Selmer stability in families of congruent Galois representations

For an abelian variety $A$ over a number field $F$, we prove that the average rank of the quadratic twists of $A$ is bounded, under the assumption that the multiplication-by-3 isogeny on $A$ factors as a composition of 3-isogenies over $F$. This is the first such boundedness result for an absolutely simple abelian variety $A$ of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension one, we deduce that if $E/F$ is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded. If $F$ is totally real, we moreover show that a positive proportion of twists have rank 0 and a positive proportion have $3$-Selmer rank 1. These results on bounded average ranks in families of quadratic twists represent new progress towards Goldfeld's conjecture -- which states that the average rank in the quadratic twist family of an elliptic curve over $\mathbb{Q}$ should be $1/2$ -- and the first progress towards the analogous conjecture over number fields other than $\mathbb{Q}$. Our results follow from a computation of the average size of the $\phi$-Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny $\phi$.

The purpose of this course is to give an introduction to the theory of p-adic integration with values in spaces of modular forms (elliptic modular forms, Siegel modular forms, . . .). We show that very general p-adic families of modular forms can be constructed as moments of certain p-adic measures on a profinite group Y = lim ←− Yi with values in a formal q-expansion ring like Zp[[q ]] where B is an additive semi-group, and q = {q |ξ ∈ B} the corresponding formally written multiplicative semi-group (for example B = Bn = {ξ = ξ ∈ Mn(Q)|ξ ≥ 0, ξ half-integral} is the semi-group, important for the theory of Siegel modular forms). We discuss some applications of this theory to the construction of certain new p-adic families of modular forms (families of Klingen-Eisenstein series, families of theta-series with spherical polynomials. . .). Main sources of this theory are: • Serre’s theory of p-adic forms as certain formal q-expansions (J.-P. Serre, Formes modulaires et fonctions zeta p-adiques, LNM 350 (1973) 191-268) [Se73]. • Hida’s theory of p-adic modular forms and p-adic Hecke algebras (H. Hida, Elementary theory of L-functions and Eisenstein series, Cambridge University Press, 1993 [Hi93]). • Construction of p-adic Siegel-Eisenstein series by the author, see [PaSE]. As an application, we describe a solution of a problem of Coleman-Mazur in [PaTV], using the RankinSelberg method and the p-adic integration in a Banach algebra A. An introductory cours given on November 29 in POSTECH (Pohang, Korea) 0 Introduction Let p be a prime number (we often assume p≥ 5). There are two different ways of introducing p-adic modular forms: the first approach uses formal q-expansions with coefficients in a p-adic ring [Se73], and the second approach is the p-adic interpolation of Galois representations attached to classical automorphic forms. The first approach was extensively developped by Katz [Ka78] for the group G = GL2 over a totally real number field, in order to construct p-adic L-functions for CM-fields using p-adic Hilbert-Eisenstein series. In general, in this q-expansion method a typical p-adic family φ of modular (automorphic) forms is an element of the Serre ring: φ ∈ Λ[[q]] where Λ = Zp[[T ]] is the Iwasawa algebra. In the second approach one considers Λ-adic Galois representations of type ρ : Gal(Q/Q) → GLm(Λ) (“Big Galois representations”, see [Hi86], [Til-U]). These two theories are essentially equivalent if we start from holomorphic automorphic forms on the group G = GL2 over a totally real field, but in other cases there is no direct link between φ and ρ. On the other hand there exist interesting examples of p-adic L-functions Lφ,p and Lρ,p attached to φ and to ρ. In general Lφ,p and Lρ,p should belong to the quotient field L = QuotΛ or to its finite extensions. If ρ interpolates a p-adic family of motives then there are conjectural general definitions of Lρ,p (see [Co-PeRi], [Colm98], [PaAdm]). It would be very interesting to formulate a general Langlands-type conjecture relating Λ-adic automorphic forms and Λ-adic Galois representations. As an application, we describe a solution of a problem of Coleman-Mazur, using the Rankin-Selberg method and the theory of p-adic integration with values in a p-adic algebra A. This problem was stated in "The Eigencurve" (1998), R.Coleman and B.Mazur stated the following as follows: Given a prime p and Coleman’s family {fk′} of cusp eigenforms of a fixed positive slope σ = ordp(αp(k )) > 0, to construct a two variable p-adic L-function interpolating on k the Amice-Velu p-adic L-functions Lp(fk′ ). Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. Such measures come from Eisenstein distributions with values in certain Banach A-modules M = M(N ;A) of families of overconvergent forms over A.

