# LINEAR RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES

Size: px
Start display at page:

Download "LINEAR RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES"

## Transcription

1 LINEAR RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES YOUNGJU CHOIE, WINFRIED KOHNEN, AND KEN ONO Appearing in the Bulletin of the London Mathematical Society Abstract. Here we generalize a classical observation of Siegel by determining all the linear relations among the initial Fourier coefficients of a modular form on SL 2 (Z). As a consequence, we identify spaces M k in which there are universal p-divisibility properties for certain p-power coefficients. As a corollary, let f(z) = a f (n)q n S k O L [[q]] n=1 (note: q := e 2πiz ) be a normalized Hecke eigenform, and let k δ(k) (mod 12), where δ(k) {4, 6, 8, 10, 14}. Reproducing earlier results of Hatada and Hida, if p is a prime for which k δ(k) (mod p 1), and p O L is a prime ideal above p, then we show that a f (p) 0 1. Introduction and statement of results If k 4 is even, then let M k (respectively S k ) denote the finite dimensional C-vector space of weight k holomorphic modular forms (respectively cusp forms) on SL 2 (Z) (see [7] for background on modular forms). As usual, we identify a modular form f(z) by its Fourier expansion f(z) = a f (n)q n, where q := e 2πiz. As is customary, let (z) S 12 be the cusp form (1.1) (z) := q n=0 (1 q n ) 24 = q 24q 2 +, and for even k 4 let E k (z) M k be the normalized Eisenstein series (1.2) E k (z) = 1 2k d k 1 q n. B k n=1 n=1 1 d n 2000 Mathematics Subject Classification. Primary 11F33; Secondary 11F11. The first author is partially supported by KOSEF R The third author is grateful for the support of a grant from the National Science Foundation, and the generous support of the Alfred P. Sloan, David and Lucile Packard, H. I. Romnes, and John S. Guggenheim Fellowships. 1

2 2 YOUNGJU CHOIE, WINFRIED KOHNEN, AND KEN ONO The rational numbers B k are the Bernoulli numbers, and they are defined by the generating function B k tk k! = t e t 1 = t t2. k=0 For convenience, we let E 0 (z) := 1. Throughout, if k 4 is even, then let (for example, see I.2 of [7]) { k/ if k 2 (mod 12), (1.3) d(k) := dim C (M k ) = k/12 if k 2 (mod 12). Furthermore, define δ(k) {0, 4, 6, 8, 10, 14} by the congruence (1.4) δ(k) k (mod 12). If N is a non-negative integer, then we let (1.5) L k,n := {(c 0, c 1,...,c ) C +1 : c ν a f (ν) = 0 f(z) = a f (n)q n M k } be the space of linear relations satisfied by the first +1 Fourier coefficients of all the forms f(z) M k. In his study of Hilbert modular forms, Siegel [8] determined the spaces L k,0. To state our results, for each g(z) M 12N, define numbers b(k, N, g; ν) by (1.6) E 14 δ(k) (z) g(z) = (z) b(k, N, g; ν)q ν + n=0 c(k, N, g; ν)q ν. The numbers b(k, N, g; ν) are the Fourier coefficients of the principal part, together with the constant term, of the modular form above. In this notation, we have the following characterization of the L k,n. Theorem 1.1. The map φ k,n : M 12N L k,n defined by ν=1 φ k,n (g(z)) = {b(k, N, g; ν) : ν = 0,..., d(k) + N} defines a linear isomorphism between M 12N and L k,n. As a corollary to Theorem 1.1, we consider the distribution of non-ordinary primes for normalized Hecke eigenforms. First we recall the following well known problem (see Gouvêa s expository article [1]). Problem. Suppose that f(z) = n=1 a f(n)q n S k is a normalized Hecke eigenform. A prime p is non-ordinary for f(z) if a f (p) 0 Are there infinitely many non-ordinary primes for f(z)? Although there are strong results on the more general problem for very special modular forms on congruence subgroups Γ 0 (M) (e.g. such as CM cusp forms, and weight 2 newforms associated to elliptic curves over Q), little is known. Using Theorem 1.1, we obtain elementary results related to this question. The following theorem applies for all forms when p = 2 and 3, and requires that δ(k) 0 for primes p 5. Theorem 1.2. Let p be prime, and suppose that f(z) = n=0 a f(n)q n M k O L [[q]], where O L denotes the algebraic integers of a number field L.

