Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

We fix a primep. In this paper, starting from a given Galois representation ϕ having values inp-adic points of a classical groupG, we study the adjoint action of ϕ on thep-adic Lie algebra of the derived group ofG. We call this new Galois representation the adjoint representation Ad(ϕ) of ϕ. Under a suitablep-ordinarity condition (and ramification conditions outsidep), we define, following Greenberg, the Selmer group Sel(Ad(ϕ))/L for each number fieldL. We scrutinize the behavior of Sel(Ad(ϕ))/E∞ as an Iwasawa module for a fixed ℤp-extensionE∞/E of a number fieldE and deduce an exact control theorem. A key ingredient of the proof is the isomorphism between the Pontryagin dual of the Selmer group and the module of Kahler differentials of the universal nearly ordinary deformation ring of ϕ. WhenG=GL(2), ϕ is a modular Galois representation and the base fieldE is totally real, from a recent result of Fujiwara identifying the deformation ring with an appropriatep-adic Hecke algebra, we conclude some fine results on the structure of the Selmer groups, including torsion-property and an exact limit formula ats=0 of the characteristic power series, after removing the trivial zero.

We prove the vanishing of the geometric Bloch-Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic zero point induced from the automorphic representation is formally smooth of the correct dimension. We do this by employing the Taylor-Wiles-Kisin patching method together with Kisin's technique of analyzing the generic fibre of universal deformation rings. Along the way we give a characterization of smooth closed points on the generic fibre of Kisin's potentially semistable local deformation rings in terms of their Weil-Deligne representations.

In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level are isomorphic and complete intersections, provided the same is true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\bar {\rho }$. If $\bar {\rho }$ is scalar at some primes dividing the discriminant of the quaternion algebra, then the Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight-2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda _f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda _f$ by computing the associated Wiles defect purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho _f$ which $f$ gives rise to by the Eichler–Shimura construction. One of the main tools used in the proof is Taylor–Wiles–Kisin patching.

We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}E$ over an arbitrary number field $K$. Under the assumption that ${\rm Gal}(K(E[2])/K) \ {\cong }\ S_3$, we show that the density (counted in a nonstandard way) of twists with Selmer rank $r$ exists for all positive integers $r$, and is given via an equilibrium distribution, depending only on a single parameter (the ‘disparity’), of a certain Markov process that is itself independent of $E$ and $K$. More generally, our results also apply to $p$-Selmer ranks of twists of two-dimensional self-dual ${\bf F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.

Let $\mathbb{F}_{q}$ be a finite field whose characteristic is relatively prime to $2$ and $3$. Let $p$ be a prime number that is coprime to $q$. Let $E$ be an elliptic curve over the global function field $K = \mathbb{F}_{q}(t)$ such that $\textrm{Gal}(K(E[p])/K)$ contains the special linear group $\textrm{SL}_{2}(\mathbb{F}_{p})$. We show that if the quadratic twist family of $E$ has an element whose Néron model has a multiplicative reduction away from $\infty $, then the average $p$-Selmer rank is $p+1$ in large $q$-limit for almost all primes $p$.

The coefficient space is a kind of resolution of singularities of the universal flat deformation space for a given Galois representation of some local field. It parametrizes (in some sense) the finite flat models for the Galois representation. The aim of this note is to determine the image of the coefficient space in the universal deformation space.

Fix a squarefree integer N and let f be a newform of weight 2 for Γ0(N); we assume that f does not have complex multiplication. It was shown in [14] and [15] that for a set of primes l of density 1 the naive deformation theory of the mod l Galois representation associated to f is unobstructed (in the sense that the universal deformation ring is a power series ring over the Witt vectors). In [31] these methods were modified to obtain results on the deformation problems studied by Taylor-Wiles. In this paper we extend the results of Flach and Mazur to the case of newforms f of weight κ ≥ 2 for Γ1(N). We now state our results more precisely. Fix l > max{5, κ+1}, let f be as above and let H be the associated l-adic representation: H is a free module of rank 2 over a certain Hecke algebra A, which itself is a finite, flat, local, Gorenstein Zl-algebra. Let T be the Tate twist EndAH(1) of the module of trace zero endomorphisms of H. Using techniques of Flach we construct a collection of cohomology classes {c} in H(Q, T ) with tightly controlled ramification. With some mild additional hypotheses, applying the methods of Kolyvagin to these classes yields a certain annihilator η ∈ A of the Selmer group H f (Q, T ∗) of the Cartier dual of T . This Selmer group is dual to the differentials ΩR⊗RA, where R is the universal minimally ramified deformation ring of the residual representation of H. In the case that η is a unit this then implies that both R and A are isomorphic to the ring of Witt vectors over the residue field of A. In the general case, following Mazur we show that our construction yields a derivation from A to the Selmer group H f (Q, T/ηT ); it follows by a formal argument that the natural surjection R A induces an isomorphism ΩR⊗RA ∼= ΩA. Although not the strongest possible result, this does provide a great deal of information on the structure of the ring R. (It is possible that any such map R A must be an isomorphism, although as far as I know this question remains open.) We also show that the isomorphism ΩR⊗RA ∼= ΩA is characterized by the fact that ΩA ∼= ΩR⊗RA ∼= HomZl ( H f (Q, T ),Ql/Zl )

We study G-valued Galois deformation rings with prescribed properties, where G is an arbitrary (not necessarily connected) reductive group over an extension of Z_l for some prime l. In particular, for the Galois groups of p-adic local fields (with p possibly equal to l) we prove that these rings are generically smooth, compute their dimensions, and show that functorial operations on Galois representations give rise to well-defined maps between the sets of irreducible components of the corresponding deformation rings. We use these local results to prove lower bounds on the dimension of global deformation rings with prescribed local properties. Applying our results to unitary groups, we improve results in the literature on the existence of lifts of mod l Galois representations, and on the weight part of Serre's conjecture.

