Modular Forms Research Articles

On effective multiplicity one for modular forms of half-integral weight

On effective multiplicity one for modular forms of half-integral weight

On modular forms and Domb numbers

On modular forms and Domb numbers

A Quadratic Curve Analogue of the Taniyama-Shimura Conjecture

For quadratic curves over finite fileds, the number of solutions, which is governed by an analogue of the Mordell-Weil group, is expressed with the Legendre symbol of a coefficient of quadratic curves. Focusing on the number of solutions, a quadratic curve analogue of the modular form in the Taniyama-Shimura conjecture is proposed. This modular form yields the Gaussian sum and also possesses some modular transformation structure.

Lucas congruences using modular forms

AbstractIn this work, we prove that many Apéry‐like sequences arising from modular forms satisfy the Lucas congruences modulo any prime. As an implication, we completely affirm four conjectural Lucas congruences that were recently posed by S. Cooper and reinterpret a number of known results.

About Filtration in a Geophysical Bridge with a Seepage Site

Based on the representation of a modular elliptic function in the form of a combination of simple algebraic formulas that conformally map the area of the velocity hodograph (curved triangle) onto a half-plane, a direct definition of filtration rates in a geophysical bridge is given. For the first time, a family of isotope lines of equal filtration rates was constructed for the inner area of the bridge in the absence of water in the downstream. For special cases, the values of the proposed formulas almost completely coincide with the hydromechanical solution of Masket M. (for 4 cases) and the numerical calculations of Khairullin Z.E. (for 2 cases). The well-known Nelson-Skornyakov F.B. formula for the output filtration rates through the lower face of the jumper, adopted by analogy with the flow outlet from the base of the flatbed into the horizontal drainage, gives an underestimation of the results by up to 45%, and therefore cannot be recommended for practical use.

Hilbert Poincaré series and kernels for products of L-functions

We study Hilbert Poincaré series associated to general seed functions and construct Cohen’s kernels and double Eisenstein series as series of Hilbert Poincaré series. Then we calculate the Rankin–Cohen brackets of Hilbert Poincaré series and Hilbert modular forms and extend Zagier’s kernel formula to totally real number fields. Finally, we show that the Rankin–Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant.

Simple vertex algebras arising from congruence subgroups

For any congruence subgroup Γ, we consider the vertex algebra of Γ-invariant global sections of chiral de Rham complex on the upper half plane that are meromorphic at the cusps. We give a description of the linear structure of the Γ-invariant vertex algebra by exhibiting a linear basis determined by meromorphic modular forms, and generalize the Rankin-Cohen bracket of modular forms to meromorphic modular forms. We also show that the Γ-invariant vertex algebra is simple.

Addressing imperfect symmetry: A novel symmetry-learning actor-critic extension

Symmetry, a fundamental concept to understand our environment, often oversimplifies reality from a mathematical perspective. Humans are a prime example, deviating from perfect symmetry in terms of appearance and cognitive biases (e.g. having a dominant hand). Nevertheless, our brain can easily overcome these imperfections and efficiently adapt to symmetrical tasks. The driving motivation behind this work lies in capturing this ability through reinforcement learning. To this end, we introduce Adaptive Symmetry Learning (ASL) — a model-minimization actor-critic extension that addresses incomplete or inexact symmetry descriptions by adapting itself during the learning process. ASL consists of a symmetry fitting component and a modular loss function that enforces a common symmetric relation across all states while adapting to the learned policy. The performance of ASL is compared to existing symmetry-enhanced methods in a case study involving a four-legged ant model for multidirectional locomotion tasks. The results show that ASL can recover from large perturbations and generalize knowledge to hidden symmetric states. It achieves comparable or better performance than alternative methods in most scenarios, making it a valuable approach for leveraging model symmetry while compensating for inherent perturbations.

L${L}$‐functions of Kloosterman sheaves

AbstractIn this article, we study a family of motives associated with the symmetric power of Kloosterman sheaves constructed by Fresán, Sabbah, and Yu. They demonstrated that for , the ‐functions of extend meromorphically to and satisfy the functional equations conjectured by Broadhurst and Roberts. Our work aims to extend these results to the ‐functions of some of the motives , with , as well as other related two‐dimensional motives. In particular, we prove several conjectures of Evans type, which relate moments of Kloosterman sheaves and Fourier coefficients of modular forms.

