Self-Dual Holomorphic Cusp Form Rigidity

• Self-dual holomorphic cusp forms are normalized newforms with trivial or quadratic nebentypus that ensure all Fourier coefficients are real.
• The principal theorem proves that the eventual sign patterns of Fourier coefficients uniquely determine the form up to scaling under precise level and weight conditions.
• The study employs analytic methods such as Rankin–Selberg convolution, Sato–Tate equidistribution, and Ramakrishnan’s lift, highlighting its impact on modular forms and L-function theory.

A self-dual holomorphic cusp form is a normalized newform $f \in S_k^{\mathrm{new}}(\Gamma_0(M))$, lying in the space of weight-$k$ holomorphic cusp forms on $\Gamma_0(M)$ (a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$), with trivial or quadratic nebentypus, for which the associated automorphic representation $\pi_f$ of $\mathrm{GL}_2(\mathbb{A}_\mathbb{Q})$ is isomorphic to its contragredient. This self-duality condition equivalently asserts that the central character $\omega_f$ of $\pi_f$ satisfies $\omega_f^2=1$, thus $\omega_f$ is at most quadratic and all Fourier coefficients $a_f(n)$ are real for all $n$.

1. Formal Definition and Context

Given integers $M \geq 1$ and $k \geq 2$, the space $S_k(\Gamma_0(M))$ consists of holomorphic cusp forms of weight $k$ and level $M$. Its subspace $S_k^{\mathrm{new}}(\Gamma_0(M))$ comprises newforms, that is, primitive Hecke eigenforms. For $f(z) = \sum_{n \geq 1} a_f(n) e^{2\pi i n z}$, $a_f(n) \in \mathbb{C}$, self-duality of $f$ is equivalent to $\pi_f \simeq \pi_f^\vee$ or, in terms of nebentypus, $\omega_f^2=1$. Forms with trivial or quadratic nebentypus therefore satisfy self-duality, and in this case all $a_f(n)$ are real. Self-dual holomorphic cusp forms play a foundational role in the arithmetic theory of modular forms and the analytic study of automorphic $L$-functions and representations (Booker, 5 Dec 2025).

2. Principal Theorem: Determination by Sign Patterns

Let $f \in S_k^{\mathrm{new}}(\Gamma_0(M))$ and $g \in S_\ell^{\mathrm{new}}(\Gamma_0(N))$ be nonzero normalized newforms with $a_f(n), a_g(n) \in \mathbb{R}$. The principal result, proven by Booker, establishes a rigidity phenomenon for self-dual holomorphic cusp forms under sign conditions:

Theorem (Booker):

Suppose $M, N \geq 1$ are such that $\mathrm{lcm}(M, N)$ is not divisible by $2^4$ nor by the square of any odd prime. Then, the following are equivalent:

1. $a_f(n)a_g(n) \geq 0$ for all sufficiently large $n \in \mathbb{N}$.
2. $(M, k) = (N, \ell)$ and $f(z) = c g(z)$ for some $c > 0$.

Consequently, if $f$ and $g$ are not proportional, then both sets $\{ n \in \mathbb{N} : a_f(n)a_g(n) > 0 \}$ and $\{ n \in \mathbb{N} : a_f(n)a_g(n) < 0 \}$ are infinite. This result demonstrates that—under mild, explicit conditions on the level—a self-dual newform is determined up to scaling by the eventual signs of its Fourier coefficients (Booker, 5 Dec 2025).

3. Hypotheses and Technical Conditions

Three technical conditions underlie the theorem:

• Newform and Non-CM Condition: Both forms $f$ and $g$ must be newforms (primitive Hecke eigenforms) and must not admit complex multiplication (CM), ensuring applicability of Rankin–Selberg and Sato–Tate methods.
• Level Restriction: $\mathrm{lcm}(M, N)$ must not be divisible by $2^4$ or by the square of any odd prime, excluding certain twist-minimal pathologies (see Remark (1) in (Booker, 5 Dec 2025)).
• Sign Assumption: There exists $N_0$ such that $a_f(n)a_g(n) \geq 0$ for all $n \geq N_0$ (eventual nonnegativity). In fact, the required assumption can be weakened to hold only along an arithmetic progression of positive density.