3 RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES 3 (1) If p {2, 3}, and b 1 is an integer for which 12p b 2 k, then a f (p b ) 0 (2) Suppose that p 5, and that δ(k) {4, 6, 8, 10, 14}. If b 1 is an odd integer, and a 0 is an integer for which then where k = (δ(k) 2)p b + 2 a(p 1), a f (p b ) (24 + α k )a f (0) α k := 2(14 δ(k)) B 14 δ(k) Z. (mod p), Remark. Theorem 1.2 does not include cases where p 5 is prime and δ(k) = 0. The condition on k in the statement of Theorem 1.2 (2) never holds when δ(k) = 0. More to the point, the conclusion of Theorem 1.2 (2) does not always hold. For example, p = 13 is an ordinary prime for (z). Theorem 1.2 allows us to re-prove some results of Hatada [2] (in the case where p = 2 and 3) and Hida [3, 4, 5] (for primes p 5) on non-ordinary primes. Corollary 1.3. Let p be prime, and suppose that f(z) = n=1 a f(n)q n S k is a normalized Hecke eigenform. Let L f be the number field generated by the coefficients of f(z), and let p O Lf be any prime ideal above p. (1) If p = 2 or 3, then a f (p) 0 (2) If p 5 is prime, δ(k) {4, 6, 8, 10, 14} and k δ(k) (mod p 1), then a f (p) 0 Remark. For primes p 5, it is well known that (see page 164 of [7]) E p 1 (z) 1 Consequently, a prime p 5 is non-ordinary for an eigenform f(z) S k if and only if it is non-ordinary for every mod p eigenform f(z) E p 1 (z) a, where a is a non-negative integer. Since δ(k) often does not equal δ(k + a(p 1)), we are able to apply Corollary 1.3 and find all of the small non-ordinary primes for eigenforms as in Hida s work. For example, the primes underlined in the table (see page 203 of [1]) below are examples of Corollary 1.3 applied to low weight eigenforms f(z) and their congruent companions f(z) E p 1 (z) a. Eigenform f(z) Primes p 10 6 for which a f (p) 0 (mod p) (z) 2, 3, 5, 7, 2411 (z)e 4 (z) 2, 3, 5, 7, 11, 13, 59, 15271, (z)e 6 (z) 2, 3, 5, 7, 11, 13 (z)e 8 (z) 2, 3, 5, 7, 11, 13, 17, 3371, (z)e 10 (z) 2, 3, 5, 7, 13, 17, 19 (z)e 14 (z) 2, 3, 5, 7, 11, 13, 17, 19, 23

4 4 YOUNGJU CHOIE, WINFRIED KOHNEN, AND KEN ONO Acknowledgements The authors thank Scott Ahlgren for his comments on an early version of this paper. 2. Proof of Theorem 1.1 The proof is a generalization of a method of Siegel [8] where the N = 0 case is treated. Proof of Theorem 1.1. Let us first show that (2.1) b(k, N, g; ν)a f (ν) = 0 for all g M 12N and all f(z) = a f(ν)q ν M k. If we let G(z) := E 14 δ(k) g(z), (z) then (2.1) is equivalent to the assertion that the constant term of the series Gf is zero. The dimension formula (1.3) implies that fg E δ(k) 1 is a meromorphic modular function on SL 2 (Z) of weight zero. Since k δ(k) (mod 12), we find from the valence formula (for example, see page 6 of [7]) that f/e δ(k) is holomorphic on H. Therefore it follows that fg E δ(k) 1 is a polynomial in the Hauptmodul j(z): j(z) = E 4(z) 3 (z) = q On the other hand, an easy calculation (for example, see equation (8) of [8]) reveals that Moreover, we have that 1 d 2πi dz j = E 14. j m d dz j = 1 d m + 1 dz jm+1 (m Z, m 0). d Since the constant term in the Fourier expansion of dz jm+1 is zero, by linearity it follows that fg E14 Gf = E δ(k) 1 has constant term zero. This confirms (2.1) since E 14 δ(k) (z)e δ(k) (z) = E 14 (z). Suppose that f(z) = n=0 a f(n)q n M k. The map φ k,n is clearly linear, and it is injective, since a modular form of weight l 2 (and vanishing at infinity if l = 0) is identically zero. To complete the proof, recall the well-known fact (see part I, chap. III,

5 RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES 5 sect. 4 of [7]) that the d(k) functionals {a f (0), a f (1),...,a f (d(k) 1)} form a basis of the dual space Mk. Therefore, it follows that This proves the theorem. dim L k,n = N + 1 = dimm 12N. Remark. If k 2 (mod 4), then E 14 δ(k) is either 1, E 4 or E 8 and hence has positive Fourier coefficients. Therefore, taking N = 0, we obtain a linear relation d(k) c ν a f (ν) = 0 between the first d(k) + 1 Fourier coefficients of modular forms in M k where all the c ν are strictly positive (this was observed in [8]). In particular, this implies that for k 2 (mod 4) the first sign change of the Fourier coefficients of a non-zero cusp form f M k with real Fourier coefficients already occurs among the first d(k) + 1 coefficients (and this bound is sharp, too, as is easily seen). If k 0 (mod 4), the above reasoning breaks down. To our knowledge, an answer to the corresponding question on the first sign change remains open in these cases. For some general results about sign changes of cusp forms on rather general subgroups of SL 2 (R), we refer to [6]. Remark. A similar result as stated in the Theorem 1.1 can certainly be proved for modular forms on genus zero subgroups of SL 2 (Z) (and in particular for half-integral weight modular forms of level 4). 3. Proof of Theorem 1.2 and Corollary 1.3 We begin by restating one of the main conclusions of Theorem 1.1 in a convenient form. Theorem 3.1. If f(z) M k and g(z) M 12N, then the constant term of is zero. E 14 δ(k) (z)g(z) (z) f(z) Proof of Theorem 1.2. First we prove Theorem 1.2 (2). Define g(z) by (3.1) g(z) := E 14 δ(k) (z) pb 1 E p 1 (z) a. The fact that b 1 is odd implies that g(z) M 12N, where (3.2) N = (pb 1)(14 δ(k)) + a(p 1) Z To see this, observe that the given representation of k implies that a(p 1) = (k 2) + (δ(k) 2)p b (p 1)(k 2) (mod 12). Formula (3.2) combined with (1.3) and the given representation of k implies that d(k) + N = p b. Theorem 3.1, combined with the fact (see page 164 of [7]) that E p 1 (z) 1 (mod p)

