Modular symbols and applications
Lead Research Organisation:
University of Nottingham
Department Name: Sch of Mathematical Sciences
Abstract
One of the most effective ways to gain insight into difficult arithmetic questions in modern Number Theory is by visualising them through a certain type of curves, called elliptic curves. These are curves with a natural form and a simple defining equation but their structure has far reaching consequences, famously including the proof of Fermat's Last Theorem by Wiles et al. Some of the deepest properties of elliptic curves, in turn, can be formulated in terms of functions on the complex plane called L-functions (a fact at the heart of Wiles' proof). For example, there are infinitely many rational solutions of the equation defining an elliptic curve if the corresponding L-function vanishes upon evaluation at 1. This is a special case of one of the 'Millennium Problems' for whose solution a 1 million USD prize is available.
Understanding better the vanishing of a certain L-function at 1 should also answer a question recently asked by Mazur and Rubin, two of the most eminent number theorists in the last hundred years. Instead of the set of rational numbers, consider a so-called number field, a larger set of numbers with properties mirroring those of the rationals. The question they asked is: How do the solutions of the equation defining an elliptic curve change if we ask for those solutions to belong to varying number fields?
They noticed that the non-vanishing problem behind this question can be expressed in terms of a fundamental invariant called modular symbol. This motivated them to study the statistics of modular symbols and especially their "mean" and "variance". Upon analysing numerical data, they, together with W. Stein, formulated conjectures describing precisely the large scale behaviour of those means and variances.
In a recent paper, we fully proved one of those two conjectures. With that breakthrough achieved, the three main aims of the proposed project then are:
1. To establish the full proof of the conjecture by Mazur et al. that deals with the "variance" of modular symbols.
2. To formulate and prove analogous conjectures for Theta coefficients and Theta elements, which are important algebraic counterparts of the objects featuring in the above conjectures.
3. To use the outcomes of Aims 1 and 2 to derive information about vanishing of L-functions.
The significance of these aims goes beyond the geometric setting they originated in, because questions about vanishing of L-functions is an area of huge significance for analytic number theory. The project will not deal directly with applications to arithmetic and geometric problems but the methods introduced will be pivotal for progress in arithmetic and geometric as well as analytic aspects. Likewise, although applications outside Mathematics is not part of the proposal, results from the project will indirectly have potential impact on Elliptic Curve Cryptography.
Understanding better the vanishing of a certain L-function at 1 should also answer a question recently asked by Mazur and Rubin, two of the most eminent number theorists in the last hundred years. Instead of the set of rational numbers, consider a so-called number field, a larger set of numbers with properties mirroring those of the rationals. The question they asked is: How do the solutions of the equation defining an elliptic curve change if we ask for those solutions to belong to varying number fields?
They noticed that the non-vanishing problem behind this question can be expressed in terms of a fundamental invariant called modular symbol. This motivated them to study the statistics of modular symbols and especially their "mean" and "variance". Upon analysing numerical data, they, together with W. Stein, formulated conjectures describing precisely the large scale behaviour of those means and variances.
In a recent paper, we fully proved one of those two conjectures. With that breakthrough achieved, the three main aims of the proposed project then are:
1. To establish the full proof of the conjecture by Mazur et al. that deals with the "variance" of modular symbols.
2. To formulate and prove analogous conjectures for Theta coefficients and Theta elements, which are important algebraic counterparts of the objects featuring in the above conjectures.
3. To use the outcomes of Aims 1 and 2 to derive information about vanishing of L-functions.
The significance of these aims goes beyond the geometric setting they originated in, because questions about vanishing of L-functions is an area of huge significance for analytic number theory. The project will not deal directly with applications to arithmetic and geometric problems but the methods introduced will be pivotal for progress in arithmetic and geometric as well as analytic aspects. Likewise, although applications outside Mathematics is not part of the proposal, results from the project will indirectly have potential impact on Elliptic Curve Cryptography.
Planned Impact
Who will benefit from this research?
1. Postgraduate students and young mathematicians.
2. Wider public.
3. Security systems companies (indirectly).
How will they benefit from this research?
1. The project will have a strong training element for the PDRA and other young mathematicians connected with the project. It will provide sophisticated experience into analytic techniques motivated by an important conjecture and, especially in Aims 2 and 3, it will involve a mix of conceptually and technically challenging work. In addition, the project includes significant input from algebraic and geometric subjects, further increasing its training value for younger mathematicians.
