecke–Maass form on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and $\varphi $ runs over the Hecke–Maass newforms on $\Gamma _0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke–Maass newforms on $\Gamma _0(p) \backslash \mathbb {H}$ of bounded eigenvalue have very uniformly distributed mass after pushforward to $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ . Our main result turns out to be closely related to estimates such as $$\begin{align*}\sum_{|n| < p} L(\Psi \otimes \chi_{n p},\tfrac{1}{2}) \ll p, \end{align*}$$ where the sum is over those n for which $n p$ is a fundamental discriminant and $\chi _{n p}$ denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke–Iwaniec.

On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

Graded rings of modular forms of rational weights

In a previous paper, for any odd integer $$N\ge 5$$ we constructed $$(N-1)/2$$ modular forms of $$\varGamma (N)$$ of rational weight $$(N-3)/2N$$ and proved that the graded rings of modular forms of weight $$\ell (N-3)/2N$$ ($$\ell \in {\mathbb {Z}}_{\ge 0}$$) are generated by our forms for $$N=5$$, 7, 9. The proof was given by a direct calculation of the structure of the ring. In this paper, we generalize the result to cases when $$N=11$$ and 13 by using Castelnuovo–Mumford criterion on normal generation and Fujita criterion on relations of sections of invertible sheaves. For this purpose, it is needed to handle modular forms of small weight where Riemann Roch theorem has cohomological obstruction. Both rings for $$N=11$$ and 13 are generated by 5 and 6 generators with 15 and 35 concrete fundamental relations, respectively. These relations also give equations of the corresponding modular varieties. We will show that the similar claim does not hold for $$N=15$$ and $$N=23$$. We also give remarks on relations of some theta constants and further problems.

Proof of a Conjecture of Farkas and Kra

In this paper we prove a conjecture of Farkas and Kra, which is a modular equation involving a half sum of certain modular form of weight $1$ for congruence subgroup $\Gamma_1(k)$ with any prime $k$. We prove that their conjecture holds for all odd integers $k \geq 3$. A new modular equation of Farkas and Kra type is also established.

Half-integral weight modular forms and modular forms for Weil representations

We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $$\Gamma _0(4)$$ with Kohnen’s plus condition and modular forms for the Weil representation associated to the lattice with Gram matrix (2). With such an isomorphism, we prove the Zagier duality and express the Borcherds lifts in the case of $$\text {O}(2,1)$$ explicitly.

Neutrino masses and mixing from double covering of finite modular groups

We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level N can be arranged into irreducible representations of the homogeneous finite modular group {Gamma}_N^{prime } which is the double covering of ΓN. The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of {Gamma}_3^{prime } ≅ T′. The modular forms of weights 2, 3, 4, 5 and 6 are presented. We build a model of lepton masses and mixing based on T′ modular symmetry.

Newforms of half-integral weight: the minus space of Sk+1/2(Γ0(8M))

We compute generators and relations for a certain 2-adic Hecke algebra of level 8 associated with the double cover of SL2 and a 2-adic Hecke algebra of level 4 associated with PGL2. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the 2-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight k + 1/2 and level 8M where M is odd and square-free. Using these operators and our previous results on half-integral weight forms of level 4M we define a subspace of the space of half-integral weight forms as a common -1 eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace, which we call the minus space, is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight 2k and level 4M. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.

Bilateral series and Ramanujan radial limits of mock (false) theta functions

Ramanujan gave a list of seventeen functions which he called mock theta functions. For one of the third-order mock theta functions [Formula: see text], he claimed that as [Formula: see text] approaches an even order [Formula: see text] root of unity [Formula: see text], then [Formula: see text] He also pointed at the existence of similar properties for other mock theta functions. Recently, [J. Bajpai, S. Kimport, J. Liang, D. Ma and J. Ricci, Bilateral series and Ramanujan’s radial limits, Proc. Amer. Math. Soc. 143(2) (2014) 479–492] presented some similar Ramanujan radial limits of the fifth-order mock theta functions and their associated bilateral series are modular forms. In this paper, by using the substitution [Formula: see text] in the Ramanujan’s mock theta functions, some associated false theta functions in the sense of Rogers are obtained. Such functions can be regarded as Eichler integral of the vector-valued modular forms of weight [Formula: see text]. We find two associated bilateral series of the false theta functions with respect to the fifth-order mock theta functions are special modular forms. Furthermore, we explore that the other two associated bilateral series of the false theta functions with respect to the third-order mock theta functions are mock modular forms. As an application, the associated Ramanujan radial limits of the false theta functions are constructed.

