Non-commutative Iwasawa theory for modular forms

Let p be an odd prime. Let $\mathcal{G}$ be a compact p-adic Lie group with a quotient isomorphic to ℤ p . We give an explicit description of K 1 of the Iwasawa algebra of $\mathcal{G}$ in terms of Iwasawa algebras of Abelian subquotients of $\mathcal{G}$ . We also prove a result about K 1 of a certain canonical localisation of the Iwasawa algebra of $\mathcal{G}$ , which occurs in the formulation of the main conjectures of noncommutative Iwasawa theory. These results predict new congruences between special values of Artin L-functions, which we then prove using the q-expansion principle of Deligne-Ribet. As a consequence we prove the noncommutative main conjecture for totally real fields, assuming a suitable version of Iwasawa’s conjecture about vanishing of the cyclotomic μ-invariant. In particular, we get an unconditional result for totally real pro-p p-adic Lie extension of Abelian extensions of ℚ.

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

Plus/minus p-adic L-functions for Hilbert modular forms

We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

The equivariant main conjecture of Iwasawa theory is shown to hold for a Galois extension K/k of totally real number fields with Galois group an l -adic pro- l Lie group of dimension 1 containing an abelian subgroup of index l , provided that Iwasawa's \mu -invariant \mu(K/k) vanishes.

We construct a bipartite Euler systemin the sense ofHoward forHilbertmodular 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 inmany cases one divisibility of the associated dihedral or anticyclotomicmain 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 CM fields via the technique of Skinner and Urban.

Letbe a real quadratic field. It is well known that if 3 divides the class number ofk, then 3 divides the class number of, and thus it dividesB1,χω−1, where χ and ω are characters belonging to the fieldskandrespectively. In general, the main conjecture of Iwasawa theory implies that if an odd primepdivides the class number ofk, thenpdividesB1,χω−1, where ω is the Teichmüller character forp.The aim of this paper is to examine its converse whenpsplits ink. Letk∞be the ℤp-extension ofk=k0andhnbe the class number ofkn, thenth layer of the ℤp-extension. We shall prove that ifp|B1,χω−1, thenp|hnfor alln≥ 1. In terms of Iwasawa theory, this amounts to saying that ifM∞/k∞, is nontrivial, thenL∞/k∞is nontrivial, whereM∞andL∞are the maximal abelianp-extensions unramified outsidepand unramified everywhere respectively.

Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, with ring of integers $\mathcal{O}$ and Hilbert class field $H$. Suppose $p\nmid [H:K]$ is a prime number which splits in $K$, say $p\mathcal{O}=\mathfrak{p}\mathfrak{p}^*$. Let $H_\infty=HK_\infty$ where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. Write $M(H_\infty)$ for the maximal abelian $p$-extension of $H_\infty$ unramified outside the primes above $\mathfrak{p}$, and set $X(H_\infty)=\mathrm{Gal}(M(H_\infty)/H_\infty)$. In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_\infty)$. As a consequence, we have that if $X(H_\infty)=0$, the relevant $L$-values are $\mathfrak{p}$-adic units. In addition, the main conjecture for $X(H_\infty)$ has implications toward (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak $\mathfrak{p}$-adic Leopoldt conjecture.

In this paper we prove a Gross-Zagier type formula for the anticyclotomic p-adic L-function of an elliptic modular form f of higher weight and of multiplicative type at p. For such f we also decribe explicitely the local Galois representation attached to it at p and compare the Fontaine-Mazur and Teitelbaum L-invariants.

Let f and g be two cuspidal modular forms and let ${\mathcal {F}}$ be a Coleman family passing through f, defined over an open affinoid subdomain V of weight space $\mathcal {W}$ . Using ideas of Pottharst, under certain hypotheses on f and $g,$ we construct a coherent sheaf over $V \times \mathcal {W}$ that interpolates the Bloch–Kato Selmer group of the Rankin–Selberg convolution of two modular forms in the critical range (i.e, the range where the p-adic L-function $L_p$ interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$ .

We prove the “main conjecture” of equivariant Iwasawa theory when μ = 0 \mu =0 .