6 6 YOUNGJU CHOIE, WINFRIED KOHNEN, AND KEN ONO shows that the constant term of E 14 δ(k) (z) pb E p 1 (z) a f(z) E 14 δ(k)(p b z) f(z) (z) pb (p b z) ( )( ) q pb q pb α k q pb + f(z) ( ( ) ) q pb + (24 + α k ) + a f (n)q n (mod p) is zero modulo p. The second line in the displayed formula above is obtained by explicitly computing the first three terms of 1/ (p b z). The conclusion of Theorem 1.2 (2) follows immediately. To prove Theorem 1.2 (1), one argues as in the δ(k) = 14 and N = 0 case above. In this case, we have E 14 δ(k) (z) = E 0 (z) = 1. One simply replaces E p 1 (z) a by E 12p b +2 k(z) M 12p b +2 k in (3.1). Here we require that 12p b + 2 k 4. The congruence for E p 1 (z) is replaced by the universal congruence E k (z) 1 (mod 24). Proof of Corollary 1.3. Suppose that f(z) = n=1 a f(n)q n S k is a normalized Hecke eigenform. We begin by establishing, in each case, that there is a positive integer b for which (3.3) a f (p b ) 0 If p = 2 or 3, then Theorem 1.2 (1) implies (3.3) since a f (0) = 0. If p 5 is prime, then there are integers 1 b 1 (mod 2) and a 0 for which k = (δ(k) 2)p b + 2 a(p 1) = (δ(k) 2)(p 1 + 1) b + 2 a(p 1) δ(k) (mod p 1). By Theorem 1.2 (2), since a f (0) = 0, we obtain (3.3). The definition of the Hecke operators implies, for every non-negative integer n, that a f (p)a f (p n ) = a f (p n+1 ) + p k 1 a f (p n 1 ) a f (p n+1 ) By induction, we have that a f (p b ) a f (p) b Corollary 1.3 follows immediately from the truth of (3.3). References n=0 [1] F. Q. Gouvêa, Non-ordinary primes: A story, Exp. Math. 6, (1997), pages [2] K. Hatada, Eigenvalues of Hecke operators on SL 2 (Z), Math. Ann. 239 (1979), pages [3] H. Hida, Galois representations into GL 2 (Z p [[x]]) attached to ordinary cusp forms, Invent. Math. 85 (1986), pages [4] H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup., (4) 19 (1986), pages [5] H. Hida, Theory of p-adic Hecke algebras and Galois representations, Sugaka Expositions 2 (1989), pages

7 RELATIONS BETWEEN MODULAR FORM COEFFICIENTS AND NON-ORDINARY PRIMES 7 [6] M. Knopp, W. Kohnen, and W. Pribitkin, On the signs of Fourier coefficients of cusp forms, The Ramanujan J. 7 (2003), pages [7] S. Lang, Introduction to modular forms, Springer-Verlag, Berlin, [8] C. L. Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Göttingen Math.-Phys. KL, II, (1969), pages Department of Mathematics, Pohang Institute of Science and Technology, Pohang Korea address: Universität Heidelberg, Mathematisches Institut, INF 288, D-69120, Heidelberg, Germany address: Department of Mathematics, University of Wisconsin, Madison, WI address:

### arxiv: v1 [math.nt] 28 Jan 2010

NON VANISHING OF CENTRAL VALUES OF MODULAR L-FUNCTIONS FOR HECKE EIGENFORMS OF LEVEL ONE D. CHOI AND Y. CHOIE arxiv:00.58v [math.nt] 8 Jan 00 Abstract. Let F(z) = n= a(n)qn be a newform of weight k and

### THE ARITHMETIC OF BORCHERDS EXPONENTS. Jan H. Bruinier and Ken Ono

THE ARITHMETIC OF BORCHERDS EXPONENTS Jan H. Bruinier and Ken Ono. Introduction and Statement of Results. Recently, Borcherds [B] provided a striking description for the exponents in the naive infinite

### Representations of integers as sums of an even number of squares. Özlem Imamoḡlu and Winfried Kohnen

Representations of integers as sums of an even number of squares Özlem Imamoḡlu and Winfried Kohnen 1. Introduction For positive integers s and n, let r s (n) be the number of representations of n as a

### THE ARITHMETIC OF THE COEFFICIENTS OF HALF INTEGRAL WEIGHT EISENSTEIN SERIES. H(1, n)q n =

THE ARITHMETIC OF THE COEFFICIENTS OF HALF INTEGRAL WEIGHT EISENSTEIN SERIES ANTAL BALOG, WILLIAM J. MCGRAW AND KEN ONO 1. Introduction and Statement of Results If H( n) denotes the Hurwitz-Kronecer class

### DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS. George E. Andrews and Ken Ono. February 17, Introduction and Statement of Results

DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS George E. Andrews and Ken Ono February 7, 2000.. Introduction and Statement of Results Dedekind s eta function ηz, defined by the infinite product ηz

### REPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES. Ken Ono. Dedicated to the memory of Robert Rankin.

REPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES Ken Ono Dedicated to the memory of Robert Rankin.. Introduction and Statement of Results. If s is a positive integer, then let rs; n denote the number of

### DIVISIBILITY AND DISTRIBUTION OF PARTITIONS INTO DISTINCT PARTS

DIVISIBILITY AND DISTRIBUTION OF PARTITIONS INTO DISTINCT PARTS JEREMY LOVEJOY Abstract. We study the generating function for (n), the number of partitions of a natural number n into distinct parts. Using

### SIMULTANEOUS SIGN CHANGE OF FOURIER-COEFFICIENTS OF TWO CUSP FORMS

SIMULTANEOUS SIGN CHANGE OF FOURIER-COEFFICIENTS OF TWO CUSP FORMS SANOLI GUN, WINFRIED KOHNEN AND PURUSOTTAM RATH ABSTRACT. We consider the simultaneous sign change of Fourier coefficients of two modular

### NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES

NON-VANISHING OF THE PARTITION FUNCTION MODULO SMALL PRIMES MATTHEW BOYLAN Abstract Let pn be the ordinary partition function We show, for all integers r and s with s 1 and 0 r < s, that #{n : n r mod

### PARITY OF THE COEFFICIENTS OF KLEIN S j-function

PARITY OF THE COEFFICIENTS OF KLEIN S j-function CLAUDIA ALFES Abstract. Klein s j-function is one of the most fundamental modular functions in number theory. However, not much is known about the parity

### Sign changes of Fourier coefficients of cusp forms supported on prime power indices

Sign changes of Fourier coefficients of cusp forms supported on prime power indices Winfried Kohnen Mathematisches Institut Universität Heidelberg D-69120 Heidelberg, Germany E-mail: winfried@mathi.uni-heidelberg.de

### QUADRATIC CONGRUENCES FOR COHEN - EISENSTEIN SERIES.

QUADRATIC CONGRUENCES FOR COHEN - EISENSTEIN SERIES. P. GUERZHOY The notion of quadratic congruences was introduced in the recently appeared paper [1]. In this note we present another, somewhat more conceptual

### Abstract. Gauss s hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form

GAUSS S 2 F HYPERGEOMETRIC FUNCTION AND THE CONGRUENT NUMBER ELLIPTIC CURVE AHMAD EL-GUINDY AND KEN ONO Abstract Gauss s hypergeometric function gives a modular parameterization of period integrals of

### CONGRUENCE PROPERTIES FOR THE PARTITION FUNCTION. Department of Mathematics Department of Mathematics. Urbana, Illinois Madison, WI 53706

CONGRUENCE PROPERTIES FOR THE PARTITION FUNCTION Scott Ahlgren Ken Ono Department of Mathematics Department of Mathematics University of Illinois University of Wisconsin Urbana, Illinois 61801 Madison,

### On Rankin-Cohen Brackets of Eigenforms

On Rankin-Cohen Brackets of Eigenforms Dominic Lanphier and Ramin Takloo-Bighash July 2, 2003 1 Introduction Let f and g be two modular forms of weights k and l on a congruence subgroup Γ. The n th Rankin-Cohen

### REDUCTION OF CM ELLIPTIC CURVES AND MODULAR FUNCTION CONGRUENCES

REDUCTION OF CM ELLIPTIC CURVES AND MODULAR FUNCTION CONGRUENCES NOAM ELKIES, KEN ONO AND TONGHAI YANG 1. Introduction and Statement of Results Let j(z) be the modular function for SL 2 (Z) defined by

### HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS.

HECKE OPERATORS ON CERTAIN SUBSPACES OF INTEGRAL WEIGHT MODULAR FORMS. MATTHEW BOYLAN AND KENNY BROWN Abstract. Recent works of Garvan [2] and Y. Yang [7], [8] concern a certain family of half-integral

### GUO-NIU HAN AND KEN ONO

HOOK LENGTHS AND 3-CORES GUO-NIU HAN AND KEN ONO Abstract. Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions,

### ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3

ARITHMETIC OF THE 13-REGULAR PARTITION FUNCTION MODULO 3 JOHN J WEBB Abstract. Let b 13 n) denote the number of 13-regular partitions of n. We study in this paper the behavior of b 13 n) modulo 3 where

### AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION

AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION KATHRIN BRINGMANN AND KEN ONO 1 Introduction and Statement of Results A partition of a non-negative integer n is a non-increasing sequence of positive integers

### CONGRUENCES FOR BROKEN k-diamond PARTITIONS

CONGRUENCES FOR BROKEN k-diamond PARTITIONS MARIE JAMESON Abstract. We prove two conjectures of Paule and Radu from their recent paper on broken k-diamond partitions. 1. Introduction and Statement of Results

### THE NUMBER OF PARTITIONS INTO DISTINCT PARTS MODULO POWERS OF 5

THE NUMBER OF PARTITIONS INTO DISTINCT PARTS MODULO POWERS OF 5 JEREMY LOVEJOY Abstract. We establish a relationship between the factorization of n+1 and the 5-divisibility of Q(n, where Q(n is the number

