ON FINE SELMER GROUPS AND SIGNED SELMER GROUPS OF ELLIPTIC MODULAR FORMS

Let K be a number field, let A be an abelian variety defined over K and let $$K_\infty /K$$ be a uniform p-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules of ideal class groups on the one side and fine Selmer groups of abelian varieties on the other side. If $$K_\infty $$ contains sufficiently many p-power torsion points of A, then we can compare the ranks and the Iwasawa $$\mu $$ -invariants of these modules over the Iwasawa algebra. In several special cases (e.g. multiple $$\mathbb {Z}_p$$ -extensions), we can also prove relations between suitably generalised Iwasawa $$\lambda $$ -invariants of the two types of Iwasawa modules. In the literature, two different kinds of generalised Iwasawa $$\lambda $$ -invariants have been introduced for ideal class groups and Selmer groups. We define analogues of both concepts for fine Selmer groups and compare the resulting invariants. In order to obtain some of our main results, we prove new asymptotic formulas for the growth of ideal class groups and fine Selmer groups in multiple $$\mathbb {Z}_p$$ -extensions.

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.

We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over ℤ p -cyclotomic extensions. First, we provide a definition of residual and non-primitive Selmer groups at non-ordinary primes. This allows us to extend techniques developed by Greenberg (for p-ordinary elliptic curves) and Kim (p-supersingular elliptic curves) to show that if two p-non-ordinary modular forms are congruent to each other, then the Iwasawa invariants of their signed Selmer groups are related in an explicit manner. Our results have several applications. First of all, this allows us to relate the parity of the analytic ranks of such modular forms generalizing a recent result of the first-named author for p-supersingular elliptic curves. Second, we can prove a Kida-type formula for the signed Selmer groups generalizing results of Pollack and Weston.

For a given Coleman family of modular forms, we construct a formal model and prove the existence of a family of Galois representations associated to the Coleman family. As an application, we study the variations of Iwasawa λ-and µ-invariants of dual fine (strict) Selmer groups over the cyclotomic Zp-extension of Q in Coleman families of modular forms. This generalizes an earlier work of Jha and Sujatha for Hida families. Résumé. Nous construisons un modèle formel pour toute famille de Coleman de formes modulaires, et nous montrons l'existence d'une famille de représentations galoisiennes qui lui est attachée. Gràce à cette construction nous étudions la variation des invariants d'Iwasawa λ et µ attachés aux duaux des groupes de Selmer fins (strictes) le long de la Zp-extension cyclotomique de Q dans des familles de Coleman de formes modulaires. Nos résultats généralisent ceux obtenus par Jha et Sujatha dans le cas de familles de Hida de formes modulaires ordinaires.

We extend and refine the theory of “organizing modules” of Mazur and Rubin to construct a canonical class of matrices that encodes a range of information about natural families of complexes in arithmetic. We then describe several concrete applications of this theory including the proof of new results on the explicit structures of Galois groups, ideal class groups, and wild kernels in higher algebraic K-theory and the formulation of a range of explicit conjectures concerning both the ranks and Galois structures of Selmer groups of abelian varieties over finite (nonabelian) Galois extensions of number fields.

In this paper, we prove a generalization of a correspondence between holomorphic Jacobi cusp forms of higher degree (matrix index) and elliptic cusp forms obtained by K. Bringmann [Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms, Math. Z.253 (2006) 735–752], for forms of higher levels (for congruence subgroups). To achieve this, we make use of the method adopted by M. Manickam and the first author in Sec. 3 of [On Shimura, Shintani and Eichler–Zagier correspondences, Trans. Amer. Math. Soc.352 (2000) 2601–2617], who obtained similar correspondence in the degree one case. We also derive a similar correspondence in the case of skew-holomorphic Jacobi forms (matrix index and for congruence subgroups). Such results in the degree one case (for the full group) were obtained by N.-P. Skoruppa [Developments in the theory of Jacobi forms, in Automorphic Functions and Their Applications, Khabarovsk, 1988 (Acad. Sci. USSR, Inst. Appl. Math., Khabarovsk, 1990), pp. 168–185; Binary quadratic forms and the Fourier coefficients of elliptic and Jacobi modular forms, J. Reine Angew. Math.411 (1990) 66–95] and by M. Manickam [Newforms of half-integral weight and some problems on modular forms, Ph.D. thesis, University of Madras (1989)].

We establish the freeness of certain anticyclotomic Selmer groups of modular forms. The freeness of these Selmer groups plays a key role in the Euler system arguments introduced by Bertolini and Darmon in their work on the anticyclotomic main conjecture for modular forms. In particular, our result fills some implicit gaps which appeared in generalizations of the Bertolini-Darmon result to the case where the associated residual representation is not minimally ramified. The removal of such a minimal ramification hypothesis is essential for applications involving congruences of modular forms.

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 $L_p$ .

Constructing plus/minus local points over Iwasawa Kummer extensions, and Iwasawa theory for supersingular reduction

Let p be a prime not dividing the integer n. By an Ihara result, we mean existence of a cokernel torsion-free injection from a full lattice in the space of p-old modular forms into a full lattice in the space of all modular forms of level pn. In this paper, we will prove an Ihara result in the number field case, for Siegel modular forms. The case of elliptic modular forms is discussed in Ihara (Discrete subgroups of Lie groups and applications to moduli, Oxford University Press, Bombay, 1975). We will use a geometric formulation for the notion of p-old Siegel modular forms (Rastegar in BIMS 43(7):1–23, 2017). Then, we apply an argument by Pappas, and prove the Ihara result using density of Hecke orbits (Chai in Invent Math 121(3):439–479, 1995). This result is meant to pave the way for modularity results in higher genera.

Barsotti Symposium in Algebraic Geometry

In this section, we discuss the period relation of Riemann (Proposition 14.5) for real Nebentype elliptic modular cusp forms of weight 2. More precisely speaking, we formulate the period relation of Riemann for the Hodge structures attached real Nebentype primitive forms of weight 2. Because the category of polarized Hodge structures of weight 1 is equivalent to the opposite category of polarized abelian varieties, the formalism attaching Hodge structures of weight 1 to real Nebentype elliptic primitive cusp forms of weight 2 follows immediately from the theory of Shimura [43]. Therefore we omit some of details and proofs of this formalism. About basic facts and definition, we refer to [43].

Jacobi Forms of Several Variables and the Maaß Space

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

Periods of Hilbert modular forms, Kronecker series and cohomology