$\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$ by Will Sawin

“…To do this, we use Deligne's description of the etale cohomology of $\overline{M}_{1,n}$ and apply the theory of congruences between modular forms.…”
Get full text

Bounds for Extreme Zeros of Classical Orthogonal Polynomials Related to Birth and Death Processes by Saiful R. Mondal, Sourav Das

“…As a consequence, transition probabilities related to <i>G</i>-fractions and modular forms are derived. Results obtained in this work are new and several graphical representations and numerical computations are provided to validate the results.…”
Get full text

A 4 modular flavour model of quark mass hierarchies close to the fixed point τ = i∞ by S. T. Petcov, M. Tanimoto

“…We consider first a model in which the up-type and down-type quark mass matrices M u and M d involve modular forms of level 3 and weights 6, 4 and 2 and each depends on four constant parameters. …”
Get full text

Modularity of trianguline Galois representations by Rebecca Bellovin

“…The use of pseudorigid spaces lets us construct integral models of the trianguline varieties of [BHS17], [Che13] after bounding the slope, and we carry out a Taylor–Wiles patching argument for families of overconvergent modular forms. This permits us to construct a patched quaternionic eigenvariety and deduce our modularity results.…”
Get full text

APS η-invariant, path integrals, and mock modularity by Atish Dabholkar, Diksha Jain, Arnab Rudra

“…We show that the η-invariant for the elliptic genus of a finite cigar is related to quantum modular forms obtained from the completion of a mock Jacobi form which we compute from the noncompact path integral.…”
Get full text

Lepton masses and mixing from modular S4 symmetry by J.T. Penedo, S.T. Petcov

“…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.…”
Get full text

Zeta elements in depth 3 and the fundamental Lie algebra of the infinitesimal Tate curve by Brown, F

“…This paper draws connections between the double shuffle equations and structure of associators; universal mixed elliptic motives as defined by Hain and Matsumoto; and the Rankin-Selberg method for modular forms for $SL_2(\mathbb{Z})$. We write down explicit formulae for zeta elements $\sigma_{2n-1}$ (generators of the Tannaka Lie algebra of the category of mixed Tate motives over $\mathbb{Z}$) in depths up to four, give applications to the Broadhurst-Kreimer conjecture, and completely solve the double shuffle equations for multiple zeta values in depths two and three…”

On the Spezialschar of Maass by Bernhard Heim

“…Let 𝑀𝑘(𝑛) be the space of Siegel modular forms of degree 𝑛 and even weight 𝑘. In this paper firstly a certain subspace Spez(𝑀𝑘(2𝑛)), the Spezialschar of 𝑀𝑘(2𝑛), is introduced. …”
Get full text

KdV charges and the generalized torus partition sum in TT‾ deformation by Meseret Asrat

“…We find that the one–point functions decompose into a direct sum of two non–holomorphic modular forms. We also obtain a general differential equation that the KdV generalized torus partition sum satisfies. …”
Get full text

ZETA ELEMENTS IN DEPTH 3 AND THE FUNDAMENTAL LIE ALGEBRA OF THE INFINITESIMAL TATE CURVE by FRANCIS BROWN

“…This paper draws connections between the double shuffle equations and structure of associators; Hain and Matsumoto’s universal mixed elliptic motives; and the Rankin–Selberg method for modular forms for $\text{SL}_{2}(\mathbb{Z})$ . …”
Get full text

Vafa-Witten invariants for projective surfaces I: stable case by Tanaka, Y, Thomas, RP

“…Calculations of these on surfaces with positive canonical bundle recover the first terms of modular forms predicted by Vafa and Witten.…”

Modular properties of surface operators in N $$ \mathcal{N} $$ = 2 SU(2) SQCD by Sourav Ballav, Renjan Rajan John

“…We then use the constraints imposed by S-duality to resum the instanton contributions to the twisted superpotential into elliptic functions and (quasi-) modular forms. The resummed results match what one would obtain from the description of surface operators as the insertion of a degenerate operator in a spherical conformal block in Liouville CFT.…”
Get full text

Vafa-Witten invariants for projective surfaces II: semistable case by Tanaka, Y, Thomas, RP

“…</p> <br/> <p>For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten.…”

Exact expressions for n-point maximal U(1) Y -violating integrated correlators in SU(N) N $$ \mathcal{N} $$ = 4 SYM by Daniele Dorigoni, Michael B. Green, Congkao Wen

“…The integrated correlators are functions of N and τ = θ/(2π) + 4πi/ g YM 2 $$ {g}_{YM}^2 $$ , and are expressed as two-dimensional lattice sums that are modular forms with holomorphic and anti-holomorphic weights (w, −w) where w = n − 4. …”
Get full text

Elliptic genera and characteristic q-series of superconformal field theory by L. Bonora, A.A. Bytsenko, M. Chaichian

“…We analyze the characteristic series, the KO series and the series associated with the Witten genus, and their analytic forms as the q-analogs of classical special functions (in particular q-analog of the beta integral and the gamma function). q-Series admit an analytic interpretation in terms of the spectral Ruelle functions, and their relations with appropriate elliptic modular forms can be described. We show that there is a deep correspondence between the characteristic series of the Witten genus and KO characteristic series, on one side, and the denominator identities and characters of N=2 superconformal algebras, and the affine Lie (super)algebras on the other. …”
Get full text

A minimal modular invariant neutrino model by Gui-Jun Ding, Xiang-Gan Liu, Chang-Yuan Yao

“…We also study the soft supersymmetry breaking terms due to the modulus F-term in this minimal model, which are constrained to be the non-holomorphic modular forms. The radiative lepton flavor violation process μ → eγ is discussed.…”
Get full text

CP symmetry and symplectic modular invariance by Gui-Jun Ding, Ferruccio Feruglio, Xiang-Gan Liu

“…We discuss the transformation properties of moduli, matter multiplets and modular forms in the Siegel upper half plane, as well as in invariant subspaces. …”
Get full text

Depth-graded motivic multiple zeta values by Brown, F

“…We study the depth filtration on multiple zeta values, on the motivic Galois group of mixed Tate motives over Z and on the Grothendieck–Teichmüller group, and its relation to modular forms. Using period polynomials for cusp forms for SL2(Z), we construct an explicit Lie algebra of solutions to the linearized double shuffle equations, which gives a conjectural description of all identities between multiple zeta values modulo ζ(2) and modulo lower depth. …”

The unequal mass sunrise integral expressed through iterated integrals on M‾1,3 by Christian Bogner, Stefan Müller-Stach, Stefan Weinzierl

“…In the equal mass case our result reduces to iterated integrals of modular forms.…”
Get full text

Heterotic global anomalies and torsion Witten index by Kazuya Yonekura

“…The torsion index gives some of the invariants of SQFTs suggested by topological modular forms, and is expected to be zero for the cases that are relevant to actual heterotic string theories.…”
Get full text