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Abstract Let $K/{\mathbf{Q}}_p$K/\mathbf{Q}_{p}be unramified.Inside the Emerton–Gee stack ${\mathcal{X}}_2$\mathcal{X}_{2}, one can consider the locus of two-dimensional mod 𝑝 representations of $\mathrm{Gal}\left(\bar{K}/K\right)$\mathrm{Gal}(\overline{K}/K)having a crystalline lift with specified Hodge–Tate weights.We study the case where the Hodge–Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms.We prove that if the gap between each pair of weights is bounded by 𝑝 (the irregular analogue of a Serre weight), then this locus is irreducible.We also establish various inclusion relations between these loci. 

Let $E/\mathbf{Q}$be an elliptic curve and let $p$be an odd prime of good reduction for $E$. Let $K$be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which $p$splits. The goal of this paper is two-fold: (1) we formulate a $p$-adic BSD conjecture for the $p$-adic $L$-function $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$introduced by Bertolini–Darmon–Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148]; and (2) for an algebraic analogue $F_{\stackrel{¯<\#comment/>}{\mathfrak{p}}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$, we show that the “leading coefficient” part of our conjecture holds, and that the “order of vanishing” part follows from the expected “maximal non-degeneracy” of an anticyclotomic $p$-adic height. In particular, when the Iwasawa–Greenberg Main Conjecture $\left(F_{\stackrel{¯<\#comment/>}{\mathfrak{p}}}^{\mathrm{B}\mathrm{D}\mathrm{P}}\right)=\left(L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}\right)$is known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\mathrm{B}\mathrm{D}\mathrm{P}}$at $T=0$up to a $p$-adic unit. Moreover, by adapting the approach of Burungale–Castella–Kim [Algebra Number Theory 15 (2021), pp. 1627–1653], we prove the main conjecture for supersingular primes $p$under mild hypotheses. In the $p$-ordinary case, and under some additional hypotheses, similar results were obtained by Agboola–Castella [J. Théor. Nombres Bordeaux 33 (2021), pp 629–658], but our method is new and completely independent from theirs, and apply to all good primes. 

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Abstract Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $$p$$-adic and $$\textrm{mod}\; p$$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: (1) the analytic continuation at unramified primes $$p$$ to the whole Shimura variety of the $$\textrm{mod}\; p$$ reduction of $$p$$-adic Maass–Shimura operators a priori defined only over the $$\mu $$-ordinary locus, and (2) the construction of new $$\textrm{mod}\; p$$ theta operators that do not arise as the $$\textrm{mod}\; p$$ reduction of Maass–Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree.