--- title: Drinfeld-Hecke Eigenforms Overview url: https://www.emergentmind.com/topics/drinfeld-hecke-eigenforms type: topic --- # Drinfeld-Hecke Eigenforms Overview Drinfeld-Hecke eigenforms are Drinfeld modular forms on function-field analogues of modular curves or higher-rank period domains that are simultaneous eigenvectors for the relevant Hecke operators. In the contemporary literature, the subject spans rank-\(2\) Drinfeld-Goss modular forms, higher-rank forms on \(\Omega^r\), forms described by \(u\)-, \(t\)-, or \(A\)-expansions, vectorial Drinfeld modular forms built from twisted Eisenstein series, and \(\wp\)-adic finite-slope families. Across these settings, the theory combines explicit double-coset operators, Goss polynomials, product formulas, harmonic-cochain models, and overconvergent or Hida-theoretic methods; it also exhibits structural features not captured by a naive classical analogy, including failure of multiplicity one, nontrivial semisimple Hecke action modulo \(T\), and natural Hecke-stable subquotients arising from representation theory [2302.06316] [1207.6479] [2103.13126] [1304.0101]. ## 1. Analytic setting and Hecke correspondences For higher-rank Drinfeld modular forms in the sense of [2302.06316], the basic ground ring is \[ A=\mathbb{F}_q[t], \qquad F=\mathrm{Frac}(A), \qquad F_\infty=\mathbb{F}_q((t^{-1})), \] with \(C_\infty\) the completion of an algebraic closure of \(F_\infty\). The Drinfeld period domain \(\Omega^r\) is the complement of all \(F_\infty\)-rational hyperplanes in \(\mathbb{P}^{r-1}(C_\infty)\), and a point is written \[ w=(w_1,\dots,w_{r-1},1)^t. \] There is an action of \(GL_r(F)\) on \(\Omega^r\), and the weight-\(k\) slash operator is \[ f|_k \gamma(w)=j(\gamma,w)^{-k} f(\gamma w), \] where \(j(\gamma,w)\) is the normalization factor given by the last entry of \(\gamma w\). A weak Drinfeld modular form of rank \(r\), weight \(k\), for a congruence subgroup \(T\subset GL_r(A)\), is a holomorphic function \(f\) satisfying \(f|_k\gamma=f\) for \(\gamma\in T\); it is a modular form when it is holomorphic at the cusps, formulated via a \(u\)-expansion at infinity [2302.06316]. Hecke operators are defined by double cosets. For congruence subgroups \(I,I'\subset GL_r(F)\) and \(\delta\in GL_r(F)\), \[ T_\delta : M_k(I') \to M_k(I), \qquad T_\delta f = \sum_\gamma f|_k \gamma, \] where \(\gamma\) runs through representatives of \(I' \backslash I' \delta I\). A basic operator is attached to \[ \delta=\mathrm{diag}(p,1,\dots,1), \] for an irreducible polynomial \(p\in A\). The paper gives explicit representatives \(B_{m,b}\) for the double coset \(GL_r(A)\backslash GL_r(A)\delta GL_r(A)\), and in rank \(2\) recovers the familiar formula \[ T_p f(w_1)=f(pw_1)+\sum_{\deg b<\deg p} f(w_1+b), \] up to normalization conventions used there [2302.06316]. In the rank-\(2\) setting of \(A\)-expansions, the Hecke operator at a prime ideal \(p\) with monic generator \(\wp\) is written \[ T_p f(z)= \wp^k f(\wp z) + \sum_{\beta\in S_p} f\!