2. I will use the proposed research to inform the general public about fundamental research. The motivating questions of the project involve elliptic curves which can be visualised through the use of computers and visual technology. The actual questions studied in the proposal (e.g. statistics about the growth of important quantities) can likewise be visualised in a way which is meaningful to the general public. This has the potential to more widely engage the public, encouraging further interest in studying Pure Mathematics and appreciation of the value of the subject.
3. A potential indirect impact of the proposed research is applications to security systems. Specifically, some of the motivating problems of the project are formulated in terms of elliptic curves. Although it is not an aim of the proposal to investigate direct applications to elliptic curves, our results will provide key insight which, as discussed in "Case for Support" and "Pathway to Impact", will be valuable for mathematicians working on elliptic curves. On the other hand, advances in the theory of elliptic curves are used by Elliptic Curve Cryptography, an important modern cryptosystem. Therefore, achieving the aims of the proposal, may, indirectly, have an essential impact on Elliptic Curve Cryptography.
1. Postgraduate students and young mathematicians.
2. Wider public.
3. Security systems companies (indirectly).
How will they benefit from this research?
1. The project will have a strong training element for the PDRA and other young mathematicians connected with the project. It will provide sophisticated experience into analytic techniques motivated by an important conjecture and, especially in Aims 2 and 3, it will involve a mix of conceptually and technically challenging work. In addition, the project includes significant input from algebraic and geometric subjects, further increasing its training value for younger mathematicians.
2. I will use the proposed research to inform the general public about fundamental research. The motivating questions of the project involve elliptic curves which can be visualised through the use of computers and visual technology. The actual questions studied in the proposal (e.g. statistics about the growth of important quantities) can likewise be visualised in a way which is meaningful to the general public. This has the potential to more widely engage the public, encouraging further interest in studying Pure Mathematics and appreciation of the value of the subject.
3. A potential indirect impact of the proposed research is applications to security systems. Specifically, some of the motivating problems of the project are formulated in terms of elliptic curves. Although it is not an aim of the proposal to investigate direct applications to elliptic curves, our results will provide key insight which, as discussed in "Case for Support" and "Pathway to Impact", will be valuable for mathematicians working on elliptic curves. On the other hand, advances in the theory of elliptic curves are used by Elliptic Curve Cryptography, an important modern cryptosystem. Therefore, achieving the aims of the proposal, may, indirectly, have an essential impact on Elliptic Curve Cryptography.
Organisations
- University of Nottingham (Lead Research Organisation)
- UNIVERSITY OF OXFORD (Collaboration)
- Vanderbilt University (Collaboration)
- UNIVERSITY OF NOTTINGHAM (Collaboration)
- Ulsan National Institute of Science and Technology (Collaboration)
- Keio University (Collaboration)
- AMERICAN UNIVERSITY OF BEIRUT (Collaboration)
- Uppsala University (Collaboration)
- Brown University (Collaboration)
- University of Bristol (Collaboration)
Publications
Diamantis N
(2020)
Period functions associated to real-analytic modular forms
in Research in the Mathematical Sciences
Diamantis N
(2021)
Eichler cohomology and zeros of polynomials associated to derivatives of L-functions
in Journal für die reine und angewandte Mathematik (Crelles Journal)
Diamantis N
(2023)
L -Series of Harmonic Maass Forms and a Summation Formula for Harmonic Lifts
in International Mathematics Research Notices
Diamantis N
(2023)
Analogues of the Bol operator for half-integral weight weakly holomorphic modular forms
in Proceedings of the American Mathematical Society
Diamantis N
(2022)
Derivatives of $L$-series of weakly holomorphic cusp forms
Diamantis N
(2022)
Derivatives of L-series of weakly holomorphic cusp forms
in Research in the Mathematical Sciences
Diamantis N
(2020)
Kernels of $L$-functions and shifted convolutions
in Proceedings of the American Mathematical Society
Diamantis N
(2022)
Modular iterated integrals associated with cusp forms
in Forum Mathematicum
Diamantis N
(2020)
Additive twists and a conjecture by Mazur, Rubin and Stein
in Journal of Number Theory
Description | I have constructed an explicit family of modular iterated integrals which involves cusp forms. The construction is based on the theory of higher-order modular forms. In collaboration with M. Lee, W. Raji and L Rolen, I have initiated a theory of L-functions associated to all harmonic Maass cusp forms and proved several foundational results. This has more recently been applied to the study of values of L-functions to provide a systematic conceptual framework for and generalise results on cycle integrals by Bruinier-Funke-Imamoglu and Alfes-Schwagenscheidt. Together with my collaborators, I have further applied this theory to special values of derivatives of L-functions and towards the construction of an analogue of the Bol operator for half-integral weight cusp forms. |
Exploitation Route | The main impact will be academic, within Number Theory. Work on the grant addresses important questions on which researchers have been working and has introduced new approaches and questions in currently highly active areas (e.g. harmonic Maass forms, modular integrals etc.) Potential applications outside mathematics can emerge through grant work on modular iterated integrals which appear in Physics (modular graph functions in String Theory). |
Sectors | Education,Other |
Description | Number Theory Research Unit Associate (American University of Beirut) |
Geographic Reach | Asia |
Policy Influence Type | Participation in a guidance/advisory committee |
Impact | Increase of interest in mathematical research among students of AUB and other universities in the broader area. |
URL | https://www.aub.edu.lb/cams/Pages/Number_Theory_Unit.aspx |
Description | Training of postgraduate students and PDRAs |
Geographic Reach | National |
Policy Influence Type | Influenced training of practitioners or researchers |
Description | Partial support for workshop "Arithmetic Statistics, Automorphic Forms and Ergodic Methods" |
Amount | € 10,000 (EUR) |
Organisation | Max Planck Institute for Mathematics |
Sector | Public |
Country | Germany |
Start | 04/2023 |
End | 04/2023 |
Title | Derivatives of L-series |
Description | Examples and algorithms for computations of Derivatives of L-series of weakly holomorphic cusp forms |
Type Of Material | Computer model/algorithm |
Year Produced | 2022 |
Provided To Others? | Yes |
Impact | The algorithm was used in the paper "Derivatives of L-series of weakly holomorphic cusp forms" by Diamantis and Stromberg and it provided explicit examples of special values of derivatives of L-series associated with weakly holomorphic cusp forms. |
URL | https://github.com/fredstro/derivatives_lseries |
Description | Arithmetic Statistics of modular symbols |
Organisation | Brown University |
Country | United States |
Sector | Academic/University |
PI Contribution | I initiated and led work on an important conjecture by Mazur, Rubin and Stein on the mean of modular symbols. This led to the proof of the full conjecture and to associated results of independent interest. On the basis of this result, research on the arithmetic statistics of so-called Brjuno function has started in collaboration with HS Sun (UNIST) which is on-going. |
Collaborator Contribution | Hoffstein (Brown) is a leading expert on the analytic theory of modular forms and he contributed to the overall programme towards the resolution of the conjecture. The contribution of Lee's (Bristol) was crucial for extending the result to all cases. Kiral (Keio) contributed to various technical aspects. HS Sun (UNIST) has approached arithmetic statistics problems from a standpoint based on dynamical systems which should further strengthen the techniques used to prove the Mazur-Rubin-Stein conjecture. We try to combine those methods to understand the Brjuno function. |
Impact | Diamantis, N., Hoffstein, J., Kiral, M., Lee M. Additive twists and a conjecture by Mazur, Rubin and Stein, Journal of Number Theory-Prime 209, 1-36 (2020) |
Start Year | 2018 |
Description | Arithmetic Statistics of modular symbols |
Organisation | Keio University |
Country | Japan |
Sector | Academic/University |
PI Contribution | I initiated and led work on an important conjecture by Mazur, Rubin and Stein on the mean of modular symbols. This led to the proof of the full conjecture and to associated results of independent interest. On the basis of this result, research on the arithmetic statistics of so-called Brjuno function has started in collaboration with HS Sun (UNIST) which is on-going. |
Collaborator Contribution | Hoffstein (Brown) is a leading expert on the analytic theory of modular forms and he contributed to the overall programme towards the resolution of the conjecture. The contribution of Lee's (Bristol) was crucial for extending the result to all cases. Kiral (Keio) contributed to various technical aspects. HS Sun (UNIST) has approached arithmetic statistics problems from a standpoint based on dynamical systems which should further strengthen the techniques used to prove the Mazur-Rubin-Stein conjecture. We try to combine those methods to understand the Brjuno function. |
Impact | Diamantis, N., Hoffstein, J., Kiral, M., Lee M. Additive twists and a conjecture by Mazur, Rubin and Stein, Journal of Number Theory-Prime 209, 1-36 (2020) |
Start Year | 2018 |
Description | Arithmetic Statistics of modular symbols |
