Hilbert Cusp Forms Research Articles

On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms

Abstract In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ > 0 \epsilon>0 , there exist infinitely many natural numbers 𝑛 such that τ ⁢ ( p n ) \tau(p^{n}) has at least 2 ( 1 - ϵ ) ⁢ log ⁡ n log ⁡ log ⁡ n 2^{(1-\epsilon)\frac{\log n}{\log\log n}} distinct prime factors for almost all primes 𝑝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above.

On the Petersson norm of the Hilbert-Siegel cusp form under the Ikeda lift

In this paper, we show an arithmeticity of the ratio of the Petersson norm of the Hilbert-Siegel cusp form determined by the Ikeda lift from a Hilbert cusp form to that of the Hilbert cusp form. This is a generalization of the algebraicity of the ratio of the Petersson norm of the Siegel cusp form under the Ikeda lift of an elliptic cusp form to that of the elliptic cusp form due to Y. Choie and W. Kohnen.

$p$-adic $L$-functions of Hilbert cusp forms and the trivial zero conjecture

We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\\operatorname{GL}\_2$ over a totally real field, which is Iwahori spherical at places above $p$. In the case of a simple zero we adapt the approach of Greenberg and Stevens, based on the functional equation for the $p$-adic $L$-function of a nearly finite slope family and on improved $p$-adic $L$-functions that we construct using automorphic symbols and overconvergent cohomology. For higher order zeros we develop a conceptually new approach studying the variation of the root number in partial families and establishing the vanishing of many Taylor coefficients of the $p$-adic $L$-function of the family.

On algebraicity of special values of symmetric 4-th and 6-th power L-functions for $$\mathrm {GL}(2)$$

We prove an algebraicity of special values of symmetric 4-th and 6-th power L-functions of Hilbert cusp forms using its symmetric power lifting and several algebraicity results on special values of automorphic L-functions.

On the lifting of Hilbert cusp forms to Hilbert-Hermitian cusp forms

We construct a lifting that associates to a Hilbert cusp form a Hilbert-Hermitian cusp form. This is a generalization of the lifting of elliptic cusp forms constructed by Ikeda to arbitrary Hilbert cusp forms.

On congruence modules related to Hilbert Eisenstein series

We generalize the work of Ohta on the congruence modules attached to elliptic Eisenstein series to the setting of Hilbert modular forms. Our work involves three parts. In the first part, we construct Eisenstein series adelically and compute their constant terms by computing local integrals. In the second part, we prove a control theorem for one-variable ordinary \(\Lambda \)-adic Hilbert modular forms following Hida’s work on the space of multivariable ordinary \(\Lambda \)-adic Hilbert cusp forms. In part three, we compute congruence modules related to Hilbert Eisenstein series through an analog of Ohta’s methods.

Ordinary primes in Hilbert modular varieties

A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2,\ldots ,2)$, we show how to produce more ordinary primes by using the Sato–Tate equidistribution and combining it with the Galois theory of the Hecke field. Under the assumption of stronger forms of Sato–Tate equidistribution, we get stronger (but conditional) results. In the case of higher weights, we formulate the ordinariness conjecture for submotives of the intersection cohomology of proper algebraic varieties with motivic coefficients, and verify it for the motives whose $\ell$-adic Galois realisations are abelian on a finite-index subgroup. We get some results for Hilbert cuspforms of weight $(3,\ldots ,3)$, weaker than those for $(2,\ldots ,2)$.

On the signs of Fourier coefficients of Hilbert cusp forms

We prove that given any $\epsilon > 0$ and a primitive adelic Hilbert cusp form $f$ of weight $k=(k_1,k_2,...,k_n) \in (2 \mathbb{Z})^n$ and full level, there exists an integral ideal $\mathfrak{m}$ with $N(\mathfrak{m}) \ll_{\epsilon} Q_{f}^{9/20+ \epsilon} $ such that the $\mathfrak{m}$-th Fourier coefficient of $C_{f} (\mathfrak{m})$ of $f$ is negative. Here $n$ is the degree of the associated number field, $N(\mathfrak{m})$ is the norm of integral ideal $\mathfrak{m}$ and $Q_{f}$ is the analytic conductor of $f$. In the case of arbitrary weights, we show that there is an integral ideal $\mathfrak{m}$ with $N(\mathfrak{m}) \ll_{\epsilon} Q_{f}^{1/2 + \epsilon}$ such that $C_{f}(\mathfrak{m}) <0$. We also prove that when $k=(k_1,k_2,...,k_n) \in (2 \mathbb{Z})^n$, asymptotically half of the Fourier coefficients are positive while half are negative.

