Let $$\ell $$ be a rational prime. Previously, abelian $$\ell $$ -towers of multigraphs were introduced which are analogous to $${\mathbb {Z}}_{\ell }$$ -extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the $$\ell $$ -part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for $${\mathbb {Z}}_{\ell }$$ -extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian $$\ell $$ -towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in $${\mathbb {Z}}_{\ell }$$ and then study the special value at $$u=1$$ of the Artin-Ihara L-function $$\ell $$ -adically.

Building on previous results [17, 35], we complete the classification of compact oriented Einstein 4-manifolds with \(\det (W^+) > 0\). There are, up to diffeomorphism, exactly 15 manifolds that carry such metrics, and, on each of these manifolds, such metrics sweep out exactly one connected component of the corresponding Einstein moduli space.

In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the $$\mu $$ -invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions.

In this computational paper we verify a truncated version of the Buzzard–Calegari conjecture on the Newton polygon of the Hecke operator $$T_2$$ for all large enough weights. We first develop a formula for computing p-adic valuations of exponential sums, which we then implement to compute 2-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard–Calegari polynomial has a vertex at $$n\le 15$$ , then it agrees with the Newton polygon of $$T_2$$ up to n.

We prove an explicit integral representation—involving the pullback of a suitable Siegel Eisenstein series—for the twisted standard L-function associated to a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to L-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local integral. By specializing our integral representation to the case n=2 we are able to prove a reciprocity law—predicted by Deligne’s conjecture—for the critical values of the twisted standard L-function for vector-valued Siegel cusp forms of degree 2 and arbitrary level. This arithmetic application generalizes previously proved critical-value results for the full level case. By specializing further to the case of Siegel cusp forms obtained via the Ramakrishnan–Shahidi lift, we obtain a reciprocity law for the critical values of the symmetric fourth L-function of a classical newform.

In this note we make corrections to the above mentioned paper. First, we correct the statement of the main theorem (Theorem 1.1).

We show that all CM-types of Galois CM-fields without proper CM-subfields are nondegenerate. As a consequence, the Hodge conjecture is true for abelian varieties with complex multiplication by such CM-fields.

In this note, we introduce a new continuity path of fourth order nonlinear equations connecting the cscK equation to a second order elliptic equation, which is the critical point equation of the J-flow introduced by Donaldson (Asian J Math 3(1):1–16, 1999) and the author (Commun Anal Geom 12(4):837–852, 2004). This is a generalization of the classical Aubin continuity path for Kahler–Einstein metrics. The aim of this new path is to attack the existence problem of cscK metric. The “openness” along this continuity path is proved and a set of open problems associated with this new path is proposed.