Organisation | Ulsan National Institute of Science and Technology |
Country | Korea, Republic of |
Sector | Academic/University |
PI Contribution | I initiated and led work on an important conjecture by Mazur, Rubin and Stein on the mean of modular symbols. This led to the proof of the full conjecture and to associated results of independent interest. On the basis of this result, research on the arithmetic statistics of so-called Brjuno function has started in collaboration with HS Sun (UNIST) which is on-going. |
Collaborator Contribution | Hoffstein (Brown) is a leading expert on the analytic theory of modular forms and he contributed to the overall programme towards the resolution of the conjecture. The contribution of Lee's (Bristol) was crucial for extending the result to all cases. Kiral (Keio) contributed to various technical aspects. HS Sun (UNIST) has approached arithmetic statistics problems from a standpoint based on dynamical systems which should further strengthen the techniques used to prove the Mazur-Rubin-Stein conjecture. We try to combine those methods to understand the Brjuno function. |
Impact | Diamantis, N., Hoffstein, J., Kiral, M., Lee M. Additive twists and a conjecture by Mazur, Rubin and Stein, Journal of Number Theory-Prime 209, 1-36 (2020) |
Start Year | 2018 |
Description | Arithmetic Statistics of modular symbols |
Organisation | University of Bristol |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I initiated and led work on an important conjecture by Mazur, Rubin and Stein on the mean of modular symbols. This led to the proof of the full conjecture and to associated results of independent interest. On the basis of this result, research on the arithmetic statistics of so-called Brjuno function has started in collaboration with HS Sun (UNIST) which is on-going. |
Collaborator Contribution | Hoffstein (Brown) is a leading expert on the analytic theory of modular forms and he contributed to the overall programme towards the resolution of the conjecture. The contribution of Lee's (Bristol) was crucial for extending the result to all cases. Kiral (Keio) contributed to various technical aspects. HS Sun (UNIST) has approached arithmetic statistics problems from a standpoint based on dynamical systems which should further strengthen the techniques used to prove the Mazur-Rubin-Stein conjecture. We try to combine those methods to understand the Brjuno function. |
Impact | Diamantis, N., Hoffstein, J., Kiral, M., Lee M. Additive twists and a conjecture by Mazur, Rubin and Stein, Journal of Number Theory-Prime 209, 1-36 (2020) |
Start Year | 2018 |
Description | L-series of harmonic Maass forms |
Organisation | American University of Beirut |
Country | Lebanon |
Sector | Academic/University |
PI Contribution | I initiated and led research on L-series associated with harmonic Maass forms. Together with my collaborators we gave the first general definition of such L-series and proved their fundamental properties. We gave applications to the theory of cycle integrals, to special values of derivatives of L-functions and to analogues of the Bol operator for half-integral weight forms. I was the principal investigator in all those topics co-ordinating the different directions and applications of the project. |
Collaborator Contribution | The partners contributed to the various applications according to their expertise. L. Rolen (Vanderbilt) contributed to the parts most related to the theory of harmonic Maass forms, M. Lee (Bristol) to the analytic techniques of L-functions, F. Stromberg (Nottingham) to computational aspects and W. Raji to the parts related to half-integral weight forms. |
Impact | 1. Diamantis, N., Rolen, L. L-values of harmonic Maass forms (submitted) 2. Diamantis, N., Rolen, L. Analogues of the Bol operator for half-integral weight weakly holomorphic modular forms. (submitted) 3. Diamantis, N., Lee, M., Raji, W., Rolen, L., L-series of harmonic Maass forms and a summation formula for harmonic lifts. International Mathematics Research Notices (in Press) 4. Diamantis, N, Strömberg, F. Derivatives of L-series of weakly holomorphic cusp forms. Research in the Mathematical Sciences 9, 64 (2022) |
Start Year | 2020 |
Description | L-series of harmonic Maass forms |
Organisation | University of Bristol |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I initiated and led research on L-series associated with harmonic Maass forms. Together with my collaborators we gave the first general definition of such L-series and proved their fundamental properties. We gave applications to the theory of cycle integrals, to special values of derivatives of L-functions and to analogues of the Bol operator for half-integral weight forms. I was the principal investigator in all those topics co-ordinating the different directions and applications of the project. |
Collaborator Contribution | The partners contributed to the various applications according to their expertise. L. Rolen (Vanderbilt) contributed to the parts most related to the theory of harmonic Maass forms, M. Lee (Bristol) to the analytic techniques of L-functions, F. Stromberg (Nottingham) to computational aspects and W. Raji to the parts related to half-integral weight forms. |