### Introductory comments on the eigencurve

Introductory comments on the eigencurve Handout # 5: March 8, 2006 (These are brief indications, hardly more than an annotated list, of topics mentioned in my lectures. ) 1 The basic Hecke diagram As before

### arxiv: v3 [math.nt] 28 Jul 2012

SOME REMARKS ON RANKIN-COHEN BRACKETS OF EIGENFORMS arxiv:1111.2431v3 [math.nt] 28 Jul 2012 JABAN MEHER Abstract. We investigate the cases for which products of two quasimodular or nearly holomorphic eigenforms

### Introduction to Modular Forms

Introduction to Modular Forms Lectures by Dipendra Prasad Written by Sagar Shrivastava School and Workshop on Modular Forms and Black Holes (January 5-14, 2017) National Institute of Science Education

### DIVISIBILITY PROPERTIES OF THE 5-REGULAR AND 13-REGULAR PARTITION FUNCTIONS

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (008), #A60 DIVISIBILITY PROPERTIES OF THE 5-REGULAR AND 13-REGULAR PARTITION FUNCTIONS Neil Calkin Department of Mathematical Sciences, Clemson

### A SHORT NOTE ON P-ADIC FAMILIES OF HILBERT MODULAR FORMS

A SHORT NOTE ON P-ADIC FAMILIES OF HILBERT MODULAR FORMS AFTAB PANDE Abstract. We extend previous work of the author using an idea of Buzzard and give an elementary construction of non-ordinary p-adic

### PARITY OF THE PARTITION FUNCTION. (Communicated by Don Zagier)

ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 1, Issue 1, 1995 PARITY OF THE PARTITION FUNCTION KEN ONO (Communicated by Don Zagier) Abstract. Let p(n) denote the number

### denote the Dirichlet character associated to the extension Q( D)/Q, that is χ D

January 0, 1998 L-SERIES WITH NON-ZERO CENTRAL CRITICAL VALUE Kevin James Department of Mathematics Pennsylvania State University 18 McAllister Building University Park, Pennsylvania 1680-6401 Phone: 814-865-757

### Cusp forms and the Eichler-Shimura relation

Cusp forms and the Eichler-Shimura relation September 9, 2013 In the last lecture we observed that the family of modular curves X 0 (N) has a model over the rationals. In this lecture we use this fact

### On the generation of the coefficient field of a newform by a single Hecke eigenvalue

On the generation of the coefficient field of a newform by a single Hecke eigenvalue Koopa Tak-Lun Koo and William Stein and Gabor Wiese November 2, 27 Abstract Let f be a non-cm newform of weight k 2

### The kappa function. [ a b. c d

The kappa function Masanobu KANEKO Masaaki YOSHIDA Abstract: The kappa function is introduced as the function κ satisfying Jκτ)) = λτ), where J and λ are the elliptic modular functions. A Fourier expansion

### AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 11, November 007, Pages 3507 3514 S 000-9939(07)08883-1 Article electronically published on July 7, 007 AN ARITHMETIC FORMULA FOR THE

### A Note on the Transcendence of Zeros of a Certain Family of Weakly Holomorphic Forms

A Note on the Transcendence of Zeros of a Certain Family of Weakly Holomorphic Forms Jennings-Shaffer C. & Swisher H. (014). A Note on the Transcendence of Zeros of a Certain Family of Weakly Holomorphic

### DIVISIBILITY CRITERIA FOR CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS

IVISIBILITY CRITERIA FOR CLASS NUMBERS OF IMAGINARY QUARATIC FIELS PAUL JENKINS AN KEN ONO Abstract. In a recent aer, Guerzhoy obtained formulas for certain class numbers as -adic limits of traces of singular

### Mod p Galois representations attached to modular forms

Mod p Galois representations attached to modular forms Ken Ribet UC Berkeley April 7, 2006 After Serre s article on elliptic curves was written in the early 1970s, his techniques were generalized and extended

### EXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.

EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is

### RANKIN-COHEN BRACKETS AND VAN DER POL-TYPE IDENTITIES FOR THE RAMANUJAN S TAU FUNCTION

RANKIN-COHEN BRACKETS AND VAN DER POL-TYPE IDENTITIES FOR THE RAMANUJAN S TAU FUNCTION B. RAMAKRISHNAN AND BRUNDABAN SAHU Abstract. We use Rankin-Cohen brackets for modular forms and quasimodular forms

### Projects on elliptic curves and modular forms

Projects on elliptic curves and modular forms Math 480, Spring 2010 In the following are 11 projects for this course. Some of the projects are rather ambitious and may very well be the topic of a master

### A Motivated Introduction to Modular Forms

May 3, 2006 Outline of talk: I. Motivating questions II. Ramanujan s τ function III. Theta Series IV. Congruent Number Problem V. My Research Old Questions... What can you say about the coefficients of

### ON THE POSITIVITY OF THE NUMBER OF t CORE PARTITIONS. Ken Ono. 1. Introduction

ON THE POSITIVITY OF THE NUMBER OF t CORE PARTITIONS Ken Ono Abstract. A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. A Ferrers graph represents a

### FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS

Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ

### Congruences, graphs and modular forms

Congruences, graphs and modular forms Samuele Anni (IWR - Universität Heidelberg) joint with Vandita Patel (University of Warwick) (work in progress) FoCM 2017, UB, 12 th July 2017 The theory of congruences

### Ramanujan-type congruences for broken 2-diamond partitions modulo 3

Progress of Projects Supported by NSFC. ARTICLES. SCIENCE CHINA Mathematics doi: 10.1007/s11425-014-4846-7 Ramanujan-type congruences for broken 2-diamond partitions modulo 3 CHEN William Y.C. 1, FAN Anna

### Applications of modular forms to partitions and multipartitions

Applications of modular forms to partitions and multipartitions Holly Swisher Oregon State University October 22, 2009 Goal The goal of this talk is to highlight some applications of the theory of modular

### On the zeros of certain modular forms

On the zeros of certain modular forms Masanobu Kaneko Dedicated to Professor Yasutaka Ihara on the occasion of his 60th birthday. The aim of this short note is to list several families of modular forms

### MULTILINEAR OPERATORS ON SIEGEL MODULAR FORMS OF GENUS 1 AND 2

MULTILINEAR OPERATORS ON SIEGEL MODULAR FORMS OF GENUS 1 AND 2 YOUNGJU CHOIE 1. Introduction Classically, there are many interesting connections between differential operators and the theory of elliptic

### On the equality case of the Ramanujan Conjecture for Hilbert modular forms

On the equality case of the Ramanujan Conjecture for Hilbert modular forms Liubomir Chiriac Abstract The generalized Ramanujan Conjecture for unitary cuspidal automorphic representations π on GL 2 posits

### SOME REMARKS ON THE RESNIKOFF-SALDAÑA CONJECTURE

SOME REMARKS ON THE RESNIKOFF-SALDAÑA CONJECTURE SOUMYA DAS AND WINFRIED KOHNEN Abstract. We give some (weak) evidence towards the Resnikoff-Saldaña conjecture on the Fourier coefficients of a degree 2

### RANKIN-COHEN BRACKETS AND SERRE DERIVATIVES AS POINCARÉ SERIES. φ k M = 1 2

RANKIN-COHEN BRACKETS AND SERRE DERIVATIVES AS POINCARÉ SERIES BRANDON WILLIAMS Abstract. We give expressions for the Serre derivatives of Eisenstein and Poincaré series as well as their Rankin-Cohen brackets

### 4 LECTURES ON JACOBI FORMS. 1. Plan

4 LECTURES ON JACOBI FORMS YOUNGJU CHOIE Abstract. 1. Plan This lecture series is intended for graduate students or motivated undergraduate students. We introduce a concept of Jacobi forms and try to explain

### Ramanujan-type Congruences for Broken 2-Diamond Partitions Modulo 3

Ramanujan-type Congruences for Broken 2-Diamond Partitions Modulo 3 William Y.C. Chen 1, Anna R.B. Fan 2 and Rebecca T. Yu 3 1,2,3 Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin 300071,

### SOME CONGRUENCES FOR TRACES OF SINGULAR MODULI

SOME CONGRUENCES FOR TRACES OF SINGULAR MODULI P. GUERZHOY Abstract. We address a question posed by Ono [7, Problem 7.30], prove a general result for powers of an arbitrary prime, and provide an explanation

### Spaces of Weakly Holomorphic Modular Forms in Level 52. Daniel Meade Adams

Spaces of Weakly Holomorphic Modular Forms in Level 52 Daniel Meade Adams A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master

### SOME RECURRENCES FOR ARITHMETICAL FUNCTIONS. Ken Ono, Neville Robbins, Brad Wilson. Journal of the Indian Mathematical Society, 62, 1996, pages

SOME RECURRENCES FOR ARITHMETICAL FUNCTIONS Ken Ono, Neville Robbins, Brad Wilson Journal of the Indian Mathematical Society, 6, 1996, pages 9-50. Abstract. Euler proved the following recurrence for p(n),

### SYSTEMS OF ORTHOGONAL POLYNOMIALS ARISING FROM THE

SYSTEMS OF ORTHOGONAL POLYNOMIALS ARISING FROM THE MODULAR -FUNCTION STEPHANIE BASHA JAYCE GETZ HARRIS NOVER AND EMMA SMITH Abstract Let S p x F p [x] be the polynomial whose zeros are the -invariants

### The Galois Representation Associated to Modular Forms (Part I)

The Galois Representation Associated to Modular Forms (Part I) Modular Curves, Modular Forms and Hecke Operators Chloe Martindale May 20, 2015 Contents 1 Motivation and Background 1 2 Modular Curves 2

### 20 The modular equation

18.783 Elliptic Curves Spring 2015 Lecture #20 04/23/2015 20 The modular equation In the previous lecture we defined modular curves as quotients of the extended upper half plane under the action of a congruence

### Shifted Convolution L-Series Values of Elliptic Curves

Shifted Convolution L-Series Values of Elliptic Curves Nitya Mani (joint with Asra Ali) December 18, 2017 Preliminaries Modular Forms for Γ 0 (N) Modular Forms for Γ 0 (N) Definition The congruence subgroup

