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RESEARCH STATEMENT
PAUL M. JENKINS
Introduction
My research has focused on modular forms and Maass forms and their applications in number theory. A modular form is a holomorphic function on the complex upper half plane satisfying a transformation law
f
az + b cz + d
= ε(a, b, c, d)(cz + d)kf (z),
for all
a b c d
in a subgroup of SL 2 (Z). Here ε(a, b, c, d) has absolute value 1, the weight k is
in 12 Z, and f satisfies certain growth conditions. Writing q = e^2 πiz^ , a modular form f (z) has a Fourier expansion
f (z) =
∑^ ∞
n=n 0
a(n)qn.
Maass forms are similar to modular forms, but are not holomorphic, and are required to be eigenfunctions of the hyperbolic Laplacian. Modular forms are objects of intense interest in number theory; most famously, they played a vital role in Wiles’ proof of Fermat’s Last Theorem. The coefficients a(n) of these forms play many different roles, appearing, for example, in the study of points on elliptic curves over finite fields, representations of integers by quadratic forms, values of the partition function, class numbers, the Birch and Swinnerton-Dyer conjecture, and values of L-functions. These coefficients make a prominent and interesting appearance in work of Borcherds. To explain, recall that many modular forms have simple infinite product expansions. For example, the weight 12 modular form ∆(z) has the expansion
∆(z) = q
∏^ ∞
n=
(1 − qn)^24.
The weight 4 Eisenstein series E 4 (z) has the more complicated expansion
E 4 (z) = 1 + 240
∑^ ∞
n=
σ 3 (n)qn^ = (1 − q)−^240 (1 − q^2 )^26760 (1 − q^3 )−^4096240 · · ·.
In his 1994 ICM lecture, Borcherds [2],[3] gave a striking description of the exponents in the infinite product expansion of modular forms with Heegner divisor, of which these are two examples, showing that the exponents in such an expansion appear as coefficients in the Fourier expansion of another modular form of weight 1/2. In particular, the product expansions of ∆ and E 4 above correspond to the weight 1/2 modular forms
12 θ = 12 + 24q + 24q^4 + 24q^9 + · · · ,
G(z) = q−^3 + 4 − 240 q + 26760q^4 − 85995 q^5 + · · · − 4096240 q^9 + · · ·.
The exponents of the (1−qn) in the infinite product expansions of ∆ and E 4 equal the coefficients of qn 2 in 12θ and G(z). Borcherds also proved that this map between spaces of modular forms is an isomorphism. 1
Theorem 1 (Borcherds). Let f (z) be an integer weight meromorphic modular form on SL 2 (Z) with Heegner divisor, integer coefficients, and leading coefficient 1. Then
f (z) = q−h
∏^ ∞
n=
(1 − qn)A(n
(^2) ) ,
where A(D) is the coefficient of qD^ in a certain nearly holomorphic modular form of weight 1 / 2 for the group Γ 0 (4).
Borcherds’ results are in fact more general, using automorphic forms on orthogonal groups. One question posed in his paper is whether a proof of this theorem exists that uses only the classical theory of modular forms. Zagier [28] answered this in the affirmative, and his work has inspired my research to date. Zagier’s results are phrased in terms of singular moduli. The usual j-function is the modular function for SL 2 (Z) with Fourier expansion j(z) = q−^1 + 744 + 196884q + · · ·.
Values of j at quadratic irrational points of discriminant −d < 0 are known as singular moduli. Singular moduli are algebraic integers which have been extensively studied in number theory; for example, they generate ring class field extensions of imaginary quadratic number fields. Zagier [28] initiated the study of Tr(d), the function giving the algebraic trace of a singular modulus of discriminant −d, and its generalization Trm(d), and showed that these traces appear as coefficients of a certain weight 3/2 nearly holomorphic modular form. To make this more precise, define gD as the unique weight 3/2 modular form on Γ 0 (4) with Fourier expansion
gD(z) = q−D^ + B 1 (D, 0) +
0 <d≡ 0 , 3 (mod 4)
B 1 (D, d)qd.
