## Final Report Summary - PERIODS (Periods of modular forms)

Modular forms are central objects in number theory, being connected with all the longstanding conjectures in the field. The goal of this project has been to study the space of modular forms for congruence subgroups of the modular group, by using its connection with the space of period polynomials.

For the modular group itself, the space of period polynomials has been introduced and studied by D. Zagier and his collaborators, following earlier work of Eichler, Shimura, and Manin. The connection between modular forms and period polynomials is provided by a version of the well-known Eichler-Shimura correspondence, and the structure of the space of modular forms (Hecke operators and Petersson product) carries over to the space of period polynomials. For the full modular group, the action of Hecke operators on period polynomials has been determinedby Choie and Zagier, and the Petersson product corresponds to a pairing on period polynomials via a formula of Haberland. Haberland's formula was refined in a paper of the researcher partly supported by the Marie Curie grant [``Rational decomposition of modular forms,'' Ramanujan J., 26 (2011), 419-435].

In the paper ``Modular forms and period polynomials'' [Proc. Lond. Math. Soc. 107/4 (2013), 713-743], the researcher together with V. Pasol introduced the space of period polynomials associated with arbitrary finite index subgroups of the modular group. The mauthors generalize the action of Hecke operators, and they show that the refinement of Haberland's formula holds in this setting as well. The resulting theory is dual to the well established theory of modular symbols developed by Merel in ``Universal expansion of modular forms'' (1994). It goes beyond the theory of modular symbols by introducing (extended) period polynomials of Eisenstein series, and extending Haberland's formula to the space of extended period polynomials as well. The paper contains many interesting and fundamental results, among which: the determination of extra relations satisfied by periods of cusp forms which are independent of the period relations; numerical computation of period polynomials of newforms; convenient inverses of the Eichler-Shimura map. This paper will likely have a significant impact on the Eichler-Shimura-Manin theory and its applications.

An impressive work in progress of the researcher and Don Zagier concerns the well-known Eichler-Selberg trace formula. Twenty years ago, Don Zagier sketched an elementary proof of the trace formula for Hecke operators acting on modular forms for the modular group, using a particular Hecke operator acting on period polynomials. Combining this idea with the work mentioned above on period polynomials for congruence subgroups, the researcher shows that the same approach can be used to give a simple determination of the trace of Hecke and Atkin-Lehner operators on modular forms for congruence subgroups as well. This approach leads to simple trace formulas compared to the existing (extensive) literature on the subject. The researcher presented this work in conference talks during 2013. Since it is quite general, this approach may open the way for simple proofs of trace formulas for Hecke operators acting on spaces of modular forms for other groups, such as groups of units in orders in quaternion algebras.

Another fundamental result of the researcher with V. Pasol involves a natural extension of the Petersson scalar product to the whole space of modulars forms for Gamma_0(N) [``On the Petersson scalar product of arbitrary modular forms'', Proc. of the Amer. Math. Soc., to appear]. They show that the extended Petersson product is Hecke equivariant, and nondegenerate, generalizing results established by Don Zagier for the modular group, and by S. Bocherer and F. Chiera for N prime.

An unexpected consequence of the study of Hecke operators on period polynomials is contained in the paper of the researcher with V. Pasol, ``An algebraic property of Hecke operators and two indefinite theta series'' [Forum Math., to appear]. In the process of proving an algebraic property of Hecke operators acting on period polynomials, the authors discovered ``indefinite theta series'' associated with a quadratic form of signature (2,2), which can be expressed in terms of Eisenstein series. This can be seen as an analogue of the famous 1834 formula of Jacobi for the number of ways of writing a positive integer as the sum of four squares. An amusing consequence that is easy to state is that the number of integer solutions to x^2+y^2-z^2-t^2=p with x,y>|z|,|t| is p-3, where p>2 is a prime.

In a different direction, the researcher together with F. Boca, V. Pasol, and A. Zaharescu, have studied the angles made by closed geodesics on the modular surface passing through the imaginary unit in the hyperbolic upper half plane. Such geodesics are fundamental objects for understanding periods of modular forms and Maass forms, as the integrals of a Hecke eigenform over such geodesics can be related with special values of twisted L-functions associated with the Hecke eigenform. In a recent paper [Algebra and Number Theory, to appear], they have computed for the first time the pair correlation of the sequence of these angles in increasingly large balls. This groundbreaking result stimulated further research: first, the researcher with F. Boca, A. Zaharescu gave a formula for the pair correlation which holds for the hyperbolic modular lattice centered at elliptic points, and conjectured that the formula holds in general [arXiv:1302.5067 submitted]; second, D. Kelmer and A. Kontorovich very recently proposed a proof of this conjecture.

