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Period-like polynomials for L-series associated with half-integral weight cusp forms (2024)
Journal Article
Branch, J., Diamantis, N., Raji, W., & Rolen, L. (2024). Period-like polynomials for L-series associated with half-integral weight cusp forms. Research in the Mathematical Sciences, 11(3), Article 44. https://doi.org/10.1007/s40687-024-00455-w

Given the L-series of a half-integral weight cusp form, we construct polynomials behaving similarly to the classical period polynomial of an integral weight cusp form. We also define a lift of half-integral weight cusp forms to integral weight cusp f... Read More about Period-like polynomials for L-series associated with half-integral weight cusp forms.

L-values of harmonic Maass forms (2024)
Journal Article
Diamantis, N., & Rolen, L. (2024). L-values of harmonic Maass forms. Transactions of the American Mathematical Society, 377, 3905-3926. https://doi.org/10.1090/tran/9045

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the "central L-value" of the modular j-invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this "L-value" as the value o... Read More about L-values of harmonic Maass forms.

Analogues of the Bol operator for half-integral weight weakly holomorphic modular forms (2023)
Journal Article
Diamantis, N., Lee, M., & Rolen, L. (2023). Analogues of the Bol operator for half-integral weight weakly holomorphic modular forms. Proceedings of the American Mathematical Society, 152, 37-51. https://doi.org/10.1090/proc/16435

We define an analogue of the Bol operator on spaces of weakly holomorphic modular forms of half-integral weight. We establish its main properties and relation with other objects.

L-Series of Harmonic Maass Forms and a Summation Formula for Harmonic Lifts (2022)
Journal Article
Diamantis, N., Lee, M., Raji, W., & Rolen, L. (2023). L-Series of Harmonic Maass Forms and a Summation Formula for Harmonic Lifts. International Mathematics Research Notices, 2023(18), 15729-15765. https://doi.org/10.1093/imrn/rnac310

We introduce an L-series associated with harmonic Maass forms and prove their functional equations. We establish converse theorems for these L-series and, as an application, we formulate and prove a summation formula for the holomorphic part of a har... Read More about L-Series of Harmonic Maass Forms and a Summation Formula for Harmonic Lifts.

Derivatives of L-series of weakly holomorphic cusp forms (2022)
Journal Article
Diamantis, N., & Stromberg, F. (2022). Derivatives of L-series of weakly holomorphic cusp forms. Research in the Mathematical Sciences, 9(4), Article 64. https://doi.org/10.1007/s40687-022-00363-x

Based on the theory of L-series associated with weakly holomorphic modular forms in Diamantis et al. (L-series of harmonic Maass forms and a summation formula for harmonic lifts. arXiv:2107.12366), we derive explicit formulas for central values of de... Read More about Derivatives of L-series of weakly holomorphic cusp forms.

Modular iterated integrals associated with cusp forms (2021)
Journal Article
Diamantis, N. (2022). Modular iterated integrals associated with cusp forms. Forum Mathematicum, 34(1), 157-174. https://doi.org/10.1515/forum-2021-0224

We construct an explicit family of modular iterated integrals which involves cusp forms. This leads to a new method of producing modular invariant functions based on iterated integrals of modular forms. The construction will be based on an extension... Read More about Modular iterated integrals associated with cusp forms.

Kernels of L-functions and shifted convolutions (2020)
Journal Article
Diamantis, N. (2020). Kernels of L-functions and shifted convolutions. Proceedings of the American Mathematical Society, 148, 5059-5070. https://doi.org/10.1090/proc/15182

We give a characterisation of the field into which quotients of values of L-functions associated to a cusp form belong. The construction involves shifted convolution series of divisor sums and to establish it we combine parts of F. Brown's technique... Read More about Kernels of L-functions and shifted convolutions.

Period functions associated to real-analytic modular forms (2020)
Journal Article
Diamantis, N., & Drewitt, J. (2020). Period functions associated to real-analytic modular forms. Research in the Mathematical Sciences, 7(3), Article 21. https://doi.org/10.1007/s40687-020-00221-8

We define L-functions for the class of real-analytic modular forms recently introduced by F. Brown. We establish their main properties and construct the analogue of period polynomial in cases of special interest, including those of modular iterated i... Read More about Period functions associated to real-analytic modular forms.

Additive twists and a conjecture by Mazur, Rubin and Stein (2019)
Journal Article
Diamantis, N., Hoffstein, J., Kıral, M., & Lee, M. (2020). Additive twists and a conjecture by Mazur, Rubin and Stein. Journal of Number Theory, 209, 1-36. https://doi.org/10.1016/j.jnt.2019.11.016

In this paper, a conjecture of Mazur, Rubin and Stein concerning certain averages of modular symbols is proved. To cover levels that are important for elliptic curves, namely those that are not square-free, we establish results about L-functions with... Read More about Additive twists and a conjecture by Mazur, Rubin and Stein.