Applying the weight 3/2 Hecke operator T 32 (m^2 ), let
Bm(D, d) = the coefficient of qd^ in gD(z)
∣ T 3
2 (m^2 ).
Zagier proved the following theorem relating traces of singular moduli and the coefficients of the gD.
Theorem 2 (Zagier). If m ≥ 1 and −d < 0 is a discriminant, then
Trm(d) = −Bm(1, d). By exhibiting a striking duality between these coefficients Bm(D, d) and the coefficients of another sequence of modular forms, this time of weight 1/2, Zagier was able to give a new proof of Borcherds’ infinite product formulas (Theorem 1) with this theorem. However, although Theorem 2 implies Borcherds’ theorem and provides the desired “modular form” proof, it does not provide much new information about the Trm(d). For example, it does not give exact formulas for calculating the coefficients Bm(D, d), or tell us about their p-divisibility.
My results
Combining properties of Kloosterman sums with the duality between coefficients of modular forms of different weights, and using an expression for Trm(d) due to Duke [11], I gave a new proof [16] of Theorem 2, thus giving a new “modular form” proof of Borcherds’ theorem. An essential element in this new proof is an explicit expression for the coefficients B 1 (D, d) of the modular form gD. This formula was derived in my work with Bruinier and Ono [8] from certain results about weakly holomorphic Maass-Poincar´e series. More precisely, if H(d)
Theorem 6 (J. [16], Theorem 1.1). If p is an odd prime and −d, D ≡ 0 , 1 (mod 4) and n are positive integers, then
B 1 (D, p^2 nd) =pnB 1 (p^2 nD, d) +
∑^ n−^1
k=
D
p
)n−k− 1 ( B 1
D
p^2
, p^2 kd
− pk+1B 1
p^2 kD,
d p^2
n∑− 1
k=
D
p
)n−k− 1 ((( D p
−d p
pkB 1 (p^2 kD, d)
where B 1 (M, N ) = 0 if M or N is not an integer.
Taking D = 1 and recalling that Tr(d) = −B 1 (1, d), we have the following precise formula which implies Edixhoven’s congruences.
Corollary 7. If
(−d p
= 1, then Tr(p^2 nd) = −pnB 1 (p^2 n, d) ≡ 0 (mod pn).
Future research
Singular moduli may be generalized by replacing the j-function with a modular function of higher level, such as the Hauptmodul associated to other groups of genus zero, as in work of Kim [18] [19]. Osburn [26] proved that Ahlgren and Ono’s congruence extends to this case, suggesting that an analogue of Theorem 6 will also hold for higher levels and give information about the p-divisibility of traces of p^2 nd for these functions. I expect to work out such a generalization in the near future. In addition, the entire framework underlying these results may be generalized by examining modular forms of weights other than 3/2 or 1/2. Bringmann and Ono’s work [6] using Maass- Poincar´e series for these half-integral weight modular forms shows that similar duality theorems hold, and Zagier discussed the interpretation of the coefficients of these forms as traces of non- holomorphic modular functions. For example, the coefficients of certain modular forms of weights 5/2 and − 1 /2 correspond to traces of the function
K(z) =
j(z) − 576 +
E∗ 2 (z)E 4 (z)E 6 (z) 6∆(z)
where E 2 ∗ (z) is the non-holomorphic Eisenstein series of weight 2. A generalization of Theorem 6 would provide information about the p-adic properties of these traces, and should provide an arithmetic interpretation in terms of isomorphism classes of elliptic curves with complex multiplication. My preliminary work in this direction seems to confirm that an analogue of Theorem 6 is true in much greater generality, and I hope to clarify this speculation. Such a generalization would involve values of exceptional harmonic weak Maass forms at CM points. Recently, Bruinier and Funke [7] obtained results on traces of modular functions that hold in much greater generality; specifically, they removed the restriction to modular curves of genus zero inherent in Zagier’s work. Their method involves integrating arbitrary modular functions of weight zero against certain theta series to obtain modular forms whose positive Fourier coefficients are the desired traces. In addition, they obtain geometric interpretations for the constant term and negative Fourier coefficients, and recover Zagier’s results as special cases. I am interested in using the techniques and ideas I have developed in my research thus far to study the p-adic properties of these traces and their arithmetic interpretations. Although my work to date has been mostly related to Borcherds’ and Zagier’s work and to singular moduli, in the near future I plan to explore other questions. I am interested in the rela- tionship between central critical values of quadratic twists of even weight modular L-functions and the coefficients of associated cusp forms. This relates to the Birch and Swinnerton-Dyer
conjecture, one of the Clay Mathematics Institute Millennial Prize problems. The conjecture is as follows.