\left(\frac{z+\beta}{\wp}\right), \] where \(S_p=\{\beta\in A : \deg \beta < \deg \wp\}\). This formula already makes visible the close interaction between Hecke indices and the polynomial arithmetic of \(A=\mathbb{F}_q[T]\) [1207.6479]. ## 2. Expansion theories, Goss polynomials, and coefficient extraction The higher-rank theory in [2302.06316] is organized around the \(u\)-parameter attached to the lattice generated by the last \(r-1\) coordinates. If \(A'\subset C_\infty\) is the relevant lattice, then \[ u(w_1)=e_{A'}(w_1)^{-1}, \] where \(e_{A'}\) is the exponential of the lattice \(A'\). Every weak modular form admits a \(u\)-expansion \[ f(w)=\sum_n f_n(w')\,u(w_1)^n \] with uniquely determined coefficient functions \(f_n\). The Hecke action on these expansions is governed by Goss polynomials \(G_n\), which in the paper are recalled to satisfy: \(G_n(X)\) is monic of degree \(n\); if \(n\le q\), then \(G_n(X)=X^n\); if \(n\ge 2\), then \(X^2\mid G_n(X)\); and every nonzero exponent in \(G_n(X)\) is congruent to \(n \pmod{q-1}\). Theorem 3.6 gives the central “Hecke-on-\(u\)-expansion” formula: the simple Hecke operator sends \[ f(w)=\sum_n f_n(w')\,u(w_1)^n \] to a sum whose first term replaces \(u(w_1)\) by \(u(pw_1)\), while the remaining coset representatives contribute Goss polynomials \(G_{n,L_b}(u(w_1))\). A basic corollary is \[ T_{p,k}f = T_{p,k}f_0 + T_{p,k-1}f_1\,u + \cdots. \] The same paper proves that Hecke operators map modular forms to modular forms, cusp forms to cusp forms, and double cusp forms to double cusp forms; it also proves complete multiplicativity for the natural operators \[ T_{p^n}=\sum_{e_1+\cdots+e_r=n} T(p^{e_1},\dots,p^{e_r}), \] namely \(T_{p^n}=(T_p)^n\) and \(T_M T_N=T_{MN}\) for coprime \(M,N\) [2302.06316]. Petrov’s \(A\)-expansions replace the usual indexing by natural numbers with indexing by monic polynomials in \(A\). An \(A\)-expansion of exponent \(n\) has the form \[ f = c_0(f) + \sum_{a \in A_+} c_a(f)\, G_n(t_a), \] where \(A_+\) is the set of monic polynomials in \(A\), \(t_a=t(az)\), and \(G_n(X)\) is the \(n\)-th Goss polynomial attached to the Carlitz module. For fixed exponent \(n\), the \(A\)-expansion is unique. Its importance for Hecke theory is that if \(f\) has an \(A\)-expansion with exponent \(n\) and is an eigenform for \(T_p\), then \[ T_p f = \wp^n f, \qquad c_p(f)=\wp^{k-n} c_1(f), \] and more generally, if \(a=\prod_i \wp_i^{e_i}\), then \[ c_a(f)=a^{k-n}c_1(f). \] The Hecke eigenvalue is therefore exactly \(\wp^n\), and the eigensystem is encoded directly in the expansion index [1207.6479]. For level-\(1\) double-cuspidal Drinfeld-Goss eigenforms with degree-one Hecke eigenvalues of power type, \[ T_{p,k}(f)=p^N f, \] the coefficient behavior is subtler than in the classical normalized-eigenform setting. The paper [1705.09795] isolates coefficients \(a_{1+qv}\) and proves the closed formula \[ \ell!\, a_{1+qv} = \sum_{\sigma\in S_\ell} \prod_{i=1}^{\ell}\bigl(\theta^{q^{N_{\sigma(i)}}}-\theta^{q^{v_i}}\bigr), \] together with the exact vanishing criterion \[ a_{1+qv}=0 \quad\Longleftrightarrow\quad N_i0\) satisfying that \(k-2n\) is a positive multiple of \(q-1\) and \(n \le p^{v_p(k-n)}\). The paper proves that \[ f_{k,n}\in S_{k,m}(GL), \qquad m \equiv n \pmod{q-1}, \] and that \[ T_p f_{k,n} = \wp^n f_{k,n} \qquad\text{for all primes }p. \] A notable special subfamily is \[ f_s := f_{q+1+s(q-1),1} = \sum_{a\in A_+} a^{q+s(q-1)}t_a, \] whose members are cuspidal, not double-cuspidal, and occur in the quotient \(S_{k,1}(GL)/S_{k,1}^2(GL)\), where the eigenforms all have eigenvalues \(T_p=\wp\). Petrov also proves a restrictive multiplicity-one statement inside the class of forms with \(A\)-expansions: if an eigenform \(f\) has an \(A\)-expansion with exponent \(n\), then \[ f=\sum_{a\in A_+} a^{k-n}G_n(t_a), \] so within that class the weight and the eigensystem determine the form [1207.6479]. Twisting by Dirichlet characters produces another explicit mechanism. For square-free monic \(n\) and a Dirichlet character \(\chi\in (A/nA)^\times\), the projection operator \[ \widehat{\chi} f := \sum_{|\beta|<|n|} \chi^{-1}(\beta)\, f\bigg| \begin{pmatrix} 1 & \beta/n \\ 0 & 1 \end{pmatrix} \] sends forms to forms with character and preserves cuspidality. Its central Hecke property is \[ T_q(\widehat{\chi}f)=\chi(q)\,\widehat{\chi}(T_q f), \] for monic irreducible \(q\) coprime to the relevant level, so if \(f\) is a Hecke eigenform with \(T_q f=\lambda_q f\) and \(\widehat{\chi}f\neq 0\), then \[ T_q(\widehat{\chi}f)=\chi(q)\lambda_q\,\widehat{\chi}f. \] The same paper emphasizes that the effect on \(u\)-expansions and on Petrov’s \(A\)-expansions is more complicated than coefficientwise twisting, because the translated arguments \(u(z+\beta/n)\) are related to \(u(z)\) by nontrivial rational expressions involving the Carlitz exponential. It also constructs Eisenstein series with character \(E^{(k)}_\chi\) for irreducible level \(\mathfrak p\), proves that they and their Fricke transforms are Hecke eigenforms, and gives the eigenvalue formulas \[ T_q E^{(k)}_\chi = q^k \chi(q)\, E^{(k)}_\chi, \qquad T_q\left(E^{(k)}_\chi\big|_k[W_\mathfrak p]\right) = q^k \chi^{-1}(q)\, \left(E^{(k)}_\chi\big|_k[W_\mathfrak p]\right) \] for \(q\neq \mathfrak p\) [1705.04471]. The determinant construction of [2509.20895] extends this constructive program to arbitrary rank by assembling twisted Eisenstein series into a matrix \(\mathcal E(z,t)\), extracting the \((r,r)\)-cofactor \(\mathcal H_r(z,t)\), and embedding it into a vectorial Drinfeld modular form \(\mathcal G_r\). The resulting specializations \(\mathcal H_r(z,\theta^{q^n})\) provide a higher-rank tower of scalar Hecke eigenforms with explicit weight, type, and eigenvalue data [2509.20895]. ## 5. Rigidity, failure phenomena, and Hecke-stable building blocks A recurring theme is that Drinfeld-Hecke eigenforms do not satisfy a naive analogue of classical multiplicity one. Petrov emphasizes that forms such as \[ g,\quad g^q,\quad h \] can share the same eigensystem, so multiplicity one fails in general. The restrictive theorem for \(A\)-expansion forms therefore applies only inside a very special class, and the same paper explicitly notes that there are eigenforms that do not seem to admit any \(A\)-expansion [1207.6479]. The literature also gives direct non-eigenform criteria. In higher rank, [2302.06316] proves that if \(\mathrm{char}(\mathbb F_q)\neq 2\) and \(r>2\), then \(g_2\) is not an eigenform for a certain Hecke operator \(T_t\). Modulo \(T\), the paper [1304.0101] shows that many powers of the normalized cuspidal form \(h\) cannot be eigenforms: if \(\mathfrak p\) has degree \(d\) and \(1\le j\le q-1\), then \[ h^{q^d+j} \] is not an eigenform for \(T_{\mathfrak p}\). It also proves that for \[ \mathfrak p=T^d+\alpha_{d-1}T^{d-1}+\cdots+\alpha_1T+\alpha_0, \] the form \(h^{q+j}\) is an eigenform modulo \(T\) precisely when \(\alpha_1=0\). These statements exhibit explicit arithmetic obstructions to eigenform status [2302.06316] [1304.0101]. The Hecke algebra itself behaves differently from the classical mod-\(2\) picture. For the Hecke algebra \(\mathcal H_{k,m}\) acting on \(M_{k,m}\), the semisimple quotient modulo \(T\) is described explicitly by \[ \mathcal H_{k,m}^{ss}\cong \mathbb F_q[(\mathbb F_q^\ast)^m], \] with \((T_{\mathfrak p}\bmod