arXiv subjects
Publications and source records attributed to Wen-Ching Winnie Li.
We prove the Sato--Tate distribution of Kloosterman sums over function fields with explicit error terms, when the places vary in arithmetic progressions or short intervals. A joint Sato--Tate distribution of two ``different" exponential sums is also proved.
In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions are in terms of hypergeometric character sums over finite fields, a theory developed largely by Greene, Katz, Beukers-Cohen-Mellit, and Fuselier-Long-Ramakrishna-Swisher-Tu. Our approach, in contrast to the previous works, is uniform and more geometric, and it works equally well for forms on elliptic modular curves and Shimura curves. The same method can be applied to obtain eigenvalues of Hecke operators as well.
A well-known formula of Whipple relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $HD$, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data $HD$ are primitive and self-dual. If the data are also defined over $\mathbb Q$, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of $\ell$-adic representations of the absolute Galois group of $\mathbb Q$ attached to $HD$. For specialized choices of $HD$, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values $_7F_6(1)$ in Whipple's formula to the periods of these modular forms.
Given two distinct number fields $K$ and $M$, and finite order Hecke characters $χ$ of $K$ and $η$ of $M$ respectively, we say that the pairs $(χ, K)$ and $(η, M)$ are arithmetically equivalent if the associated L-functions coincide: $$L(s, χ, K) = L(s, η, M) .$$ When the characters are trivial, this reduces to the question of fields with the same Dedekind zeta function, investigated by Gassman in 1926, who found such fields of degree 180, and by Perlis (1977) and others, who showed that there are no nonisomorphic fields of degree less than $7$. We construct infinitely many such pairs where the fields are quadratic. This gives dihedral automorphic forms induced from characters of different quadratic fields. We also give a classification of such characters of order 2 for the quadratic fields of our examples, all with odd class number.
In this expository paper, we illustrate two explicit methods which lead to special $L$-values of certain modular forms admitting complex multiplication (CM), motivated in part by properties of $L$-functions obtained from Calabi-Yau manifolds defined over $\mathbb Q$.
In this paper, we consider Galois representations of the absolute Galois group $\text{Gal}(\overline {\mathbb Q}/\mathbb Q)$ attached to modular forms for noncongruence subgroups of $\text{SL}_2(\mathbb Z)$. When the underlying modular curves have a model over $\mathbb Q$, these representations are constructed by Scholl and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over $\mathbb Q$. As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments in the automorphy lifting theorem, we obtain various automphy and potential automorphy results for potentially $\text{GL}_2$-type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. Especially, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions.
In this paper we give a unified approach in categorical setting to the problem of finding the Galois closure of a finite cover, which includes as special cases the familiar finite separable field extensions, finite unramified covers of a connected undirected graph, finite covering spaces of a locally connected topological space, finite étale covers of a smooth projective irreducible algebraic variety, and finite covers of normal varieties. We present two algorithms whose outputs are shown to be desired Galois closures. An upper bound of the degree of the Galois closure under each algorithm is also obtained.
In this paper, we study relations between Langlands L-functions and zeta functions of geodesic walks and galleries for finite quotients of the apartments of G=PGL3 and PGSp4 over a nonarchimedean local field with q elements in its residue field. They give rise to an identity (Theorem 5.3) which can be regarded as a generalization of Ihara's theorem for finite quotients of the Bruhat-Tits trees. This identity is shown to agree with the q=1 version of the analogous identities for finite quotients of the building of G established in (KL1, KLW, FLW), verifying the philosophy of the field with one element by Tits. A new identity for finite quotients of the building of PGSp4 involving the standard $L$-function (Theorem 6.3), complementing the one in (FLW) which involves the spin L-function, is also obtained.
This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \in {\bf C}$ a two-dimensional vector space spanned by linear combinations of Lerch zeta functions is characterized as a maximal space of simultaneous eigenfunctions for this family of Hecke operators. This result is an analogue of a result of Milnor for the Hurwitz zeta function. We also relate these functions to a linear partial differential operator in the $(a, c)$-variables having the Lerch zeta function as an eigenfunction.
Atkin and Swinnerton-Dyer congruences are special congruence recursions satisfied by coefficients of noncongruence modular forms. These are in some sense $p$-adic analogues of Hecke recursion satisfied by classic Hecke eigenforms. They actually appeared in different context and sometimes can be obtained using the theory of formal groups. In this survey paper, we introduce the Atkin and Swinnerton-Dyer congruences, and discuss some recent progress on this topic.
The zeta function attached to a finite complex $X_Γ$ arising from the Bruhat-Tits building for $\PGL_3(F)$ was studied in \cite{KL}, where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of $X_Γ$. In this paper we reprove the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed chambers, respectively.
Let $F$ be a non-archimedean local field with a finite residue field. To a 2-dimensional finite complex $X_Γ$ arising as the quotient of the Bruhat-Tits building $X$ associated to $\Sp_4(F)$ by a discrete torsion-free cocompact subgroup $Γ$ of $\PGSp_4(F)$, associate the zeta function $Z(X_Γ, u)$ which counts geodesic tailless cycles contained in the 1-skeleton of $X_Γ$. Using a representation-theoretic approach, we obtain two closed form expressions for $Z(X_Γ, u)$ as a rational function in $u$. Equivalent statements for $X_Γ$ being a Ramanujan complex are given in terms of vertex, edge, and chamber adjacency operators, respectively. The zeta functions of such Ramanujan complexes are distinguished by satisfying the Riemann Hypothesis.
In this paper we study the compatible family of degree-4 Scholl representations $ρ_{\ell}$ associated with a space $S$ of weight $κ> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $κ$ is odd, $ρ_\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term Atkin and Swinnerton-Dyer congruences.
In this paper we obtain a closed form expression of the zeta function $Z(X_Γ, u)$ of a finite quotient $X_Γ= Γ\backslash PGL_3(F)/PGL_3(O_F)$ of the Bruhat-Tits building of $PGL_3$ over a nonarchimedean local field $F$. Analogous to a graph zeta function, $Z(X_Γ, u)$ is a rational function and it satisfies the Riemann hypothesis if and only if $X_Γ$ is a Ramanujan complex.
Given a finite index subgroup of $SL_2(\mathbb Z)$ with modular curve defined over $\mathbb Q$, under the assumption that the space of weight $k$ ($ \ge 2$) cusp forms is $1$-dimensional, we show that a form in this space with Fourier coefficients in $\mathbb Q$ has bounded denominators if and only if it is a congruence modular form.
In this paper, we study the Drinfeld cusp forms for $Γ_1(T)$ and $Γ(T)$ using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for $Γ_1(T)$ of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for $Γ_1(T)$ of large weights, and not for $Γ(T)$ even of small weights. The Hecke eigenvalues on cusp forms for $Γ(T)$ with small weights are determined and the eigenspaces characterized.
In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the $p$-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigen forms. We also show that there is an automorphic $L$-function over $\mathbb Q$ whose local factors agree with those of the $l$-adic Scholl representations attached to the space of noncongruence cusp forms.
In this paper we study Cayley graphs on $\PGL_2(\mathbb F_q)$ mod the unipotent subgroup, the split and nonsplit tori, respectively. Using the Kirillov models of the representations of $\PGL_2(\mathbb F_q)$ of degree greater than one, we obtain explicit eigenvalues of these graphs and the corresponding eigenfunctions. Character sum estimates are then used to conclude that two types of the graphs are Ramanujan, while the third is almost Ramanujan. The graphs arising from the nonsplit torus were previously studied by Terras et al. We give a different approach here.