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A superelliptic curve over a discrete valuation ring
$\mathscr{O}$
of residual characteristic p is a curve given by an equation
$\mathscr{C}\;:\; y^n=\,f(x)$
, with
$\textrm{Disc}(\,f)\neq 0$
. The purpose of this article is to describe the Galois representation attached to such a curve under the hypothesis that f(x) has all its roots in the fraction field of
$\mathscr{O}$
and that
$p \nmid n$
. Our results are inspired on the algorithm given in Bouw and WewersGlasg (Math. J.59(1) (2017), 77–108.) but our description is given in terms of a cluster picture as defined in Dokchitser et al. (Algebraic curves and their applications, Contemporary Mathematics, vol. 724 (American Mathematical Society, Providence, RI, 2019), 73–135.).
We settle a part of the conjecture by Bandini and Valentino [‘On the structure and slopes of Drinfeld cusp forms’, Exp. Math.31(2) (2022), 637–651] for
$S_{k,l}(\Gamma _0(T))$
when
$\mathrm {dim}\ S_{k,l}(\mathrm {GL}_2(A))\leq 2$
. We frame and check the conjecture for primes
$\mathfrak {p}$
and higher levels
$\mathfrak {p}\mathfrak {m}$
, and show that a part of the conjecture for level
$\mathfrak {p} \mathfrak {m}$
does not hold if
$\mathfrak {m}\ne A$
and
$(k,l)=(2,1)$
.
Let f be a primitive Hilbert modular form over F of weight k with coefficient field
$E_f$
, generated by the Fourier coefficients
$C(\mathfrak {p}, f)$
for
$\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)$
. Under certain assumptions on the image of the residual Galois representations attached to f, we calculate the Dirichlet density of
$\{\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)| E_f = \mathbb {Q}(C(\mathfrak {p}, f))\}$
. For
$k=2$
, we show that those assumptions are satisfied when
$[E_f:\mathbb {Q}] = [F:\mathbb {Q}]$
is an odd prime. We also study analogous results for
$F_f$
, the fixed field of
$E_f$
by the set of all inner twists of f. Then, we provide some examples of f to support our results. Finally, we compute the density of
$\{\mathfrak {p} \in \mathrm {Spec}(\mathcal {O}_F)| C(\mathfrak {p}, f) \in K\}$
for fields K with
$F_f \subseteq K \subseteq E_f$
.
We prove new cases of the Inverse Galois Problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight
$3$
newforms will show that there are Galois extensions of
${\mathbb Q}$
with Galois group
$\operatorname {PSL}_2({\mathbb F}_p)$
for all primes p and
$\operatorname {PSL}_2({\mathbb F}_{p^3})$
for all odd primes
$p \equiv \pm 2, \pm 3, \pm 4, \pm 6 \ \pmod {13}$
.
In this paper, we compute the Fourier expansion of the Shintani lift of nearly holomorphic modular forms. As an application, we deduce modularity properties of generating series of cycle integrals of nearly holomorphic modular forms.
Let G be a p-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra—the endomorphism algebra of a pro-generator of the given component. Using Heiermann’s construction of these algebras, we describe the Bernstein components of the Gelfand–Graev representation for $G=\mathrm {SO}(2n+1)$, $\mathrm {Sp}(2n)$, and $\mathrm {O}(2n)$.
In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\mathbb {Q}(\sqrt {-d})$, where $d=1,7,19,43,67$ whenever p is bigger than this bound. During the course of the proof, we prove various results about the irreducibility of Galois representations, image of inertia groups, and Bianchi newforms.
We use a linear algebra interpretation of the action of Hecke operators on Drinfeld cusp forms to prove that when the dimension of the $\mathbb {C}_\infty $-vector space $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ is one, the Hecke operator $\mathbf {T}_t$ is injective on $S_{k,m}(\mathrm {{GL}}_2(\mathbb {F}_q[t]))$ and $S_{k,m}(\Gamma _0(t))$ is a direct sum of oldforms and newforms.