Cpvlib: A comprehensive open-source tool for modeling CPV systems

The design and simulation of concentrator photovoltaic (CPV) systems necessitate precise modeling tools, for which some commercial and open-source options exist. However, when new technologies or applications need to be modeled, they can present some limitations: lack of documentation transparency and inability to extend existing models, or little flexibility to do it. For instance, the novel hybrid CPV/flat-plate module, conceived by Insolight and developed within the HIPERION project, required the ability to model integrated tracking and dual use of incident irradiance, which was not possible with existing tools. Addressing these issues, cpvlib is introduced as a comprehensive, open-source tool offering modular and adaptable functionalities for CPV-based systems, built as an extension of the popular pvlib python library.cpvlib's design enables the simulation of various CPV-based configurations, incorporating advanced architectures such as integrated tracking and hybrid CPV-flat plate modules. The library uses PVSyst's utilization factors to model deviations from the single-diode model, accounting for spectral and thermal effects. Its class structure leverages object-oriented programming principles, ensuring ease of use and extension.The validation of cpvlib is carried out through the modeling and long-term monitoring of Insolight's hybrid Si/III-V translucent planar micro-tracking modules, achieving a root mean square error of 3.5 % in case of Si cells and 2.7 % for III-V CPV cells. The tool accounts for complex behaviors like air mass impact on CPV performance, angle of incidence limits, and light spillage. The annual energy yield for a hybrid module is computed using typical meteorological year data, showcasing cpvlib's practical application.

Facile Mechanochemical Functionalization of Hydrophobic Substrates for Single-Walled Carbon Nanotube Based Optical Reporters of Hydrolase Activity.

Single walled carbon nanotubes (SWCNT) have recently been demonstrated as modular, near-infrared (nIR) probes for reporting hydrolase activity; however, these have been limited to naturally amphipathic substrate targets used to noncovalently functionalize the hydrophobic nanoparticles. Many relevant substrate targets are hydrophobic (such as recalcitrant biomass) and pose a challenge for modular functionalization. In this work, a facile mechanochemistry approach was used to couple insoluble substrates, such as lignin, to SWCNT using l-lysine amino acid as a linker and tip sonication as the mechanochemical energy source. The proposed coupling mechanism is ion pairing between the lysine amines and lignin carboxylic acids, as evidenced by FTIR, NMR, SEM, and elemental analyses. The limits of detection for the lignin-lysine-SWCNT (LLS) probe were established using commercial enzymes and found to be 0.25 ppm (volume basis) of the formulated product. Real-world use of the LLS probes was shown by evaluating soil hydrolase activities of soil samples gathered from different corn root proximal locations and soil types. Additionally, the probes were used to determine the effect of storage temperature on the measured enzyme response. The modularity of this mechanochemical functionalization approach is demonstrated with other substrates such as zein and 9-anthracenecarboxylic acid, which further corroborate the mechanochemical mechanism.

Rankin–Cohen type differential operators on Hermitian modular forms

Rankin–Cohen type differential operators on Hermitian modular forms

Convolution identities for divisor sums and modular forms

We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums [Formula: see text]where [Formula: see text] is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to [Formula: see text], do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms.

An end-to-end model of active electrosensation.

1Weakly electric fish localize and identify objects by sensing distortions in a self-generated electric field. Fish can determine the resistance and capacitance of an object, for example, even though the field distortions being sensed are small and highly-dependent on object distance and size. Here we construct a model of the responses of the fish's electroreceptors on the basis of experimental data, and we develop a model of the electric fields generated by the fish and the distortions due to objects of different resistances and capacitances. This provides us with an accurate and efficient method for generating large artificial data sets simulating fish interacting with a wide variety of objects. Using these sets, we train an artificial neural network (ANN), representing brain areas downstream of electroreceptors, to extract the 3D location, size, and electrical properties of objects. The model performs best if the ANN operates in two stages: first estimating object distance and size and then using this information to extract electrical properties. This suggests a specific form of modularity in the electrosensory system that can be tested experimentally and highlights the potential of end-to-end modeling for studies of sensory processing.

Double Eisenstein series and modular forms of level 4

Double Eisenstein series and modular forms of level 4

Developing Porous Protein Cage Nanoparticles as Cargo-Loadable and Ligand-Displayable Modular Delivery Nanoplatforms.