On the lifting of Hilbert cusp forms to Hilbert-Siegel cusp forms

Starting from a Hilbert cusp form of weight 2κ, we will construct a Hilbert-Siegel cusp form of weight κ+ m 2 and degree m and its transfer to inner forms of symplectic groups.

A note on lower bounds of heights of non-zero Fourier coefficients of Hilbert cusp forms

It is a conjecture of Atkin and Serre that for any $$\epsilon > 0$$, there exists a constant $$c(\epsilon ) > 0$$ such that the Ramanujan $$\tau $$-function satisfies $$\begin{aligned} |\tau (p)| \ge c(\epsilon ) ~p^{9/2 - \epsilon } \end{aligned}$$for all sufficiently large primes. This in particular would settle Lehmer’s folklore conjecture for almost all primes and hence is widely open. In an interesting and elegant work, R. Murty, K. Murty, and T. Shorey showed that if $$\tau (n)$$ is odd, then $$|\tau (n)| > (\log n)^{\delta }$$ for some effective absolute constant $$\delta > 0$$. In this short note, building upon their work, we derive lower bounds for heights of certain Fourier coefficients of primitive Hilbert cusp forms. The extension to Hilbert modular forms brings in some new difficulties which are absent for the Ramanujan $$\tau $$-function and compel us to abandon absolute values and work with Weil heights. This brings in the further caveat that the bounds on prime powers no longer lead to bounds on arbitrary Fourier coefficients as the height inequalities are no longer compatible to derive such lower bounds. A major point in this circle of questions is the issue of non-vanishing of the eigenvalues $$c({\mathfrak {p}})$$ which are associated to the prime ideals $${\mathfrak {p}}$$ of the ambient number field. In the final section, we indicate a somewhat curious link between this and the oscillation of the real sequence $$\{c(\mathfrak {p^n})\}_{n\in {{\mathbb {N}}}}$$.

Simultaneous behaviour of the Fourier coefficients of two Hilbert modular cusp forms

In this article, we study the simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different integral weights. We also study the simultaneous non-vanishing of Fourier coefficients, of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights, at powers of a fixed prime ideal.

On Doi–Naganuma and Shimura liftings

In this paper, we extend the Doi–Naganuma lifting to higher levels by following the methods of Zagier and Kohnen. We prove that there is a Hecke-equivariant linear map from the space of elliptic cusp forms of integer weight k, level $$N, ((N,D)=1)$$ to Hilbert cusp forms of weight k, level N associated to a real quadratic field of discriminant D ( $$D\equiv 1\pmod {4}$$ ) with class number one. The above lifting is obtained by computing the explicit image of Poincare series of weight k, level N for the cusp at $$\infty $$ . Finally, we see that the above lifting is closely related to the Dth Shimura lift on the Kohnen plus space.

Determining Hilbert modular forms by central values of Rankin-Selberg convolutions: The level aspect

In this paper, we prove that a primitive Hilbert cusp form $\mathbf{g}$ is uniquely determined by the central values of the Rankin-Selberg $L$-functions $L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$, where $\mathbf{f}$ runs through all primitive Hilbert cusp forms of level $\mathfrak{q}$ for infinitely many prime ideals $\mathfrak{q}$. This result is a generalization of a theorem of Luo to the setting of totally real number fields.