We construct p-adic L-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions via the Jacquet–Langlands correspondence, generalizing works of Bertolini–Darmon, Vatsal and others. The construction given here is adelic, which allows us to deduce a precise interpolation formula from a Waldspurger-type theorem, as well as a formula for the dihedral μ-invariant. We also make a note of Howard's non-vanishing criterion for these p-adic L-functions, which can be used to reduce the associated Iwasawa main conjecture to a certain non-triviality criterion for families of p-adic L-functions.

We construct p-adic L-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions via the Jacquet-Langlands correspondence, generalizing works of Bertolini-Darmon, Vatsal and others. The construction given here is adelic, which allows us to deduce a precise interpolation formula from a Waldspurger type formula, as well as a formula for the dihedral mu-invariant. We also make a note of Howard's nonvanishing criterion for these p-adic L-functions, which can be used to reduce the associated Iwasawa main conjecture to a certain nontriviality criterion for families of p-adic L-functions.

We prove a stronger version of the keystone result of Dasgupta and Kakde (Ann Math 197:289–388, 2023) on the $$\mathbb Z[G(H/F)]^-$$ Z [ G ( H / F ) ] - -Fitting ideals of certain Selmer modules $$Sel_S^T(H)^-$$ S e l S T ( H ) - associated with an abelian, CM extension H / F of a totally real number field F , and use this to compute the $$\mathbb Z_p[[G(H_\infty /F)]]^-$$ Z p [ [ G ( H ∞ / F ) ] ] - -Fitting ideal of the Iwasawa module analogues $$Sel_S^T(H_\infty )_p^-$$ S e l S T ( H ∞ ) p - of these Selmer modules, where $$H_\infty $$ H ∞ is the cyclotomic $${\mathbb {Z}}_p$$ Z p -extension of H , for an odd prime p . Our main Iwasawa theoretic result states that the $$\mathbb Z_p[[G(H_\infty /F)]]^-$$ Z p [ [ G ( H ∞ / F ) ] ] - -module $$Sel_S^T(H_\infty )_p^-$$ S e l S T ( H ∞ ) p - is of projective dimension 1 (unlike the $${\mathbb {Z}}[G(H/F)]^-$$ Z [ G ( H / F ) ] - -module $$Sel_S^T(H)^-$$ S e l S T ( H ) - which could have infinite projective dimension), is quadratically presented, and that its Fitting ideal is principal, generated by an equivariant p -adic L -function $$\Theta _S^T(H_\infty /F)$$ Θ S T ( H ∞ / F ) . Further, we establish a perfect duality pairing between $$Sel_S^T(H_\infty )_p^-$$ S e l S T ( H ∞ ) p - and a certain $$\mathbb Z_p[[G(H_\infty /F)]]^-$$ Z p [ [ G ( H ∞ / F ) ] ] - -module $${\mathcal {M}}_S^T(H_\infty )^-$$ M S T ( H ∞ ) - , essentially introduced by Greither and the second author in (J Algebraic Geom 24:629–692, 2015). As a consequence, we recover the Equivariant Main Conjecture for the Tate module $$T_p(\mathcal M_S^T(H_\infty ))^-$$ T p ( M S T ( H ∞ ) ) - , proved in loc.cit. under the hypothesis that the classical Iwasawa $$\mu $$ μ -invariant associated with H and p vanishes. As a further consequence, we give an unconditional proof of the refined Coates–Sinnott Conjecture, proved in loc.cit. under the same $$\mu =0$$ μ = 0 hypothesis, and also proved unconditionally but with different methods by Johnston and Nickel in (An unconditional proof of the abelian equivariant Iwasawa main conjecture and applications, arxiv:2010.03186), regarding the $$\mathbb Z[G(H/F)]$$ Z [ G ( H / F ) ] -Fitting ideals of the higher Quillen K -groups $$K_{2n-2}({\mathcal {O}}_{H,S})$$ K 2 n - 2 ( O H , S ) , for all $$n\ge 2$$ n ≥ 2 . Finally, we combine the techniques developed in the process with the method of “Taylor–Wiles primes” (introduced by Wiles (Ann Math (2) 131: 555–565, 1990) and refined by Greither in (Math Z 233: 515–534, 2000)) to strengthen further the keystone result in Dasgupta and Kakde (Ann Math 197:289–388, 2023) and prove, as a consequence, a conjecture of Burns–Kurihara–Sano on Fitting ideals of Selmer groups of CM number fields.