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.

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 carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic p, generalising the author’s prior work with Sasaki to the case where p is ramified in the totally real field. In particular, we use the partial Hasse invariants and Kodaira–Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren’s construction of partial $\Theta $ -operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore, we describe the kernels of partial $\Theta $ -operators in terms of images of geometrically constructed partial Frobenius operators. Finally, we apply our results to prove a partial positivity result for minimal weights of mod p Hilbert modular forms.

We prove the explicit version of the Buzzard–Diamond–Jarvis conjecture formulated by Dembele et al. (Serre weights and wild ramification in two-dimensional Galois representations, Preprint (2016), arXiv:1603.07708 [math.NT]). More precisely, we prove that it is equivalent to the original Buzzard–Diamond–Jarvis conjecture, which was proved for odd primes (under a mild Taylor–Wiles hypothesis) in earlier work of the third author and coauthors.

We consider mod $p$ Hilbert modular forms associated to a totally real field of degree $d$ in which $p$ is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a $d$-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren–Oort stratification on mod $p$ Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.

We prove modularity of some two-dimensional, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$-adic Galois representations over a totally real field that are nearly ordinary at all places above $2$ and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the $2$-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above $2$.

In this paper, we construct a generalization of the Kohnen plus space for Hilbert modular forms of half-integral weight. The Kohnen plus space can be characterized by the eigenspace of a certain Hecke operator. It can be also characterized by the behavior of the Fourier coefficients. For example, in the parallel weight case, a modular form of weight $\kappa + (1/ 2)$ with $\xi \mathrm{th} $ Fourier coefficient $c(\xi )$ belongs to the Kohnen plus space if and only if $c(\xi )= 0$ unless $\mathop{(- 1)}\nolimits ^{\kappa } \xi $ is congruent to a square modulo $4$. The Kohnen subspace is isomorphic to a certain space of Jacobi forms. We also prove a generalization of the Kohnen–Zagier formula.

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini–Darmon, Longo, Nekovar, Pollack–Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over $\text{CM}$ fields via the technique of Skinner and Urban.

In this article we refine the method of Bertolini and Darmon $\left[ \text{BD}1 \right],\,\left[ \text{BD2} \right]$ and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two.

For p=3 and p=5, we exhibit a finite nonsolvable extension of ℚ which is ramified only at p, proving in the affirmative a conjecture of Gross. Our construction involves explicit computations with Hilbert modular forms.

For the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.

Using a p-adic analogue of the convolution method of Rankin–Selberg and Shimura, we construct the two-variable p-adic L-function of a Hida family of Hilbert modular eigenforms of parallel weight. It is shown that the conditions of Greenberg–Stevens [R. Greenberg and G. Stevens, p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111 (1993), 407–447] are satisfied, from which we deduce special cases of the Mazur–Tate–Teitelbaum conjecture in the Hilbert modular setting.

We introduce an action of a discrete subgroup $\varGamma$ of $SL(2,\mathbb{R})^n$ on the space of pseudodifferential operators of $n$ variables, and construct a map from the space of Hilbert modular forms for $\varGamma$ to the space of pseudodifferential operators invariant under such a $\varGamma$-action, which is a lifting of the symbol map of pseudodifferential operators. We also obtain a necessary and sufficient condition for a certain type of pseudodifferential operator to be $\varGamma$-invariant.

AMS 2000 Mathematics subject classification: Primary 11F41; 35S05

Let f be a primitive Hilbert modular cusp form of arbitrary level and parallel weight k, defined over a totally real number field F. We define a finite set of primes $\cal S$ that depends on the weight and level of f, the field F, and the torsion in the boundary cohomology groups of the Borel–Serre compactification of the underlying Hilbert-Blumenthal variety. We show that, outside $\cal S$, any prime that divides the algebraic part of the value at s=1 of the adjoint L-function of f is a congruence prime for f. In special cases we identify the ‘boundary primes’ in terms of expressions of the form $N_{F/{\mathbb{Q}}}(\epsilon^{k-1} - 1)$, where ε is a totally positive unit of F.