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In this paper, we prove one divisibility of the Iwasawa–Greenberg main conjecture for the Rankin–Selberg product of a weight two cusp form and an ordinary complex multiplication form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm {GU}(3,1)$, generalizing an earlier result of the third-named author to allow nonordinary cusp forms. The main result is a key input in the third-named author’s proof of Kobayashi’s $\pm $-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semiordinary Hida theory along an appropriate smaller weight space and a study of the semiordinary Eisenstein family.
In [14], Jacquet–Piatetskii-Shapiro–Shalika defined a family of compact open subgroups of p-adic general linear groups indexed by nonnegative integers and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of nonnegative integers. For the proof, we introduce the Rankin–Selberg integrals for Speh representations.
We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.
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 prove Fermat’s Last Theorem over
$\mathbb {Q}(\sqrt {5})$
and
$\mathbb {Q}(\sqrt {17})$
for prime exponents
$p \ge 5$
in certain congruence classes modulo
$48$
by using a combination of the modular method and Brauer–Manin obstructions explicitly given by quadratic reciprocity constraints. The reciprocity constraint used to treat the case of
$\mathbb {Q}(\sqrt {5})$
is a generalization to a real quadratic base field of the one used by Chen and Siksek. For the case of
$\mathbb {Q}(\sqrt {17})$
, this is insufficient, and we generalize a reciprocity constraint of Bennett, Chen, Dahmen, and Yazdani using Hilbert symbols from the rational field to certain real quadratic fields.
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
$p \equiv \pm 2, \pm 3, \pm 4, \pm 6 \ \pmod {13}$
.
We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type
$\frac {1}{4}(1,1)$
. The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type associated with a unimodular lattice of signature
$(2,10-k)$
. We prove that the invariant, viewed as a function of the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension
$2$
. As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case.
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.
We discuss the
$\ell $
-adic case of Mazur’s ‘Program B’ over
$\mathbb {Q}$
: the problem of classifying the possible images of
$\ell $
-adic Galois representations attached to elliptic curves E over
$\mathbb {Q}$
, equivalently, classifying the rational points on the corresponding modular curves. The primes
$\ell =2$
and
$\ell \ge 13$
are addressed by prior work, so we focus on the remaining primes
$\ell = 3, 5, 7, 11$
. For each of these
$\ell $
, we compute the directed graph of arithmetically maximal
$\ell $
-power level modular curves
$X_H$
, compute explicit equations for all but three of them and classify the rational points on all of them except
$X_{\mathrm {ns}}^{+}(N)$
, for
$N = 27, 25, 49, 121$
and two-level
$49$
curves of genus
$9$
whose Jacobians have analytic rank
$9$
.
Aside from the
$\ell $
-adic images that are known to arise for infinitely many
${\overline {\mathbb {Q}}}$
-isomorphism classes of elliptic curves
$E/\mathbb {Q}$
, we find only 22 exceptional images that arise for any prime
$\ell $
and any
$E/\mathbb {Q}$
without complex multiplication; these exceptional images are realised by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on
$X_{\mathrm {ns}}^+(\ell )$
with
$\ell \ge 19$
, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the
$\ell $
-adic images of Galois for any elliptic curve over
$\mathbb {Q}$
.
In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on
$\Gamma _1(N)$
are of
$\operatorname {GL}_2$
-type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of
$X_H$
.
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 prove an asymptotic expansion of the second moment of the central values of the
$\mathrm {GL}(n)\times \mathrm {GL}(n)$
Rankin–Selberg L-functions
$L(1/2,\pi \otimes \pi _0)$
for a fixed cuspidal automorphic representation
$\pi _0$
over the family of
$\pi $
with analytic conductors bounded by a quantity that is tending to infinity. Our proof uses the integral representations of the L-functions, period with regularised Eisenstein series and the invariance properties of the analytic newvectors.
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.
We use the method of Bruinier–Raum to show that symmetric formal Fourier–Jacobi series, in the cases of norm-Euclidean imaginary quadratic fields, are Hermitian modular forms. Consequently, combining a theorem of Yifeng Liu, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension for unitary Shimura varieties defined in these cases.
A new reciprocity formula for Dirichlet L-functions associated to an arbitrary primitive Dirichlet character of prime modulus q is established. We find an identity relating the fourth moment of individual Dirichlet L-functions in the t-aspect to the cubic moment of central L-values of Hecke–Maaß newforms of level at most
$q^{2}$
and primitive central character
$\psi ^{2}$
averaged over all primitive nonquadratic characters
$\psi $
modulo q. Our formula can be thought of as a reverse version of recent work of Petrow–Young. Direct corollaries involve a variant of Iwaniec’s short interval fourth moment bound and the twelfth moment bound for Dirichlet L-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms.
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.