T)^{ss}\mapsto \mathfrak p^m\). Thus the semisimple part modulo \(T\) is nonzero, in sharp contrast with the classical nilpotence phenomenon cited in the paper. The same work conjectures that for each \(q\) there exist large \(k\) and suitable \(m\) such that \[ H^2(\mathcal H_{k,m},M_{k,m})\neq 0, \] which would imply that the Hecke algebra is not smooth in large weight [1304.0101]. A more structural explanation for the failure of a naive Maeda-type picture appears in [2103.13126]. Using an adelic extension of Teitelbaum’s isomorphism, \[ S_{k,l}(\mathcal K)\cong C^{ad}_{har}(V_{k,l},\mathcal K)^{\mathrm{GL}_2(F)}\otimes_F \mathbb C_\infty, \] together with \[ C^{ad}_{har}(N,\mathcal K)^{\mathrm{GL}_2(F)} \cong St_\mathcal K\otimes_{\mathrm{GL}_2(F)}N, \] the paper proves that the functor \(N\mapsto St_\mathcal K\otimes_{\mathrm{GL}_2(F)}N\) is exact on finite-dimensional \(F[\mathrm{GL}_2(F)]\)-modules. Any filtration of the coefficient representation \(V_{k,l}\) therefore induces a Hecke-stable filtration on cusp forms, and the associated graded pieces are harmonic-cochain spaces for the simple composition factors \(L_{k'}\otimes\det^{m'}\). Numerical data for \(A=\mathbb F_3[t]\) then show a conjecturally infinite supply of \(\mathbb F_3(t)\)-rational eigenforms in weights \(k_n=1+3^n\), even after the obvious representation-theoretic obstructions are removed [2103.13126]. ## 6. \(\wp\)-adic families, finite slope, and classicity The finite-slope theory studies Drinfeld-Hecke eigenforms varying with the weight. In [1904.08618], for spaces \[ S_k(\Gamma_1(\mathfrak n\wp^r)), \] the key hypotheses are a degree-one prime factor condition, one-dimensionality of the generalized slope-\(a\) eigenspace, and a sufficiently small slope bound. Under these assumptions, fixing a slope-\(a\) Hecke eigenform \(F_k\), the paper constructs a family \[ \{F_{k'} \mid v_p(k'-k)\ge \log_p(p^n+a)\} \] of Hecke eigenforms \(F_{k'}\in S_{k'}(\Gamma_1(\mathfrak n\wp^r))\) of the same slope \(a\). For any \(Q\in A\), their Hecke eigenvalues satisfy the congruence estimate \[ v_\wp\bigl(\lambda_Q(F_{k'})-\lambda_Q(F_k)\bigr) > p^{v_p(k'-k)}-p^n-a. \] The construction uses weight-reduction maps on coefficient modules, local constancy of slope dimensions, and perturbation lemmas for Hecke matrices [1904.08618]. A broader higher-rank framework is developed in [1805.08793]. There, Hida theory is built for rank-\(r\) Drinfeld modular forms of slope zero for a suitably defined \(\mathrm U_{\mathfrak p}\)-operator. The ordinary projector is \[ e := \lim_{n\to\infty} \mathrm U_{\mathfrak p}^{\,n!}, \] and the ordinary part is finite over the relevant weight algebra. In the finite-slope setting, the paper constructs a Fredholm determinant \[ F_{\mathrm U_T}(X)\in C(\mathbb Z_p,\mathcal O[1/\varpi])\{X\}, \] a spectral variety \[ Z = V(F_{\mathrm U_T}(X)) \subset \mathbb A^1_{\mathbb Z_p}, \] and a Hecke variety \(\mathcal C\to Z\) parametrizing Hecke eigensystems on finite-slope overconvergent forms. It also proves a classicity criterion: if an overconvergent Drinfeld modular form of weight \(k\) is a \(\mathrm U_T\)-eigenform with \[ \mathrm{slope}(f) < k-r+1, \] then \(f\) is classical. This establishes a function-field analogue of the ordinary and finite-slope eigenform machine familiar from classical \(p\)-adic modular forms, but in a distinctly Drinfeld setting [1805.08793]. Source: https://www.emergentmind.com/topics/drinfeld-hecke-eigenforms