These conditions are necessary for both the analytic arguments and the rigidity conclusions; the level constraint is sharp, as explicit counterexamples can be constructed with quadratic twists otherwise.

4. Proof Strategy and Analytical Framework

The proof bifurcates depending on equality of weights/levels:

Case A: $(M, k) \neq (N, \ell)$

If $a_f(n)a_g(n) \geq 0$ for a set of primes $p$ of Dirichlet density near $1$, the Rankin–Selberg $L$-series attached to $f$ and $g$ produces analytic contradictions. The analysis is founded on:

• Hecke eigenbases decompositions of $f$ and $g$ into non-CM, twist-minimal newforms,
• Ramakrishnan's lift from $\mathrm{GL}_2 \times \mathrm{GL}_2$ to $\mathrm{GL}_4$, providing distinct automorphic representations,
• Rankin–Selberg estimates: $\sum_{p \leq x} \lambda_f(p)\lambda_g(p)/p = O(1)$ and $$\sum_{p \leq x} (\lambda_f(p)\lambda_g(p))^2/p = C \sum_{p \leq x} 1/p + O(1)$$ for $C>0$,
• Application of Cauchy–Schwarz and Deligne's bounds to derive a contradiction from the sign constraint.

Case B: $(M, k) = (N, \ell)$

Here, $f$ and $g$ reside in the same newform space of dimension $d\geq1$. The key innovation is a "dense sign-pattern" argument:

• By joint Sato–Tate equidistribution, for any two non-CM, twist-inequivalent newforms, the vector of signs of their Fourier coefficients at powers of suitable primes becomes dense in projective space $\mathbb{RP}^{d-1}$,
• This density allows, via linear-algebraic separation of the coefficient vectors, the explicit construction of $n$ where $a_f(n)$ and $a_g(n)$ differ in sign, unless $f$ and $g$ are proportional,
• Overall, eventual agreement of sign patterns enforces proportionality.

5. Examples, Corollaries, and Level Sharpness

• For distinct non-CM normalized newforms $f, g$ of the same weight and level, both the sets where $a_f(n)a_g(n) > 0$ and $a_f(n)a_g(n) < 0$ are infinite.
• If $f, g$ differ in weight or level, the same conclusion holds if $a_f(p)a_g(p) \geq 0$ for almost all primes $p$.
• The level condition is optimal: If $\Delta$ divides $M$ to square-power, it is possible to construct $$g = 2f + f \otimes \left( \frac{\Delta}{\cdot} \right)$$ so $a_f(n)a_g(n) \geq 0$ for all $n$, even though $g \neq c f$ for any scalar $c$.
• No effective bound exists for the first sign difference between $f$ and $g$ when the newspace is multidimensional, as $g_\varepsilon \to f$ allows the first disagreement to be arbitrarily delayed.

6. Significance and Connections to Automorphic Theory

The determination of self-dual holomorphic cusp forms by sign patterns of their Fourier coefficients provides a sharp analogue to "multiplicity one" phenomena in representation theory: the sign-sequence captures all information about the form up to scaling. The proof leverages deep arithmetic and analytic properties—most notably Deligne's bounds, functoriality via Ramakrishnan’s transfer, Rankin–Selberg convolution theory, and equidistribution results of Sato–Tate for joint eigenangles (contributions of Barnet-Lamb, Gee, Geraghty, and Wong). The density-and-linear-algebra method introduced for the equal-weight case offers potential applications to other sign-rigidity questions in automorphic forms.

The result addresses outstanding questions on sign changes of Fourier coefficients, complementing previous works (KLSW, Matomäki, GKR) by demonstrating that, in the generic (non-pathological) setting, the sign pattern not only oscillates but characterizes the form itself (Booker, 5 Dec 2025).