In this paper, we study the variation of Selmer groups in families of modular Galois representations that are congruent modulo a fixed prime $p \geq 5$ . Motivated by analogies with Goldfeld’s conjecture on ranks in quadratic twist families of elliptic curves, we investigate the stability of Selmer groups defined over $\mathbb{Q}$ via Greenberg’s local conditions under congruences of residual Galois representations. Let X be a positive real number. Fix a residual representation $\bar{\rho}$ and a corresponding modular form f of weight 2 and optimal level. We count the number of level-raising modular forms g of weight 2 that are congruent to f modulo p , with level $N_g\leq X$ , such that the p -rank of the Selmer groups of g equals that of f . Under some mild assumptions on $\bar{\rho}$ , we prove that this count grows at least as fast as $X (\log X)^{\alpha - 1}$ as $X \to \infty$ , for an explicit constant $\alpha \gt 0$ . The main result is a partial generalisation of theorems of Ono and Skinner on rank-zero quadratic twists to the setting of modular forms and Selmer groups.

In this paper, we compare the structure of Selmer groups of certain classes of Galois representations over an admissible $p$-adic Lie extension. Namely, we show that the $\pi$-primary submodules of the Pontryagin dual of the Selmer groups of two Galois representations have the same elementary representations when the two Galois representations in question are either Tate dual to each other or are congruent to each other. In the first situation, our result gives a partial answer to the question of Greenberg on whether the Pontryagin dual of the Selmer groups of two Galois representations that are Tate dual to each other are pseudo-isomorphic (up to a twist of the Iwasawa algebra). In the second situation, our result will be applied to study the variation of the $\pi$-primary submodules of the dual Selmer groups of certain specialization of a big Galois representation. One of the important ingredient in our proofs is an asymptotic formula for $\pi$-primary modules over a noncommutative Iwasawa algebra which can be viewed as a generalization of a weak analog of the classical Iwasawa asymptotic formula.

This collection of survey and research articles brings together topics at the forefront of the theory of L-functions and Galois representations. Highlighting important progress in areas such as the local Langlands programme, automorphic forms and Selmer groups, this timely volume treats some of the most exciting recent developments in the field. Included are survey articles from Khare on Serre's conjecture, Yafaev on the André-Oort conjecture, Emerton on his theory of Jacquet functors, Venjakob on non-commutative Iwasawa theory and Vigneras on mod p representations of GL(2) over p-adic fields. There are also research articles by: Böckle, Buzzard, Cornut and Vatsal, Diamond, Hida, Kurihara and R. Pollack, Kisin, Nekovář, and Bertolini, Darmon and Dasgupta. Presenting the very latest research on L-functions and Galois representations, this volume is indispensable for researchers in algebraic number theory.

Wiles’ proof [17] of the modularity of semistable elliptic curves over Q relies on a construction of Taylor and Wiles [16] showing that certain Hecke algebras are complete intersections. These Hecke algebras are defined by considering the action of Hecke operators on spaces of modular forms of “minimal level”, or equivalently, on homology groups or Jacobians of modular curves. Taylor and Wiles proceed, roughly speaking, by “patching” algebras arising from forms of different levels. One of the deep results used in their construction was the fact that the homology of the modular curve becomes a free module (of rank two) over the Hecke algebra upon localization at certain maximal ideals. This result, known as a “multiplicity one” result, is a generalization of a theorem of Mazur [11]. Its proof relied on the q-expansion principle of Deligne-Rapoport and Katz, and the comparison of mod ` Betti and de Rham cohomologies (see Sect. 2.1 of [17]). Multiplicity one was thought to be a crucial ingredient of the Taylor-Wiles construction as well as other parts of Wiles’ proof. The purpose of this paper is to explain how to alter the arguments of [16] and [17] so that multiplicity one results are a byproduct rather than an ingredient. The key conceptual change underlying this improvement is the following: Rather than prove that (after localization) the Hecke algebra can be identified with the universal deformation ring of a mod ` Galois representation, we prove that the homology of the modular curve is a free module over this deformation ring. To carry this out, we change the Taylor-Wiles construction1 by 1) “patching” the modules as well as the algebras, and 2) applying the Auslander-Buchsbaum

Conjecturally, the Galois representations that are attached to essentially self-dual regular algebraic cuspidal automorphic representations are Zariski-dense in a polarized Galois deformation ring. We prove new results in this direction in the context of automorphic forms on definite unitary groups over totally real fields. This generalizes the infinite fern argument of Gouvêa–Mazur and Chenevier and relies on the construction of nonclassical p-adic automorphic forms and the computation of the tangent space of the space of trianguline Galois representations. This boils down to a surprising statement about the linear envelope of intersections of Borel subalgebras.

Deformations and the rigidity method

This study investigates the asymptotic behavior of the ranks of Selmer groups associated with elliptic curves possessing a rational 2-torsion point defined over the integers. The Selmer group plays a central role in understanding the Mordell–Weil group and the Birch and Swinnerton-Dyer conjecture. The arithmetic of elliptic curves with torsion points has long attracted significant interest, with foundational results tracing back to the work of Mordell, Selmer, and later refinements by Cassels and others. In particular, the behavior of 2-Selmer groups provides insights into the distribution of ranks and the structure of rational points. Building upon previous methods developed for quadratic twists and leveraging tools from Galois cohomology, we demonstrate that the upper bounds on the size of these Selmer groups are unbounded within certain infinite families of elliptic curves. Our approach highlights the interplay between local conditions at primes and global properties of the curve, offering new perspectives on how torsion influences Selmer ranks.