Protein cage nanoparticles, self-assembled from protein subunits, provide distinct exterior and interior spaces and can carry diagnostic and/or therapeutic cargo agents through chemical conjugation, in vitro disassembly/reassembly process, or assembly-mediated encapsulation. Here, we developed porous SpyCatcher-mi3 (SC-mi3) as modular delivery nanoplatforms, capable of loading cargos through pores and displaying targeting ligands using SpyCatchers (SC) as anchors for SpyTagged (ST) ligands. Fluorescent dyes (F5M and A647) and a pH-sensitive prodrug (Aldox) were conjugated to the interior surface cysteines of SC-mi3, forming F5M@SC-mi3, A647@SC-mi3, and Aldox@SC-mi3. Subsequently, EGFR-binding affibody molecules (EGFRAfb) were displayed on the exterior surface of F5M@SC-mi3 and Aldox@SC-mi3 using the SC/ST protein ligation system, forming F5M@mi3/EGFRAfb and Aldox@mi3/EGFRAfb, respectively. F5M@mi3/EGFRAfb selectively bound to EGFR-overexpressing MDA-MB-468 cells, visualizing the target cancer cells, while Aldox@mi3/EGFRAfb selectively delivered doxorubicin, leading to target-specific cancer cell death. To encapsulate large proteins within SC-mi3, biotins were initially conjugated to the interior surface (BPM@SC-mi3) and mSA2-fused protein cargo molecules (mSA2-HaloTag and mSA2-yCD) were successfully introduced through the pores and securely encapsulated, forming TMR-H@SC-mi3 and yCD@SC-mi3, respectively. Subsequent display of EGFRAfb on their surface allowed the visualization of target cancer cells using fluorescent HaloTag ligand labeling and facilitated the killing of target cancer cells by converting the prodrug 5-FC to the cytotoxic drug 5-FU. Modular functionalization of the two distinct spaces in porous SC-mi3 may offer opportunities for developing target-specific functional cargo-delivery nanoplatforms in biomedical fields.

Windings of Prime Geodesics

Abstract The winding of a closed oriented geodesic around the cusp of the modular orbifold is computed by the Rademacher symbol, a classical function from the theory of modular forms. In this article, we introduce a new construction of winding numbers to record the winding of closed oriented geodesics about a prescribed cusp of a general cusped hyperbolic orbifold. For various arithmetic families of surfaces, this winding number can again be expressed by a Rademacher symbol, and access to the spectral theory of automorphic forms yields statistical results on the distribution of closed (primitive) oriented geodesics with respect to their winding.

Modular forms on $\Gamma_0(2)$ with all zeros on a specific geodesic

Modular forms on $\Gamma_0(2)$ with all zeros on a specific geodesic

On Shafarevich-Tate groups and analytic ranks in families of modular forms, I. Hida families

Let $f$ be a newform of weight $2$, square-free level and trivial character, let $A_f$ be the abelian variety attached to $f$ and for every good ordinary prime $p$ for $f$ let $\boldsymbol f^{(p)}$ be the $p$-adic Hida family through $f$. We prove that, for all but finitely many primes $p$ as above, if $A_f$ is an elliptic curve such that $A_f(\mathbb Q)$ has rank $1$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\mathbb Q$ is finite then all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite ($p$-primary) Shafarevich-Tate group and $1$-dimensional image of the relevant $p$-adic \'etale Abel-Jacobi map. Analogous results are obtained also in the rank $0$ case. As a second contribution, with no restriction on the dimension of $A_f$ but assuming the non-degeneracy of certain height pairings \`a la Gillet-Soul\'e between Heegner cycles, we show that if $f$ has analytic rank $1$ then, for all but finitely many $p$, all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have analytic rank $1$. This result provides some evidence in rank $1$ and weight larger than $2$ for a conjecture of Greenberg predicting that the analytic ranks of even weight modular forms in a Hida family should be as small as allowed by the functional equation, with at most finitely many exceptions.

The light we can see: extracting black holes from weak Jacobi forms

We quantify how constraints on light states affect the asymptotic growth of heavy states in weak Jacobi forms. The constraints we consider are sparseness conditions on the Fourier coefficients of these forms, which are necessary to interpret them as gravitational path integrals. Using crossing kernels, we extract the leading and subleading behavior of these coefficients and show that the leading Cardy-like growth is robust in a wide regime of validity. On the other hand, we find that subleading corrections are sensitive to the constraints placed on the light states, and we quantify their imprint on the asymptotic growth of states. Our approach is tested against the generating function of symmetric product orbifolds, where we provide new insights into the factors contributing to the asymptotic growth of their Fourier coefficients. Finally, we use our methods to revisit the UV/IR connection that relates black hole microstate counting to modular forms. We provide a microscopic interpretation of the logarithmic corrections to the entropy of BPS black holes in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2, 4 ungauged supergravity in four and five dimensions, and tie it to consistency conditions in AdS3/CFT2.