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- Periods of modular forms and Jacobi theta functions
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- On the periods of modular forms
- Modular form
Periods of modular forms and Jacobi theta functions
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.
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.
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 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. 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. This can be seen as an analogue of the famous formula of Jacobi for the number of ways of writing a positive integer as the sum of four squares.
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 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.
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Matrix representations of Hecke operators on classical holomorphic cusp forms and corresponding period polynomials are well known. In this paper we define Hecke operators on period functions introduced recently by Lewis and Zagier and show how they are related to the Hecke operators on Maass cusp forms. Moreover, we give an explicit general compatibility criterion for formal sums of matrices to represent Hecke operators on period functions. An explicit example of such matrices with only nonnegative entries is constructed. Most users should sign in with their email address. If you originally registered with a username please use that to sign in. To purchase short term access, please sign in to your Oxford Academic account above.
PDF | It is known that there is an one-to-one correspondence among space of cusp forms, space of homogeneous period polynomials and.
On the periods of modular forms
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In mathematics , a modular form is a complex analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group , and also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology , sphere packing , and string theory. Instead, modular functions are meromorphic that is, they are almost holomorphic except for a set of isolated points. Modular form theory is a special case of the more general theory of automorphic forms , and therefore can now be seen as just the most concrete part of a rich theory of discrete groups. Modular forms can also be interpreted as sections of a specific line bundles on modular varieties. The dimensions of these spaces of modular forms can be computed using the Riemann—Roch theorem.
Skip to Main Content. A not-for-profit organization, IEEE is the world's largest technical professional organization dedicated to advancing technology for the benefit of humanity. Use of this web site signifies your agreement to the terms and conditions. Hecke operators on period functions for the full modular group Abstract: Matrix representations of Hecke operators on classical holomorphic cusp forms and corresponding period polynomials are well known. In this paper we define Hecke operators on period functions introduced recently by Lewis and Zagier and show how they are related to the Hecke operators on Maass cusp forms. Moreover, we give an explicit general compatibility criterion for formal sums of matrices to represent Hecke operators on period functions.
We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's.