Shintani lifts for weakly holomorphic modular forms
Final Report Abstract
The interplay between different spaces of modular forms was central to the proposed research. Modular forms are, in the words of Barry Mazur, functions on the complex upper half plane that are inordinately symmetric. A number of remarkable theorems have been proven by forming a link between two different spaces of modular forms and then exploiting known facts on either side of this connection. In some examples, lifts from one space to another uncover or explain “exceptional" subspaces. A good illustration of this arises from the Shimura lift. The fact that the so-called “unary theta functions" are counterexamples to the Ramanujan-Petersson Conjecture may be explained by the fact that while they are cusp forms, their images under the Shimura lift are not. In other cases, lifts supply a powerful machinery that leads to important new spaces of modular objects where the modularity, holomorphicity, and growth conditions may be loosened. To give an example, Bruinier realized that the famous Borcherds lift is not surjective onto the target space. Finding an appropriate preimage led to harmonic Maass forms. Harmonic Maass forms have symmetries like modular forms but are annihilated by some differential operator. Such functions are linked to classical modular forms through a differential operator. This allows one to reformulate classical conjectures in terms of harmonic Maass forms and then use the additional structure to forge a new attack on these problems. To give another example of the importance of the Shimura lift, celebrated work of Waldspurger used the relationship between integral and half-integral weight modular forms to solve open problems in both spaces. Empirical data indicated that the central value of the L-function of an integral weight modular form is essentially a square. On the other hand, Shintani showed that the Fourier coefficients of half-integral weight modular forms are "cycle integrals" of integral weight modular forms. Waldspurger fused these two statements into a coherent theory by proving that the central value of the L-function of integral weight modular forms is proportional to the square of a coefficient of a half-integral weight modular form. Kohnen and Zagier then explicitly computed the constant of proportionality. In particular, by determining that it is positive, they showed that the central value of the L-function is nonnegative. The key to this work was to construct two variable kernel functions for the Shimura lift, which is related to quadratic forms. These kernel functions satisfying integral weight modularity in one variable and half-integral weight modularity in the other, hence explaining the link between the two spaces. At the heart of many of the above applications are the "Hecke Operators", which act on spaces of modular forms. The Hecke operators "commute" with the lifts. Investigating growth towards the cusps, one can easily show that “purely" weakly holomorphic Hecke eigenforms, in the naive sense, cannot exist. However, in the integral weight case, Guerzhoy viewed weakly holomorphic Hecke eigenforms as elements in a quotient space by factoring out by a subspace originating from a lift. While lifts give a general framework upon which quite broad theorems may be proven, a deeper understanding of individual cases has held the key to applications. Along this vein, the main objective of the proposal was to extend the Shintani lift to weakly holomorphic modular forms. This goal was successfully achieved and along the way I obtained further results that are of independent interest. (1) Jointly with Guerzhoy and Kane, I extended the classical Shintani lift to include weakly holomorphic modular forms. Along the way we obtained the following results of independent interest: - A Hecke theory of half-integral weight weakly holomorphic modular forms. - An extension of the Zagier lift. (2) We used the newly developed Hecke theory for half-integral weight modular forms to prove congruences. (3) Using harmonic Maass forms we showed that the cycles of weakly holomorphic modular forms agree with those of cusp forms. (4) I further investigated certain distinguished meromorphic modular forms and investigated the following applications. - Taking an inner product of certain meromorphic modular forms yields an evaluation of Green's function (joint with Kane and v. Pippich). - For a proper understanding of meromorphic modular form of arbitrary weight and level, one needs to determine their divisor, i.e. the locations of their zeros and poles. In joint work with Kane, Löbrich, Ono, and Rolen, I investigated this problem using harmonic Maass forms. - Hardy and Ramanujan considered Fourier coefficients of examples of meromorphic modular forms. Jointly with Kane, I built a theory to find formulas for Fourier coefficients of such forms.
Publications
- Shintani lifts and fractional derivatives of harmonic weak Maass forms, Adv. Math. 255 (2014), 641–671
K. Bringmann, P. Guerzhoy, and B. Kane
(See online at https://doi.org/10.1016/j.aim.2014.01.015) - Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular forms, Res. Number Theory (2015), 1–26
K. Bringmann, P. Guerzhoy, and B. Kane
(See online at https://doi.org/10.1007/s40993-015-0027-1) - On cycle integrals of weakly holomorphic modular forms, Math. Proc. Camb. Phil. Soc. 58 (2015), 439–449
K. Bringmann, P. Guerzhoy, and B. Kane
(See online at https://doi.org/10.1017/S0305004115000055) - Modular local polynomials, Math. Res. Letters 23 (2016), 973–987
K. Bringmann and B. Kane
(See online at https://doi.org/10.4310/MRL.2016.v23.n4.a2) - Linear Incongruences for Generalized Eta-Quotients, Res. Number Theory 3 (2017), 8
S. Löbrich
(See online at https://doi.org/10.1007/s40993-017-0082-x) - Number-theoretic generalization of the Monster denominator formula, J. Phys. A, 50 (2017), 473001
K. Bringmann, B. Kane, S. Lörich, K. Ono, and L. Rolen
(See online at https://doi.org/10.1088/1751-8121/aa8f5d) - Radial Limits of the Universal Mock Theta Function g3 , Proc. Amer. Math. Soc. 145 (2017), 925–935
M. Jang and S. Löbrich
(See online at https://doi.org/10.1090/proc/13065) - Special Values of Motivic L-Functions and Zeta-Polynomials for Symmetric Powers of Elliptic Curves, Res. Math. Sci. 4 (2017), 16
S. Löbrich, W. Ma, and J. Thorner
(See online at https://doi.org/10.1186/s40687-017-0114-0) - Niebur-Poincaré Series and Traces of Singular Moduli, J. Math. Anal. Appl. 465 (2018), 673–689
S. Löbrich
(See online at https://doi.org/10.1016/j.jmaa.2018.05.034) - On divisors of modular forms, Adv. Math., 329 (2018), 541–554
K. Bringmann, B. Kane, S. Lörich, K. Ono, and L. Rolen
(See online at https://doi.org/10.1016/j.aim.2018.02.001)