Conjecture 8 (Birch and Swinnerton-Dyer). For an elliptic curve E/Q,
ords=1 L(E, s) = rank(E).
Moreover, if Etor is the torsion subgroup of E over Q, X(E) is the Shafarevich-Tate group of E, Tam(E) is the Tamagawa number for E, Ω(E) is the real period of E, and R(E) is the regulator of E/Q, then
lim s→ 1
(s − 1)−^ rank(E)L(E, s)
2 rank(E)^ · |X(E)| · Tam(E) · Ω(E) · R(E) |Etor^2 |
Although this problem remains open, there have been some significant results in its direction. The most famous result is due to Kolyvagin [23], who proved that if the order of the L-function at s = 1 is 0 or 1, then
ords=1(L(E, s)) = rank(E) and |X(E)| < ∞.
This result was at first conditional, but the restrictions were removed by work of Gross and Zagier [14] and of Bump, Friedberg and Hoffstein [9] on the nonvanishing of central critical values of quadratic twists of L-functions. Further work in this direction has been done by Murty and Murty [24], Iwaniec [15], Waldspurger [27], and Ono and Skinner [25]. Gross and Zagier’s work relates the central value of an L-function to the canonical height of a Galois trace of a Heegner point; a natural question, then, would be whether the central critical values themselves can be interpreted as traces. Work of Kohnen and Zagier [21], [20], [22] and Waldspurger shows that central critical values of quadratic twists of L-functions can be written in terms of the squares of certain Fourier coefficients of half integral weight cusp forms. Using recent work of Bringmann and Ono [6] as a prototype, it should be possible to adapt theorems on Maass-Poincar´e series to cuspidal Poincar´e series to describe these L-values in terms of traces of some sort of modular invariant evaluated at CM points or their equivalent. I intend to explore this relationship thoroughly. I am also interested in learning more about equidistribution results, such as those arising in Duke’s work [12]. His theorems have interesting applications to the asymptotic behavior of Tr(d). More specifically, we may approximate the j-function by the q−^1 term in its Fourier expansion and examine the average difference G(d H)−(dTr() d)between traces of singular moduli and
their approximations G(d) arising from this truncation. Some care is required to define G(d) properly, as some “small” CM points are actually counted with multiplicity two. Since H(163) = 1, computing this difference for d = 163 gives the famous classical observation that
eπ
√ (^163) − 262537412640768744 = − 0. 0000000000007499...
and suggests that this average should be “small”. Duke was able to use methods he developed in [12] concerning the distribution of CM points to prove the following theorem [11], where ζ(s) is the Riemann zeta function.
Theorem 9 (Duke). As −d ranges over negative fundamental discriminants, we have
lim −d→−∞
Tr(d) − G(d) H(d)
ζ(−1)
I plan to develop several results of this kind with respect to generic CM points and singular moduli problems; the idea will be to determine when such limiting distributions come from values of zeta functions.
