Notes On Hecke Eigenforms

In mathematics,in particular in the theory of modular forms,a Hecke operator is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.Mordell（1917）used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan,ahead of the general theory given by Hecke（1937）.For a fixed integer k and any positive integer n,the Hecke operator T_n is defined on the set Mk of entire modular forms of weight k by Hecke operators have many great properties,such as the associativity and commuta-tivity of its multiplication（therefore Hecke operators present commutative operator algebras,which are called Hecke algebras）.Moreover,they are self-adjoint with respect to the Petersson inner product and multiplicative,and so on.An Hecke eigenform f（z）（meaning simultaneous eigenform with modular group SL₂（Z））is a modular form which is an eigenvector for all Hecke operators.That’s to say,for every positive integer n there is an complex number A（n）for which(T_nf（z）= λ（n）f（z）.A common example of an eigenform,and the only non-cuspidal eigenforms,are the Eisenstein series.Another example is the △ Function.The Hecke eigenform f（z）has Fourier seriesIf a₀= 0,it is called a cusp form;if a₁=1,it is said to be normalized.With out lose of generality,the Hecke eigenforms shown in this paper are normalized.Hecke eigenforms fall into the realm of number theory,but can be found in other areas of math and science such as analysis,combinatorics,and physics.It a hotspot among mathematicians.Recently,Ahmad,Wissam,Holowinsky,Luo,Ono,Soundararajan,Sarnak and many other outstanding mathematicians achieve excellent results on this aspect（see[6,13,20,21]etc.）.In this paper,we study two problems for Hecke cusp forms（i.e.,a₀=0 in the Fourier expansion）.They are the non-ordinary primes for Hecke eigenforms and the distribution of zeros of the period polynomials for Hecke eigenforms.We start with the non-ordinary primes for Hecke eigenforms.If k≥4 is even,then let M_k（resp.S_k）denote the finite dimensional C-vector space of weight k holomorphic modular forms（resp.cusp forms）on SL₂（Z）.Furthermore,let M_k^! denote the infinite dimensional space of weakly holomorphic modular forms of weight k with respect to SL₂（Z）（see[18]）.Recall that a mero-morphic modular form is weakly holomorphic if its poles（if any）are supported at cusps.We shall identify a modular form on SL₂（Z）by its Fourier expansion at infinitySuppose that OL is the ring of integers of a number field L,and suppose that is a normalized Hecke eigenform for SL₂（Z）.We say that f is non-ordinary at a prime p if there is a prime ideal p（?）OL above p for which.We recall the following well-known open problem about the distribution of non-ordinary primes（see Gouvea’s expository article[7]）.Problem Are there infinitely many non-ordinary primes for a generic normalized Hecke eigenform f（z）?Although there are strong results on the more general problem for very special modular forms on congruence subgroups of SL₂（Z）（e.g.such as such as CM cusp forms,and weight 2 newforms associated to elliptic curves over Q）,little is known.In 2005,Choie,Kohnen and Ono[2]get a theorem applies for all forms when P＝2,3,and requires that δ（k）≠0 for primes p≥5.We do not solve this problem here.It remains open.However,we establish the following related result based on the work of Choie,Kohnen and Ono.Theorem 1 If S is a finite set of primes,then there are infinitely many normalized Hecke eigenforms for SL₂（Z）which are non-ordinary for each p∈S.We go on with the distribution of zeros of the period polynomials for Hecke eigenforms.One natural and useful object attached to a modular form is its Eichler integral defined byAlthough ε_f（z）is not a modular form,it nonetheless behaves nicely under modular transformations in a way that gives rise to the so-called period function,which is another important object attached to f（z）.The period function is given by The even part r+f（z）and odd part r-f（z）of rf（z）are defined byIn particular,let г be the discrete subgroup of PSL₂（R）with parabolic cusp at i∞.Then the period polynomial for f（z）∈ S_k（г）,k ∈2Z_≥o turns to be It is not hard to see that r_f（z）is a polynomial of degree k-2,whose coefficients involve special values of the L-functions of f（z）（L（f,1）,L（f,2）,...,L（f,k-1））.Hence period polynomials are the generating functions for these values（see[17]）.Thus,r_f（z）provides a general description connecting Eichler integrals to those L-values.Critical values of L-functions are objects of central importance in arithmetic geometry and number theory.For general facts on period polynomials,the readers are encouraged to see[3,16,17,25,32];other papers broadly related to the themes of this paper are[8,23].In analogy with the Riemann hypothesis,we may ask whether all of the zeros of the period polynomials lie on the circle.We call it the Riemann hypothesis for period polynomials of modular forms.In 2013,Conrey,Farmer and Imamoglu[4]proved that if f∈S_k（SL₂（Z））is a Hecke eigenform,then the odd period polynomial has simple zeros at 0,±2,± 1/2 and double zeros at ±1;the rest of its zeros are complex numbers on the unit circle.In 2014,El-Guindy and Raji[6]proved that if f（z）∈ S_k（SL₂（Z））is a Hecke eigenform,then r_f（z）has all of its zeros on the unit circle.In this paper,we focus on the zeros of period polynomials for Hecke eigen-forms on arithmetic Hecke groups and Hecke newforms on SL₂（г₀（N））.In addition to a generalization of the result of El-Guindy and Raji,we show that ask → ∞,the zeros are equidisributed.Our result concerns Hecke eigenforms on arithmetic Hecke groups is follow-ing.Theorem 2 Let г be one of the Hecke groups H₃,H₄,H₆ and H_∞.If a Hecke eigenform has sufficiently large weight k,then all of the zeros of rf（z）are on the unit circle.Moreover,as →∞,the zeros are equidis-tributed.Modulo a strong form of the Ramanujan-Petersson Conjecture,the conclusion of the first half of Theorem 2 applies to Hecke eigenforms on discrete subgroups T of PSL₂（R）which contain.For г₀（N）,we have a similar result for Hecke newforms,which form a sub-space of the space for Hecke eigenforms.A newform in S_k^new（r_o（N））is a normal-ized cusp form of all the Hecke operators and all of the Atkin-Lehner involutions ,for primes p｜N,and Fricke involution ｜_kW（N）.Theorem 3 Let f（z）∈S_k（Γ0（N））be a newform.For any even integer k≥4 and any level N,all of the zeros of the period polynomial r_f（X）are on the circle .If we restrict r_f（z）to the circle,we get a trigonometric polynomial,and our proof of Theorem 3 proceeds by finding the right number of sign changes as z varies over the unit circle.If the weight or level is large enough,then the zeros of r_f（z）are regularly spaced on the circle.We establish the following result on the location of the roots.Theorem 4 For newform f（z）∈S_k（г₀（N））,the following are true.（i）Suppose that k 4.If ∈（f）=-1,then the zeros of r_f（z）are.If∈（f）= 1 and N is sufficiently large,then the zeros of r_f（z）are located at.（ii）If k ≥ 6 and either N or k is large enough,then the roots of r_f（z）may be written as where for 0 ≤l ≤ 2m-1 we denote by θl the unique solution in[0,2π)to the equation...