Let K be a number field. For which primes p does there exist an elliptic curve
$E / K$
admitting a K-rational p-isogeny? Although we have an answer to this question over the rationals, extending this to other number fields is a fundamental open problem in number theory. In this paper, we study this question in the case that K is a quadratic field, subject to the assumption that E is semistable at the primes of K above p. We prove results both for families of quadratic fields and for specific quadratic fields.
The purpose of this paper is to extend the explicit geometric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of the enveloping algebra.
A (folklore?) conjecture states that no holomorphic modular form
$F(\tau )=\sum _{n=1}^{\infty } a_nq^n\in q\mathbb Z[[q]]$
exists, where
$q=e^{2\pi i\tau }$
, such that its anti-derivative
$\sum _{n=1}^{\infty } a_nq^n/n$
has integral coefficients in the q-expansion. A recent observation of Broadhurst and Zudilin, rigorously accomplished by Li and Neururer, led to examples of meromorphic modular forms possessing the integrality property. In this note, we investigate the arithmetic phenomenon from a systematic perspective and discuss related transcendental extensions of the differentially closed ring of quasi-modular forms.
Let f be an elliptic modular form and p an odd prime that is coprime to the level of f. We study the link between divisors of the characteristic ideal of the p-primary fine Selmer group of f over the cyclotomic
$\mathbb {Z}_p$
extension of
$\mathbb {Q}$
and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms.
Hilbert schemes are an object arising from geometry and are closely related to physics and modular forms. Recently, there have been investigations from number theorists about the Betti numbers and Hodge numbers of the Hilbert schemes of points of an algebraic surface. In this paper, we prove that Göttsche's generating function of the Hodge numbers of Hilbert schemes of $n$ points of an algebraic surface is algebraic at a CM point $\tau$ and rational numbers $z_1$ and $z_2$. Our result gives a refinement of the algebraicity on Betti numbers.
Let p be a rational prime. Let F be a totally real number field such that F is unramified over p and the residue degree of any prime ideal of F dividing p is
$\leq 2$
. In this paper, we show that the eigenvariety for
$\mathrm {Res}_{F/\mathbb {Q}}(\mathit {GL}_{2})$
, constructed by Andreatta, Iovita, and Pilloni, is proper at integral weights for
$p\geq 3$
. We also prove a weaker result for
$p=2$
.
We determine the local deformation rings of sufficiently generic mod $l$ representations of the Galois group of a $p$-adic field, when $l \neq p$, relating them to the space of $q$-power-stable semisimple conjugacy classes in the dual group. As a consequence, we give a local proof of the $l \neq p$ Breuil–Mézard conjecture of the author, in the tame case.
We determine reductions of
$2$
-dimensional, irreducible, semistable, and non-crystalline representations of
$\mathrm {Gal}\left (\overline {\mathbb {Q}}_p/\mathbb {Q}_p\right )$
with Hodge–Tate weights
$0 < k-1$
and with
$\mathcal L$
-invariant whose p-adic norm is sufficiently large, depending on k. Our main result provides the first systematic examples of the reductions for
$k \geq p$
.
In this paper, we study the extreme values of the Rankin–Selberg L-functions associated with holomorphic cusp forms in the vertical direction. Assuming the generalised Riemann hypothesis (GRH), we prove that
We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer
$n \neq 0,4,7 \,(\textrm{mod}\ 8)$
is represented as
$n= x_1^2 + x_2^2 + x_3^3$
for integers
$x_1,x_2,x_3$
such that the product
$x_1x_2x_3$
has at most 72 prime divisors.
In this paper, we study lower-order terms of the one-level density of low-lying zeros of quadratic Hecke L-functions in the Gaussian field. Assuming the generalized Riemann hypothesis, our result is valid for even test functions whose Fourier transforms are supported in $(-2, 2)$. Moreover, we apply the ratios conjecture of L-functions to derive these lower-order terms as well. Up to the first lower-order term, we show that our results are consistent with each other when the Fourier transforms of the test functions are supported in $(-2, 2)$.