Determining Hilbert modular forms by central values of Rankin–Selberg convolutions: the weight aspect

The purpose of this paper is to prove that a primitive Hilbert cusp form $\mathbf{g}$ is uniquely determined by the central values of the Rankin-Selberg $L$-functions $L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$, where $\mathbf{f}$ runs through all primitive Hilbert cusp forms of weight $k$ for infinitely many weight vectors $k$. This work is a generalization of a result of Ganguly, Hoffstein, and Sengupta to the setting of totally real number fields, and it is a weight aspect analogue of the authors recent work.

On Doi-Naganuma lifting

In this paper, we extend the Doi-Naganuma lifting as suggested by Kudla [4], on the lines of Zagier's work [6]. For each fundamental discriminant D associated with a real quadratic field, we prove that there exists a Hecke-equivarient map ιD which maps the mth Poincare series of weight k, level M and character χD = (./D) into a Hilbert cusp form of weight k, level M/D associated with the real quadratic field of discriminant D of class number one. Through this, we get its adjoint ι*D with respect to the Petersson inner product.

Classical weight one forms in Hida families: Hilbert modular case

The purpose of this paper is to investigate the number of classical weight one specializations of a non-CM ordinary Hida family of parallel weight Hilbert cusp forms. It is known that a specialization of a primitive ordinary Hida family at any arithmetic points of weight at least two is a classical holomorphic Hilbert cusp form. However, this is not always the case for weight one specializations. Balasubramanyam, Ghate and Vatsal proved that such a Hida family admits infinitely many classical weight one specializations if and only if it is of CM type. We give an explicit upper bound on the number of classical weight one specializations of a non-CM primitive ordinary Hida family.

Existence of Hilbert Cusp Forms with Non-vanishingL-values

AbstractWe develop a derivative version of the relative trace formula on PGL(2) studied in our previous work, and derive an asymptotic formula of an average of central values (derivatives) of automorphicL-functions for Hilbert cusp forms. As an application, we prove the existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are arbitrarily large.

On a congruence prime criterion for cusp forms on GL2 over number fields

Abstract Let f be a normalized Hecke eigenform on GL2 over a number field F and let 𝔓 be a prime ideal of a number field which contains the Galois closure of the number field which is generated by all Fourier coefficients of f over F. In this paper, we give a sufficient condition for 𝔓 to be a congruence prime for f. This criterion is a generalization of congruence prime criteria which were known for the case of elliptic cusp forms by Hida, for the case where F is an imaginary quadratic field by Urban and for the case of Hilbert cusp forms by Ghate and Dimitrov to arbitrary number fields.

Fermat-type equations of signature $$(13,13,p)$$ via Hilbert cuspforms

In this paper we prove that for \(p > 13649\) equations of the form \(x^{13} + y^{13} = Cz^{p}\) have no non-trivial primitive solutions \((a,b,c)\) such that \(13 \not \mid c\) for an infinite family of values for \(C\). Our method consists on relating a solution \((a,b,c)\) to the previous equation to a solution \((a,b,c_1)\) of another Diophantine equation with coefficients in \(\mathbb Q (\sqrt{13})\). Then we attach to \((a,b,c_1)\) a Frey curve \(E_{(a,b)}\) defined over \(\mathbb Q (\sqrt{13})\) that is not a \(\mathbb Q \)-curve. We prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at \(3\). This theorem, in particular, implies modularity of \(E_{(a,b)}\). This enables us to use level lowering results and apply the modular approach via Hilbert cuspforms over \(\mathbb Q (\sqrt{13})\) to prove the non-existence of \((a,b,c_1)\) and, consequently, of \((a,b,c)\).

On p-adic families of Hilbert cusp forms of finite slope

Let p be a prime number and F a totally real field. In this article, we obtain a p-adic interpolation of spaces of totally definite quaternionic automorphic forms over F of finite slope, and construct p-adic families of automorphic forms parametrized by affinoid Hecke varieties. Further, as an application to the case where [ F : Q ] is even, we obtain p-adic analytic families of Hilbert eigenforms having fixed finite slope parametrized by weights. This is an analogue of Coleman's analytic families in [R.F. Coleman, p-Adic Banach spaces and families of modular forms, Invent. Math. 127 (1997) 417–479].