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.

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(phi) of a two-dimensional modular Galois representation phi. We start with the p-adic Galois representation phi0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(phi0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(phi0)) from the proof of Wiles of the Shimura-Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let phi denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting phi0, and the characteristic power series of the Selmer group Sel(ad(phi)) is given by a p-adic L-function interpolating L(1, ad(phik)) for weight k + 2 specialization phik of phi. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(phi) [symbol, see text] nu-1), where nu is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

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

We prove (under certain assumptions) the irreducibility of the limit sigma _2 of a sequence of irreducible essentially self-dual Galois representations sigma _k: G_{{mathbf {Q}}} rightarrow {{,mathrm{GL},}}_4(overline{{mathbf {Q}}}_p) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to 1 oplus rho oplus chi with rho irreducible, two-dimensional of determinant chi , where chi is the mod p cyclotomic character. More precisely, we assume that sigma _k are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as krightarrow 2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to rho ) which we assume are non-zero.

This thesis provides congruences between unstable and stable automorphic forms for the symplectic similitude group $GSp(4)$. More precisely, we raise the level of certain CAP representations $Pi$ of Saito-Kurokawa type, arising from classical modular forms $f in S_4(Gamma_0(N))$ of square-free level and root number $epsilon_f=-1$. We first transfer $Pi$ to a suitable inner form $G$ such that $G(R)$ is compact modulo its center. This is achieved by viewing $G$ as a similitude spin group of a definite quadratic form in five variables, and then $ heta$-lifting the whole Waldspurger packet for $widetilde{SL}(2)$ determined by $f$. Thereby we obtain an automorphic representation $pi$ of $G$. For the inner form we prove a precise level-raising result, inspired by the work of Bellaiche and Clozel, and relying on computations of Schmidt. Thus we obtain a $ ilde{pi}$ congruent to $pi$, with a local component that is irreducibly induced from an unramified twist of the Steinberg representation of the Klingen Levi subgroup. To transfer $ ilde{pi}$ back to $GSp(4)$, we use Arthur's stable trace formula and the exhaustive work of Hales on Shalika germs and the fundamental lemma in this case. Since $ ilde{pi}$ has a local component of the above type, all endoscopic error terms vanish. Indeed, by Weissauer, we only need to show that such a component does not participate in the $ heta$-correspondence with any $GO(4)$. This is an exercise in using Kudla's filtration of the Jacquet modules of the Weil representation. Thus we get a cuspidal automorphic representation $ tilde{Pi}$ of $GSp(4)$ congruent to $Pi$, which is neither CAP nor endoscopic. In particular, its Galois representations are irreducible by work of Ramakrishnan. It is crucial for our application that we can arrange for $ ilde{Pi}$ to have vectors fixed by the non-special maximal compact subgroups at all primes dividing $N$. Since $G$ is necessarily ramified at some prime $r$, we have to show a non-special analogue of the fundamental lemma at $r$. Fortunately, by work of Kottwitz we can compare the involved orbital integrals to twisted orbital integrals over the unramified quadratic extension of $Q_r$. The inner form $G$ splits over this extension, and the comparison of the twisted orbital integrals can be done by hand. Finally we give an application of our main result to the Bloch-Kato conjecture. Assuming a conjecture of Skinner and Urban on the rank of the monodromy operators at the primes dividing $N$, we construct a torsion class in the Selmer group of the motive $M_f(2)$.

In this article, we set up a strategy to prove one divisibility towards the main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint modular Galois representations associated to Hida families. This conjecture asserts the equality of the p-adic L-function interpoling the critical values of the symmetric square of the modular forms in these families and the characteristic ideal of the associated Selmer group. The idea is to introduce a third characteristic ideal containing informations on the congruences between cuspidal Siegel modular forms of genus 2 and the Klingen type Eisenstein series and to prove the two divisibilities: The p-adic L-function divides the Eisenstein ideal and that the Eisenstein ideal divides the characteristic ideal of the Selmer group. In that paper we proved the latter divisibility.