Impact | 1. Diamantis, N., Rolen, L. L-values of harmonic Maass forms (submitted) 2. Diamantis, N., Rolen, L. Analogues of the Bol operator for half-integral weight weakly holomorphic modular forms. (submitted) 3. Diamantis, N., Lee, M., Raji, W., Rolen, L., L-series of harmonic Maass forms and a summation formula for harmonic lifts. International Mathematics Research Notices (in Press) 4. Diamantis, N, Strömberg, F. Derivatives of L-series of weakly holomorphic cusp forms. Research in the Mathematical Sciences 9, 64 (2022) |
Start Year | 2020 |
Description | L-series of harmonic Maass forms |
Organisation | University of Nottingham |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I initiated and led research on L-series associated with harmonic Maass forms. Together with my collaborators we gave the first general definition of such L-series and proved their fundamental properties. We gave applications to the theory of cycle integrals, to special values of derivatives of L-functions and to analogues of the Bol operator for half-integral weight forms. I was the principal investigator in all those topics co-ordinating the different directions and applications of the project. |
Collaborator Contribution | The partners contributed to the various applications according to their expertise. L. Rolen (Vanderbilt) contributed to the parts most related to the theory of harmonic Maass forms, M. Lee (Bristol) to the analytic techniques of L-functions, F. Stromberg (Nottingham) to computational aspects and W. Raji to the parts related to half-integral weight forms. |
Impact | 1. Diamantis, N., Rolen, L. L-values of harmonic Maass forms (submitted) 2. Diamantis, N., Rolen, L. Analogues of the Bol operator for half-integral weight weakly holomorphic modular forms. (submitted) 3. Diamantis, N., Lee, M., Raji, W., Rolen, L., L-series of harmonic Maass forms and a summation formula for harmonic lifts. International Mathematics Research Notices (in Press) 4. Diamantis, N, Strömberg, F. Derivatives of L-series of weakly holomorphic cusp forms. Research in the Mathematical Sciences 9, 64 (2022) |
Start Year | 2020 |
Description | L-series of harmonic Maass forms |
Organisation | Vanderbilt University |
Country | United States |
Sector | Academic/University |
PI Contribution | I initiated and led research on L-series associated with harmonic Maass forms. Together with my collaborators we gave the first general definition of such L-series and proved their fundamental properties. We gave applications to the theory of cycle integrals, to special values of derivatives of L-functions and to analogues of the Bol operator for half-integral weight forms. I was the principal investigator in all those topics co-ordinating the different directions and applications of the project. |
Collaborator Contribution | The partners contributed to the various applications according to their expertise. L. Rolen (Vanderbilt) contributed to the parts most related to the theory of harmonic Maass forms, M. Lee (Bristol) to the analytic techniques of L-functions, F. Stromberg (Nottingham) to computational aspects and W. Raji to the parts related to half-integral weight forms. |
Impact | 1. Diamantis, N., Rolen, L. L-values of harmonic Maass forms (submitted) 2. Diamantis, N., Rolen, L. Analogues of the Bol operator for half-integral weight weakly holomorphic modular forms. (submitted) 3. Diamantis, N., Lee, M., Raji, W., Rolen, L., L-series of harmonic Maass forms and a summation formula for harmonic lifts. International Mathematics Research Notices (in Press) 4. Diamantis, N, Strömberg, F. Derivatives of L-series of weakly holomorphic cusp forms. Research in the Mathematical Sciences 9, 64 (2022) |
Start Year | 2020 |
Description | Real-analytic modular forms |
Organisation | University of Nottingham |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I initiated the study of classical number theoretic aspects of a relatively class of objects defined by F. Brown, called real-analytic modular forms, and, in particular, the modular iterated integrals. My collaborators and I defined L-series and period polynomials associated with modular iterated integrals, constructed special families of modular iterated integrals which reflects structures of classical cusp forms and , at the same time, are of interest to mathematical Physicists and, as applications, I derived algebraicity information for L-values of classical and of weakly holomorphic cusp forms. |
Collaborator Contribution | Drewitt (Nottingham) was the co-author of the work where we constructed the L-series and period polynomials for modular iterated integrals and where we established the algebraicity property for L-values of weakly holomorphic cusp forms. F. Brown (Oxford) was the founder of the theory of real-analytic modular forms, and modular iterated integrals and our conversations provided important insight towards progress in the project. Discussions with Schlotterer (Uppsala) led to awareness of the importance of this work for mathematical physics and informed the choice of the subsequent questions. |