In conclusion, the project has contributed to advancing our knowledge in a fundamental area of research in number theory. The results obtained will likely be of interest to many researchers in number theory.

For the modular group itself, the space of period polynomials has been introduced and studied by D. Zagier and his collaborators, following earlier work of Eichler, Shimura, and Manin. The connection between modular forms and period polynomials is provided by a version of the well-known Eichler-Shimura correspondence, and the structure of the space of modular forms (Hecke operators and Petersson product) carries over to the space of period polynomials. For the full modular group, the action of Hecke operators on period polynomials has been determinedby Choie and Zagier, and the Petersson product corresponds to a pairing on period polynomials via a formula of Haberland. Haberland's formula was refined in a paper of the researcher partly supported by the Marie Curie grant [``Rational decomposition of modular forms,'' Ramanujan J., 26 (2011), 419-435].

In the paper ``Modular forms and period polynomials'' [Proc. Lond. Math. Soc. 107/4 (2013), 713-743], the researcher together with V. Pasol introduced the space of period polynomials associated with arbitrary finite index subgroups of the modular group. The mauthors generalize the action of Hecke operators, and they show that the refinement of Haberland's formula holds in this setting as well. The resulting theory is dual to the well established theory of modular symbols developed by Merel in ``Universal expansion of modular forms'' (1994). It goes beyond the theory of modular symbols by introducing (extended) period polynomials of Eisenstein series, and extending Haberland's formula to the space of extended period polynomials as well. The paper contains many interesting and fundamental results, among which: the determination of extra relations satisfied by periods of cusp forms which are independent of the period relations; numerical computation of period polynomials of newforms; convenient inverses of the Eichler-Shimura map. This paper will likely have a significant impact on the Eichler-Shimura-Manin theory and its applications.

An impressive work in progress of the researcher and Don Zagier concerns the well-known Eichler-Selberg trace formula. Twenty years ago, Don Zagier sketched an elementary proof of the trace formula for Hecke operators acting on modular forms for the modular group, using a particular Hecke operator acting on period polynomials. Combining this idea with the work mentioned above on period polynomials for congruence subgroups, the researcher shows that the same approach can be used to give a simple determination of the trace of Hecke and Atkin-Lehner operators on modular forms for congruence subgroups as well. This approach leads to simple trace formulas compared to the existing (extensive) literature on the subject. The researcher presented this work in conference talks during 2013. Since it is quite general, this approach may open the way for simple proofs of trace formulas for Hecke operators acting on spaces of modular forms for other groups, such as groups of units in orders in quaternion algebras.

Another fundamental result of the researcher with V. Pasol involves a natural extension of the Petersson scalar product to the whole space of modulars forms for Gamma_0(N) [``On the Petersson scalar product of arbitrary modular forms'', Proc. of the Amer. Math. Soc., to appear]. They show that the extended Petersson product is Hecke equivariant, and nondegenerate, generalizing results established by Don Zagier for the modular group, and by S. Bocherer and F. Chiera for N prime.

An unexpected consequence of the study of Hecke operators on period polynomials is contained in the paper of the researcher with V. Pasol, ``An algebraic property of Hecke operators and two indefinite theta series'' [Forum Math., to appear]. In the process of proving an algebraic property of Hecke operators acting on period polynomials, the authors discovered ``indefinite theta series'' associated with a quadratic form of signature (2,2), which can be expressed in terms of Eisenstein series. This can be seen as an analogue of the famous 1834 formula of Jacobi for the number of ways of writing a positive integer as the sum of four squares. An amusing consequence that is easy to state is that the number of integer solutions to x^2+y^2-z^2-t^2=p with x,y>|z|,|t| is p-3, where p>2 is a prime.

In a different direction, the researcher together with F. Boca, V. Pasol, and A. Zaharescu, have studied the angles made by closed geodesics on the modular surface passing through the imaginary unit in the hyperbolic upper half plane. Such geodesics are fundamental objects for understanding periods of modular forms and Maass forms, as the integrals of a Hecke eigenform over such geodesics can be related with special values of twisted L-functions associated with the Hecke eigenform. In a recent paper [Algebra and Number Theory, to appear], they have computed for the first time the pair correlation of the sequence of these angles in increasingly large balls. This groundbreaking result stimulated further research: first, the researcher with F. Boca, A. Zaharescu gave a formula for the pair correlation which holds for the hyperbolic modular lattice centered at elliptic points, and conjectured that the formula holds in general [arXiv:1302.5067 submitted]; second, D. Kelmer and A. Kontorovich very recently proposed a proof of this conjecture.

In conclusion, the project has contributed to advancing our knowledge in a fundamental area of research in number theory. The results obtained will likely be of interest to many researchers in number theory.