### SIGN CHANGES OF COEFFICIENTS OF HALF INTEGRAL WEIGHT MODULAR FORMS

SIGN CHANGES OF COEFFICIENTS OF HALF INTEGRAL WEIGHT MODULAR FORMS JAN HENDRIK BRUINIER AND WINFRIED KOHNEN Abstract. For a half integral weight modular form f we study the signs of the Fourier coefficients

### p-adic families of modular forms

April 3, 2009 Plan Background and Motivation Lecture 1 Background and Motivation Overconvergent p-adic modular forms The canonical subgroup and the U p operator Families of p-adic modular forms - Strategies

### Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

### A weak multiplicity-one theorem for Siegel modular forms

A weak multiplicity-one theorem for Siegel modular forms Rudolf Scharlau Department of Mathematics University of Dortmund 44221 Dortmund, Germany scharlau@math.uni-dortmund.de Lynne Walling Department

### CONGRUENCES FOR POWERS OF THE PARTITION FUNCTION

CONGRUENCES FOR POWERS OF THE PARTITION FUNCTION MADELINE LOCUS AND IAN WAGNER Abstract. Let p tn denote the number of partitions of n into t colors. In analogy with Ramanujan s work on the partition function,

### SHIMURA LIFTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS ARISING FROM THETA FUNCTIONS

SHIMURA LIFTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS ARISING FROM THETA FUNCTIONS DAVID HANSEN AND YUSRA NAQVI Abstract In 1973, Shimura [8] introduced a family of correspondences between modular forms

### Eisenstein Series and Modular Differential Equations

Canad. Math. Bull. Vol. 55 (2), 2012 pp. 400 409 http://dx.doi.org/10.4153/cmb-2011-091-3 c Canadian Mathematical Society 2011 Eisenstein Series and Modular Differential Equations Abdellah Sebbar and Ahmed

### Universität Regensburg Mathematik

Universität Regensburg Mathematik On projective linear groups over finite fields as Galois groups over the rational numbers Gabor Wiese Preprint Nr. 14/2006 On projective linear groups over finite fields

### SHIMURA LIFTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS ARISING FROM THETA FUNCTIONS

SHIMURA LIFTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS ARISING FROM THETA FUNCTIONS DAVID HANSEN AND YUSRA NAQVI Abstract. In 1973, Shimura [8] introduced a family of correspondences between modular forms

### Modular forms and the Hilbert class field

Modular forms and the Hilbert class field Vladislav Vladilenov Petkov VIGRE 2009, Department of Mathematics University of Chicago Abstract The current article studies the relation between the j invariant

### Divisibility of the 5- and 13-regular partition functions

Divisibility of the 5- and 13-regular partition functions 28 March 2008 Collaborators Joint Work This work was begun as an REU project and is joint with Neil Calkin, Nathan Drake, Philip Lee, Shirley Law,

### TATE-SHAFAREVICH GROUPS OF THE CONGRUENT NUMBER ELLIPTIC CURVES. Ken Ono. X(E N ) is a simple

TATE-SHAFAREVICH GROUPS OF THE CONGRUENT NUMBER ELLIPTIC CURVES Ken Ono Abstract. Using elliptic modular functions, Kronecker proved a number of recurrence relations for suitable class numbers of positive

### A brief overview of modular and automorphic forms

A brief overview of modular and automorphic forms Kimball Martin Original version: Fall 200 Revised version: June 9, 206 These notes were originally written in Fall 200 to provide a very quick overview

### Families of modular forms.

Families of modular forms. Kevin Buzzard June 7, 2000 Abstract We give a down-to-earth introduction to the theory of families of modular forms, and discuss elementary proofs of results suggesting that

### THE ANDREWS-STANLEY PARTITION FUNCTION AND p(n): CONGRUENCES

THE ANDREWS-STANLEY PARTITION FUNCTION AND pn: CONGRUENCES HOLLY SWISHER Abstract R Stanley formulated a partition function tn which counts the number of partitions π for which the number of odd parts

### A NOTE ON THE SHIMURA CORRESPONDENCE AND THE RAMANUJAN τ(n) FUNCTION

A NOTE ON THE SHIMURA CORRESPONDENCE AND THE RAMANUJAN τ(n) FUNCTION KEN ONO Abstract. The Shimura correspondence is a family of maps which sends modular forms of half-integral weight to forms of integral

### 20 The modular equation

18.783 Elliptic Curves Lecture #20 Spring 2017 04/26/2017 20 The modular equation In the previous lecture we defined modular curves as quotients of the extended upper half plane under the action of a congruence

### A NOTE ON THE EXISTENCE OF CERTAIN INFINITE FAMILIES OF IMAGINARY QUADRATIC FIELDS

A NOTE ON THE EXISTENCE OF CERTAIN INFINITE FAMILIES OF IMAGINARY QUADRATIC FIELDS IWAO KIMURA ABSTRACT. Let l > 3 be an odd prime. Let S 0, S +, S be mutually disjoint nite sets of rational primes. For

### Rational Equivariant Forms

CRM-CICMA-Concordia University Mai 1, 2011 Atkin s Memorial Lecture and Workshop This is joint work with Abdellah Sebbar. Notation Let us fix some notation: H := {z C; I(z) > 0}, H := H P 1 (Q), SL 2 (Z)

