On the dihedral Euler characteristics of Selmer groups of abelian varieties

This note shows how to use the framework of Euler characteristic formulae to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures, and in particular how to verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this setting. When the Selmer group is cotorsion with respect to the associated Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with respect to the associated Iwasawa algebra, we give a conjectural description of the Euler characteristic of the cotorsion submodule, and explain how to deduce inequalities from the associated main conjecture divisibilities of Perrin-Riou and Howard.Comment: 26 pages. Previous discussion of two-variable setting removed, and discussion of the indefinite setting modified accordingly. To appear in the HIM "Arithmetic and Geometry" conference proceeding

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

We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori-Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate-Shafarevich group of an elliptic curve in the Z_p^2-extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange.Comment: 33 pages, to appear in Journal of Algebr

On the dihedral main conjectures of Iwasawa theory for Hilbert modular eigenforms

We construct a bipartite Euler system in the sense of Howard for Hilbert modular 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 main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main 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.Comment: 58 pages, absolute final version with very minor edits, to appear in the Canadian Journal of Mathematic

Integral presentations of the shifted convolution problem and subconvexity estimates for GL⁡n\operatorname{GL}_n-automorphic LL-functions

Fix $n \geq 2$ an integer, and $F$ be a totally real number field. We reduce the shifted convolution problem for $L$-function coefficients of $\operatorname{GL}_n({\bf{A}}_F)$-automorphic forms to the better-understood setting of $\operatorname{GL}_2({\bf{A}}_F)$. The key idea behind this reduction is to use the classical projection operator $\mathbb P^n_1 \varphi$ together with properties of its Fourier-Whittaker expansion. This allows us to derive novel integral presentations for the shifted convolution problem as Fourier-Whittaker coefficients of certain $L^2$-automorphic forms on the mirabolic subgroup $P_2({\bf{A}}_F)$ of $\operatorname{GL}_2({\bf{A}}_F)$ or its two-fold metaplectic cover $\overline{P}_2({\bf{A}}_F)$. We then construct liftings of these mirabolic forms to $\operatorname{GL}_2({\bf{A}}_F)$ and its two-fold metaplectic cover $\overline{G}({\bf{A}}_F)$ to justify expanding the underlying forms into linear combinations of Poincar\'e series. Decomposing each of the Poincar\'e series spectrally then allows us to derive completely new bounds for the shifted convolution problem in dimensions $n \geq 3$. As an application, we derive a uniform subconvexity bound for $\operatorname{GL}_n({\bf{A}}_F)$-automorphic $L$-functions twisted by Hecke characters. This uniform level-aspect subconvexity estimate appears to the the first of its kind for dimensions $n \geq 3$.Comment: Substantial revision, particularly with respect to the mirabolic form and its lifting to GL_2 via Iwasawa decomposition, 43 p

Class group twists and Galois averages of GL⁡n\operatorname{GL}_n-automorphic LL-functions

Fix $n \geq 2$ an integer, and let $F$ be a totally real number field. We derive estimates for the finite parts of the $L$-functions of irreducible cuspidal $\operatorname{GL}_n({\bf{A}}_F)$-automorphic representations twisted by class group characters or ring class characters of a totally imaginary quadratic extensions $K$ of $F$, evaluated at central values $s=1/2$ or more generally values $s \in {\bf{C}}$ within the strip $\frac{1}{2} - \frac{1}{n^2 + 1} < \Re(s) < 1$. Assuming the generalized Ramanujan conjecture at infinity, we obtain estimates for all arguments in the critical strip $0 < \Re(s) < 1$. We also derive finer nonvanishing estimates for central values $s=1/2$ twisted by ring class characters of $K$. When the dimension $n \leq 4$ is small, these give us nonvanishing estimates depending on the best known approximations towards the generalized Lindel\"of hypothesis for $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms in the level aspect, and in particular unconditional nonvanishing for $n \leq 3$ (with the case of $n=3$ being new). We derive such estimates via certain exact integral representations for the moments, and in particular for new developments of bounds on the shifted convolution problem in this context. In the setting where the cuspidal representation is cohomological of even rank $n \geq 2$, we also explain how to view these estimates in terms of recent rationality theorems towards Deligne's conjecture for automorphic motives over CM fields.Comment: Substantial revision, particularly to the discussion of the shifted convolution problem, with constraints on the rank n for applications to moments, 58 p

Nonvanishing of self-dual LL-values via spectral decomposition of shifted convolution sums

We obtain nonvanishing estimates for central values of certain self-dual Rankin-Selberg $L$-functions on $\operatorname{GL}_2({\bf{A}}_F) \times \operatorname{GL}_2({\bf{A}}_F)$, and more generally $\operatorname{GL}_r({\bf{A}}_F) \times \operatorname{GL}_2({\bf{A}}_F)$ for $r \geq 2$ an integer over $F$ a totally real number field, contingent on the best known approximations towards the generalized Lindel\"of hypothesis for $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms in the level aspect, as well as the best known approximations to the generalized Ramanujan conjecture hypothesis for $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms. We proceed by developing a spectral approach to the shifted convolution problem for coefficients of $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms, accessing he higher-rank case through the classical projection operator $\mathbb P^r_1$ and the way it respects Fourier-Whittaker expansions. In the course of deriving our results, we supply the required nonvanishing hypothesis for recent work of Darmon-Rotger to bound Mordell-Weil ranks of elliptic curves in number fields cut out by tensor products of two odd, two-dimensional Artin representations whose product of determinants is trivial. This in particular allows us to deduce bounds (on average) for Mordell-Weil ranks of elliptic curves in ring class extensions of real quadratic fields which had not been accessible previously.Comment: Substantial revision, particularly to the discussion of the shifted convolution problem, with constraints on the rank r for applications to average central values, 55 p