This chapter provides the necessary background concerning modular curves and modular forms. It covers modular curves, modular forms, lattices and modular forms, Galois representations attached to eigenforms, and Galois representations over finite fields and reduction to torsion in Jacobians.

This chapter discusses several aspects of the practical side of computing with modular forms and Galois representations. It starts by discussing computations with modular forms, and from there work towards the computation of polynomials that give the Galois representations associated with modular forms. Throughout, the chapter denotes the space of cusp forms of weight k, group Γ‎₁(N), and character ε‎ by Sₖ(N, ε‎).

In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL2(Fpn) occurs as the Galois group of some finite extension of the rational nu mbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their resid ual Galois representations are as large as a priori possible. Both results essentially use Khare’s a nd Wintenberger’s notion of gooddihedral primes. Particular care is taken in order to exclud e nontrivial inner twists. 2000 Mathematics Subject Classification: 11F80 (primary); 12F12, 11F11.

We continue the investigation of the distribution of ℓ ∞ \ell ^{\infty } -Selmer groups in degree ℓ \ell twist families of Galois modules over number fields begun by the author in [J. Amer. Math. Soc. 39-1 (2026), pp. 1–72]. Building off the work on higher Selmer groups in that part, we find conditions under which we can compute the distribution of the ℓ ∞ \ell ^{\infty } -Selmer groups for a given degree ℓ \ell twist family. Along the way, we show that the average rank in the quadratic twist family of any given abelian variety over a number field is bounded.

abstract: The Iwasawa $\mu$-invariant of the Selmer group of a residually reducible Galois representation arising from a Hecke eigencuspform is studied. Furthermore, certain Iwasawa-invariants refining the $\mu$-invariant are defined and analyzed. As an application, we show that given a reducible mod-$p$ Galois representation $\bar{\rho}$ and any choice of integer $N\geq 1$, there is a modular Galois representation lifting $\bar{\rho}$ whose associated Selmer group has $\mu$-invariant $\geq N$. This is a refinement of Serre's conjecture in the residually reducible case.

Let $p$ be a fixed odd prime number, $\mu$ be a Hida family over the Iwasawa algebra of one variable, $\rho_{\mu}$ its Galois representation, $\Bbb{Q}_\infty/\Bbb{Q}$ the $p$-cyclotomic tower and $S$ the variable of the cyclotomic Iwasawa algebra. We compare, for $n\leq 4$ and under certain assumptions, the characteristic power series $L(S)$ of the dual of Selmer groups $\textrm{Sel}(\Bbb{Q}_{\infty},\textrm{Sym}^{2n}\otimes\textrm{det}^{-n} \rho_{\mu})$ to certain congruence ideals (the case $n=1$ has been treated by H. Hida). In particular, we express the first term of the Taylor expansion at the trivial zero $S=0$ of $L(S)$ in terms of an $\scr{L}$-invariant and a congruence number. We conjecture the non-vanishing of this $\scr{L}$-invariant; this implies therefore that these Selmer groups are cotorsion. We also show that our $\scr{L}$-invariants coincide with Greenberg's $\scr{L}$-invariants calculated by R. Harron and A. Jorza.

Stable reduction of modular curves.- On p-adic families of automorphic forms.- ?-curves and abelian varieties of GL2-type from dihedral genus 2 curves.- The old subvariety of J0(NM).- Irreducibility of Galois actions on level 1 Siegel cusp forms.- On elliptic K-curves.- ?-curves and Galois representations.- On the local behaviour of ordinary modular Galois representations.- Arithmetic of ?-curves.- Serre's conjecture for mod 7 Galois representations.- Pairings in the arithmetic of elliptic curves.- Explicit parametrizations of ordinary and supersingular regions of X0(Pn).- Elliptic ?-curves with complex multiplication.- Abelian varieties over ? with large endomorphism algebras and their simple components over ?.- Abelian varieties over ? and modular forms.- Shimura curves embedded in Igusa's threefold.- Shafarevich-Tate groups of nonsquare order.