Impact | Diamantis, N., Modular iterated integrals associated with cusp forms. Forum Mathematicum 34(1): 157-174 (2022) Diamantis, N. Kernels of L-functions and shifted convolutions, Proceedings of the American Mathematical Society 148, 5059-5070 (2020) Diamantis, N., Drewitt, J. Period functions associated to real-analytic modular forms Research in the Mathematical Sciences 7, 21 (2020) |
Start Year | 2017 |
Description | Real-analytic modular forms |
Organisation | University of Oxford |
Department | Mathematical Institute Oxford |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | I initiated the study of classical number theoretic aspects of a relatively class of objects defined by F. Brown, called real-analytic modular forms, and, in particular, the modular iterated integrals. My collaborators and I defined L-series and period polynomials associated with modular iterated integrals, constructed special families of modular iterated integrals which reflects structures of classical cusp forms and , at the same time, are of interest to mathematical Physicists and, as applications, I derived algebraicity information for L-values of classical and of weakly holomorphic cusp forms. |
Collaborator Contribution | Drewitt (Nottingham) was the co-author of the work where we constructed the L-series and period polynomials for modular iterated integrals and where we established the algebraicity property for L-values of weakly holomorphic cusp forms. F. Brown (Oxford) was the founder of the theory of real-analytic modular forms, and modular iterated integrals and our conversations provided important insight towards progress in the project. Discussions with Schlotterer (Uppsala) led to awareness of the importance of this work for mathematical physics and informed the choice of the subsequent questions. |
Impact | Diamantis, N., Modular iterated integrals associated with cusp forms. Forum Mathematicum 34(1): 157-174 (2022) Diamantis, N. Kernels of L-functions and shifted convolutions, Proceedings of the American Mathematical Society 148, 5059-5070 (2020) Diamantis, N., Drewitt, J. Period functions associated to real-analytic modular forms Research in the Mathematical Sciences 7, 21 (2020) |
Start Year | 2017 |
Description | Real-analytic modular forms |
Organisation | Uppsala University |
Country | Sweden |
Sector | Academic/University |
PI Contribution | I initiated the study of classical number theoretic aspects of a relatively class of objects defined by F. Brown, called real-analytic modular forms, and, in particular, the modular iterated integrals. My collaborators and I defined L-series and period polynomials associated with modular iterated integrals, constructed special families of modular iterated integrals which reflects structures of classical cusp forms and , at the same time, are of interest to mathematical Physicists and, as applications, I derived algebraicity information for L-values of classical and of weakly holomorphic cusp forms. |
Collaborator Contribution | Drewitt (Nottingham) was the co-author of the work where we constructed the L-series and period polynomials for modular iterated integrals and where we established the algebraicity property for L-values of weakly holomorphic cusp forms. F. Brown (Oxford) was the founder of the theory of real-analytic modular forms, and modular iterated integrals and our conversations provided important insight towards progress in the project. Discussions with Schlotterer (Uppsala) led to awareness of the importance of this work for mathematical physics and informed the choice of the subsequent questions. |
Impact | Diamantis, N., Modular iterated integrals associated with cusp forms. Forum Mathematicum 34(1): 157-174 (2022) Diamantis, N. Kernels of L-functions and shifted convolutions, Proceedings of the American Mathematical Society 148, 5059-5070 (2020) Diamantis, N., Drewitt, J. Period functions associated to real-analytic modular forms Research in the Mathematical Sciences 7, 21 (2020) |
Start Year | 2017 |
Description | Zeros of polynomials associated to derivatives of L-functions |
Organisation | Vanderbilt University |
Country | United States |
Sector | Academic/University |
PI Contribution | I provided the theoretical framework for the experiments that led to a conjecture about an object previously studied by me. I also contributed to the proof of the main theorems |
Collaborator Contribution | The co-author made the numerical experiments that help us formulate the conjecture. He also contributed to the the proofs. |
Impact | 1. Diamantis, N., Rolen, L. Period polynomials, derivatives of L-functions, and zeros of polynomials. Research in the Mathematical Sciences 5, no. 1, Paper No. 9 (2018) 2. Diamantis, N., Rolen, L. Eichler cohomology and zeros of polynomials associated to derivatives of L-functions Journal fur die reine und angewandte Mathematik (Crelle's Journal) 770, 1-25 (2021) |
Start Year | 2017 |