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Let $f$ and $g$ be two modular forms which are non-ordinary at $p$. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution $f \otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on non-vanishing of $p$-adic $L$-fuctions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei-Loeffler-Zerbes. Furthermore, we define an explicit logarithmic matrix using the theory of Wach modules, and show that this describes the growth of the Euler systems and $p$-adic $L$-functions associated to $f \otimes g$ in the cyclotomic tower. This allows us to formulate "signed" Iwasawa main conjectures for $f\otimes g$ in the spirit of Kobayashi's $\pm$-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses., We have expanded the introduction and corrected a few minor imprecisions

Euler systems are certain compatible families of cohomology classes, which play a key role in studying the arithmetic of Galois representations. We briefly survey the known Euler systems, and recall a standard conjecture of Perrin-Riou predicting what kind of Euler system one should expect for a general Galois representation. Surprisingly, several recent constructions of Euler systems do not seem to fit the predictions of this conjecture, and we formulate a more general conjecture which explains these extra objects. The novel aspect of our conjecture is that it predicts that there should often be Euler systems of several different ranks associated to a given Galois representation, and we describe how we expect these objects to be related., Comment: Expository article, 13 pages. To appear in proceedings of the Iwasawa 2017 conference

This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(\Gamma) the algebra of Qp-valued distributions on \Gamma = Gal(Qp(\mu (p^\infty) / Qp). Our first result determines the H(\Gamma)-elementary divisors of the quotient of D_{cris}(V) \otimes H(\Gamma) by the H(\Gamma)-submodule generated by (\phi * N(V))^{\psi = 0}, where N(V) is the Wach module of V. By comparing the determinant of this map with that of L_V (which can be computed via Perrin-Riou's explicit reciprocity law), we obtain a precise description of the images of the Coleman maps. In the case when V arises from a modular form, we get some stronger results about the integral Coleman maps, and we can remove many technical assumptions that were required in our previous work in order to reformulate Kato's main conjecture in terms of cotorsion Selmer groups and bounded p-adic L-functions., Comment: 27 pages

For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f = sum(a_n q^n) be a normalized new modular eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f, we define two Coleman maps with values in the Iwasawa algebra of Zp^* (after extending scalars to some extension of Qp). Applying these maps to the Kato zeta elements gives a decomposition of the (generally unbounded) p-adic L-functions of f into linear combinations of two power series of bounded coefficients, generalizing works of Pollack (in the case a_p=0) and Sprung (when f corresponds to a supersingular elliptic curve). Using ideas of Kobayashi for elliptic curves which are supersingular at p, we associate to each of these power series a cotorsion Selmer group. This allows us to formulate a "main conjecture". Under some technical conditions, we prove one inclusion of the "main conjecture" and show that the reverse inclusion is equivalent to Kato's main conjecture., Comment: 44 pages. To appear in the Asian Journal of Mathematics

In this paper, we prove the “local $\varepsilon$ -isomorphism” conjecture of Fukaya and Kato for a particular class of Galois modules, obtained by interpolating the twists of a fixed crystalline representation of $G_{ \mathbf {Q}_{p}}$ by a family of characters of $G_{ \mathbf {Q}_{p}}$ . This can be regarded as a local analogue of the Iwasawa main conjecture for abelian $p$ -adic Lie extensions of $ \mathbf {Q}_{p}$ , extending earlier work of Kato for rank one modules and of Benois and Berger for the cyclotomic extension. We show that such an $\varepsilon$ -isomorphism can be constructed using the 2-variable version of the Perrin-Riou regulator map constructed by the first and third authors.