### MATH G9906 RESEARCH SEMINAR IN NUMBER THEORY (SPRING 2014) LECTURE 1 (FEBRUARY 7, 2014) ERIC URBAN

MATH G9906 RESEARCH SEMINAR IN NUMBER THEORY (SPRING 014) LECTURE 1 (FEBRUARY 7, 014) ERIC URBAN NOTES TAKEN BY PAK-HIN LEE 1. Introduction The goal of this research seminar is to learn the theory of p-adic

### MA4H9 Modular Forms: Problem Sheet 2 Solutions

MA4H9 Modular Forms: Problem Sheet Solutions David Loeffler December 3, 010 This is the second of 3 problem sheets, each of which amounts to 5% of your final mark for the course This problem sheet will

### Computer methods for Hilbert modular forms

Computer methods for Hilbert modular forms John Voight University of Vermont Workshop on Computer Methods for L-functions and Automorphic Forms Centre de Récherche Mathématiques (CRM) 22 March 2010 Computer

### Galois groups with restricted ramification

Galois groups with restricted ramification Romyar Sharifi Harvard University 1 Unique factorization: Let K be a number field, a finite extension of the rational numbers Q. The ring of integers O K of K

### A MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET. March 7, 2017

A MOD-p ARTIN-TATE CONJECTURE, AND GENERALIZED HERBRAND-RIBET DIPENDRA PRASAD March 7, 2017 Abstract. Following the natural instinct that when a group operates on a number field then every term in the

### CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)

CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit and Yee introduced partition

### Congruences for the Coefficients of Modular Forms and Applications to Number Theory

Congruences for the Coefficients of Modular Forms and Applications to Number Theory By Karl Mahlburg A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

### CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ω(q) AND ν(q)

CONGRUENCES RELATED TO THE RAMANUJAN/WATSON MOCK THETA FUNCTIONS ωq) AND νq) GEORGE E. ANDREWS, DONNY PASSARY, JAMES A. SELLERS, AND AE JA YEE Abstract. Recently, Andrews, Dixit, and Yee introduced partition

### Computing central values of twisted L-functions of higher degre

Computing central values of twisted L-functions of higher degree Computational Aspects of L-functions ICERM November 13th, 2015 Computational challenges We want to compute values of L-functions on the

### Mock modular forms and their shadows

Mock modular forms and their shadows Zachary A. Kent Emory University Classical Eichler-Shimura Theory Modular Forms Basic Definitions Classical Eichler-Shimura Theory Modular Forms Basic Definitions Notation:

### SERRE S CONJECTURE AND BASE CHANGE FOR GL(2)

SERRE S CONJECTURE AND BASE CHANGE OR GL(2) HARUZO HIDA 1. Quaternion class sets A quaternion algebra B over a field is a simple algebra of dimension 4 central over a field. A prototypical example is the

### On some congruence properties of elliptic curves

arxiv:0803.2809v5 [math.nt] 19 Jun 2009 On some congruence properties of elliptic curves Derong Qiu (School of Mathematical Sciences, Institute of Mathematics and Interdisciplinary Science, Capital Normal

### TWISTS OF ELLIPTIC CURVES. Ken Ono

TWISTS OF ELLIPTIC CURVES Ken Ono Abstract. If E is an elliptic curve over Q, then let E(D) denote the D quadratic twist of E. It is conjectured that there are infinitely many primes p for which E(p) has

### Calculation and arithmetic significance of modular forms

Calculation and arithmetic significance of modular forms Gabor Wiese 07/11/2014 An elliptic curve Let us consider the elliptic curve given by the (affine) equation y 2 + y = x 3 x 2 10x 20 We show its

### Raising the Levels of Modular Representations Kenneth A. Ribet

1 Raising the Levels of Modular Representations Kenneth A. Ribet 1 Introduction Let l be a prime number, and let F be an algebraic closure of the prime field F l. Suppose that ρ : Gal(Q/Q) GL(2, F) is

### CHIRANJIT RAY AND RUPAM BARMAN

ON ANDREWS INTEGER PARTITIONS WITH EVEN PARTS BELOW ODD PARTS arxiv:1812.08702v1 [math.nt] 20 Dec 2018 CHIRANJIT RAY AND RUPAM BARMAN Abstract. Recently, Andrews defined a partition function EOn) which

### Appendix A. Application to the three variable Rankin-Selberg p-adic L-functions. A corrigendum to [Ur14].

Appendix A. Application to the three variable Rankin-Selberg p-adic L-functions. A corrigendum to [r14]. A.1. Introduction. In [r14], the author introduced nearly overconvergent modular forms of finite

### Periods and congruences of various lifts

Miyama Conference Periods and congruences of various lifts KATSURADA Hidenori (Muroran I. T.) October 2010 1. Introduction G 12 (z) := Γ(12) 2(2π) 12 (c,d) Z 2 \{(0,0)} (cz + d) 12 the Eisenstein series

### Introduction to modular forms Perspectives in Mathematical Science IV (Part II) Nagoya University (Fall 2018)

Introduction to modular forms Perspectives in Mathematical Science IV (Part II) Nagoya University (Fall 208) Henrik Bachmann (Math. Building Room 457, henrik.bachmann@math.nagoya-u.ac.jp) Lecture notes