Modular S4 models of lepton masses and mixing

We investigate models of charged lepton and neutrino masses and lepton mixing based on broken modular symmetry. The matter fields in these models are assumed to transform in irreducible representations of the finite modular group Γ4 ≃ S4. We analyse the minimal scenario in which the only source of symmetry breaking is the vacuum expectation value of the modulus field. In this scenario there is no need to introduce flavon fields. Using the basis for the lowest weight modular forms found earlier, we build minimal phenomenologically viable models in which the neutrino masses are generated via the type I seesaw mechanism. While successfully accommodating charged lepton masses, neutrino mixing angles and mass-squared differences, these models predict the values of the lightest neutrino mass (i.e., the absolute neutrino mass scale), of the Dirac and Majorana CP violation (CPV) phases, as well as specific correlations between the values of the atmospheric neutrino mixing parameter sin2θ23 and i) the Dirac CPV phase δ, ii) the sum of the neutrino masses, and iii) the effective Majorana mass in neutrinoless double beta decay. We consider also the case of residual symmetries ℤ3ST and ℤ2S respectively in the charged lepton and neutrino sectors, corresponding to specific vacuum expectation values of the modulus.

On a q-deformation of modular forms

There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work, such partial sums are related to the radial asymptotics of infinite q-hypergeometric sums at roots of unity. Here we combine the two features to construct a hypergeometric q-deformation of two CM modular forms of weight 3 and discuss the corresponding q-congruences.

Short proof of Rademacher’s formula for partitions

This note rederives a formula for r-color partitions, $$1\le r \le 24$$ , including Rademacher’s celebrated result for ordinary partitions, from the duality between modular forms of weights $${-r}/{2}$$ and $${2+r}/{2}$$ .

On the structure of Selmer and Shafarevich–Tate groups of even weight modular forms

On the structure of Selmer and Shafarevich–Tate groups of even weight modular forms

Modular forms in the spectral action of Bianchi IX gravitational instantons

We prove a modularity property for the heat kernel and the Seeley-deWitt coefficients of the heat kernel expansion for the Dirac-Laplacian on the Bianchi IX gravitational instantons. We prove, via an isospectrality result for the Dirac operators, that each term in the expansion is a vector-valued modular form, with an associated ordinary (meromorphic) modular form of weight 2. We discuss explicit examples related to well known modular forms. Our results show the existence of arithmetic structures in Euclidean gravity models based on the spectral action functional.

A variant of multiplicity one theorems for half-integral weight modular forms

We show that signs of Fourier coefficients, on certain sub-families, determine the half-integral weight cuspidal eigenform uniquely, up to a positive constant. We also study sign change results for the product of the Fourier coefficients of two distinct half-integral weight eigenforms.

Lepton masses and mixing from modular S4 symmetry

We study models of lepton masses and mixing based on broken modular invariance. We consider invariance under the finite modular group Γ4≃S4 and focus on the minimal scenario where the expectation value of the modulus is the only source of symmetry breaking, such that no flavons need to be introduced. After constructing a basis for the lowest weight modular forms, we build two minimal models, one of which successfully accommodates charged lepton masses and neutrino oscillation data, while predicting the values of the Dirac and Majorana CPV phases.

The p-adic L-function for half-integral weight modular forms

The p-adic L-function for modular forms of integral weight is well-known. For certain weights the p-adic L-function for modular forms of half-integral weight is also known to exist, via a correspondence, established by Shimura, between them and forms of integral weight. However, we construct it here without any recourse to the Shimura correspondence, allowing us to establish its existence for all weights, including those exempt from the Shimura correspondence. We do this by employing the Rankin–Selberg method, and proving explicit p-adic congruences in the resultant Rankin–Selberg expression.