Holomorphic automorphic forms and cohomology (2018)
Journal Article
Bruggeman, R., Choie, Y. J., & Diamantis, N. (2018). Holomorphic automorphic forms and cohomology. Memoirs of the American Mathematical Society, 253(1212), 1-182. https://doi.org/10.1090/memo/1212

© 2018 by the American Mathematical Society. All rights reserved. We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this corresp... Read More about Holomorphic automorphic forms and cohomology.

Period polynomials, derivatives of L-functions, and zerosof polynomials (2018)
Journal Article
Diamantis, N., & Rolen, L. (2018). Period polynomials, derivatives of L-functions, and zerosof polynomials. Research in the Mathematical Sciences, 5, Article 9. https://doi.org/10.1007/s40687-018-0126-4

Period polynomials have long been fruitful tools for the study of values of L-functions in the context of major outstanding conjectures. In this paper, we survey some facets of this study from the perspective of Eichler cohomology. We discuss ways to... Read More about Period polynomials, derivatives of L-functions, and zerosof polynomials.

Regularized inner products and errors of modularity (2016)
Journal Article
Bringmann, K., Diamantis, N., & Ehlen, S. (2017). Regularized inner products and errors of modularity. International Mathematics Research Notices, 2017(24), 7420-7458. https://doi.org/10.1093/imrn/rnw225

© The Author(s) 2016. We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth on regularized in... Read More about Regularized inner products and errors of modularity.

Fourier coefficients of Eisenstein series formed with modular symbols and their spectral decomposition (2016)
Journal Article
Bruggeman, R., & Diamantis, N. (2016). Fourier coefficients of Eisenstein series formed with modular symbols and their spectral decomposition. Journal of Number Theory, 167, https://doi.org/10.1016/j.jnt.2016.03.009

The Fourier coefficient of a second order Eisenstein series is described as a shifted convolution sum. This description is used to obtain the spectral decomposition of and estimates for the shifted convolution sum.

A correspondence of modular forms and applications to values of L-series (2015)
Journal Article
Diamantis, N., Neururer, M., & Strömberg, F. (in press). A correspondence of modular forms and applications to values of L-series. Research in Number Theory, 1(27), https://doi.org/10.1007/s40993-015-0029-z

An interpretation of the Rogers–Zudilin approach to the Boyd conjectures is established. This is based on a correspondence of modular forms which is of independent interest. We use the reinterpretation for two applications to values of L-series and v... Read More about A correspondence of modular forms and applications to values of L-series.

A converse theorem for double Dirichlet series and Shintani zeta functions (2014)
Journal Article
Diamantis, N., & Goldfeld, D. (2014). A converse theorem for double Dirichlet series and Shintani zeta functions. https://doi.org/10.2969/jmsj/06620449

The main aim of this paper is to obtain a converse theorem for double Dirichlet series and use it to show that the Shintani zeta functions which arise in the theory of prehomogeneous vector spaces are actually linear combinations of Mellin transforms... Read More about A converse theorem for double Dirichlet series and Shintani zeta functions.

Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions (2012)
Journal Article
Bringmann, K., Diamantis, N., & Raum, M. (2013). Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions. Advances in Mathematics, 233(1), 115-134. https://doi.org/10.1016/j.aim.2012.09.025

We introduce a new technique of completion for 1-cohomology which parallels the corresponding technique in the theory of mock modular forms. This technique is applied in the context of non-critical values of L-functions of GL(2) cusp forms. We prove... Read More about Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions.

Second-Order Modular Forms with Characters (2012)
Book Chapter
Blann, T., & Diamantis, N. (2012). Second-Order Modular Forms with Characters. In Fourier Analysis and Number Theory to Radon Transforms and Geometry (55-66). Springer. https://doi.org/10.1007/978-1-4614-4075-8_4

We introduce spaces of second-order modular forms for which the relevant action involves characters. We compute the dimensions of these spaces by constructing explicit bases.

A converse theorem for double Dirichlet series (2011)
Journal Article
Diamantis, N., & Goldfeld, D. (2011). A converse theorem for double Dirichlet series. American Journal of Mathematics, 133(4), 913-938. https://doi.org/10.1353/ajm.2011.0024

We prove that a certain vector valued double Dirichlet series satisfying appropriate functional equations is a Mellin transform of a vector valued metaplectic Eisenstein series. We establish an analogous result for scalar double Dirichlet series of t... Read More about A converse theorem for double Dirichlet series.

New percolation crossing formulas and second-order modular forms (2009)
Journal Article
Diamantis, N., & Kleban, P. (2009). New percolation crossing formulas and second-order modular forms. Communications in Number Theory and Physics, 3(4), 677–696. https://doi.org/10.4310/cntp.2009.v3.n4.a4

We consider the three crossing probability densities for percolation recently found via conformal field theory [23]. We prove that all three of them (i) may be simply expressed in terms of Cardy’s [4] and Watts’ [24] crossing probabilities, (ii) are... Read More about New percolation crossing formulas and second-order modular forms.