Selmer Groups Research Articles

The average number of integral points on the congruent number curves

We show that the total number of non-torsion integral points on the elliptic curves ED:y2=x3−D2x, where D ranges over positive squarefree integers less than N, is O(N(log⁡N)−14+ϵ). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown's method on estimating the average size of the 2-Selmer groups of the curves in this family.

Statistics for 3-isogeny induced Selmer groups of elliptic curves

Given a sixth power free integer a, let Ea be the elliptic curve defined by y2=x3+a. We prove explicit results for the lower density of sixth power free integers a for which the 3-isogeny induced Selmer group of Ea over Q(μ3) has dimension ≤1. The results are proven by refining the strategy of Davenport–Heilbronn, by relating the statistics for integral binary cubic forms to the statistics for 3-isogeny induced Selmer groups.

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

We study an infinite family of j-invariant zero elliptic curves ED:y2=x3+16D and their λ-isogenous curves ED′:y2=x3−27⋅16D, where D and D′=−3D are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field KD=Q(D) always has non-trivial 3-class group. We prove a series of results related to the set of rational points ED′(Q)∖λ(ED(Q)), and the SL(2,Z)-equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of ED and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant D=48035713, and we show that every curve in these classes violates the Hasse Principle.

The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves

Abstract We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$ , and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F. The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.

Iwasawa theory of fine Selmer groups associated to Drinfeld modules

AbstractLet be a prime power and be the rational function field over , the field with elements. Let be a Drinfeld module over and be a nonzero prime ideal of . Over the constant ‐extension of , we introduce the fine Selmer group associated to the ‐primary torsion of . We show that it is a cofinitely generated module over . This proves an analogue of Iwasawa's conjecture in this setting, and provides context for the further study of the objects that have been introduced in this article.

Theta cycles and the Beilinson–Bloch–Kato conjectures

We introduce ‘canonical’ classes in the Selmer groups of certain Galois representations with a conjugate-symplectic symmetry. They are images of special cycles in unitary Shimura varieties, and defined uniquely up to a scalar. The construction is a slight refinement of one of Y. Liu, based on the conjectural modularity of Kudla's theta series of special cycles. For 2-dimensional representations, Theta cycles are (the Selmer images of) Heegner points. In general, they conjecturally exhibit an analogous strong relation with the Beilinson–Bloch–Kato conjectures in rank 1, for which we gather the available evidence.

WILES DEFECT OF HECKE ALGEBRAS VIA LOCAL-GLOBAL ARGUMENTS

Abstract In his work on modularity of elliptic curves and Fermat’s last theorem, A. Wiles introduced two measures of congruences between Galois representations and between modular forms. One measure is related to the order of a Selmer group associated to a newform $f \in S_2(\Gamma _0(N))$ (and closely linked to deformations of the Galois representation $\rho _f$ associated to f), whilst the other measure is related to the congruence module associated to f (and is closely linked to Hecke rings and congruences between f and other newforms in $S_2(\Gamma _0(N))$ ). The equality of these two measures led to isomorphisms $R={\mathbf T}$ between deformation rings and Hecke rings (via a numerical criterion for isomorphisms that Wiles proved) and showed these rings to be complete intersections. We continue our study begun in [BKM21] of the Wiles defect of deformation rings and Hecke rings (at a newform f) acting on the cohomology of Shimura curves over ${\mathbf Q}$ : It is defined to be the difference between these two measures of congruences. The Wiles defect thus arises from the failure of the Wiles numerical criterion at an augmentation $\lambda _f:{\mathbf T} \to {\mathcal O}$ . In situations we study here, the Taylor–Wiles–Kisin patching method gives an isomorphism $ R={\mathbf T}$ without the rings being complete intersections. Using novel arguments in commutative algebra and patching, we generalize significantly and give different proofs of the results in [BKM21] that compute the Wiles defect at $\lambda _f: R={\mathbf T} \to {\mathcal O}$ , and explain in an a priori manner why the answer in [BKM21] is a sum of local defects. As a curious application of our work we give a new and more robust approach to the result of Ribet–Takahashi that computes change of degrees of optimal parametrizations of elliptic curves over ${\mathbf Q}$ by Shimura curves as we vary the Shimura curve. The results we prove are not attainable using only the methods of Ribet–Takahashi.

A higher Gross–Zagier formula and the structure of Selmer groups

We describe a Kolyvagin system-theoretic refinement of Gross–Zagier formula by comparing Heegner point Kolyvagin systems with Kurihara numbers when the root number of a rational elliptic curve E E over an imaginary quadratic field K K is − 1 -1 . When the root number of E E over K K is 1, we first establish the structure theorem of the p ∞ p^\infty -Selmer group of E E over K K . The description is given by the values of certain families of quaternionic automorphic forms, which is a part of bipartite Euler systems. By comparing bipartite Euler systems with Kurihara numbers, we also obtain an analogous refinement of Waldspurger formula. No low analytic rank assumption is imposed in both refinements. We also prove the equivalence between the non-triviality of various “Kolyvagin systems” and the corresponding main conjecture localized at the augmentation ideal. As consequences, we obtain new applications of (weaker versions of) the Heegner point main conjecture and the anticyclotomic main conjecture to the structure of p ∞ p^\infty -Selmer groups of elliptic curves of arbitrary rank. In particular, the Heegner point main conjecture localized at the augmentation ideal implies the strong rank one p p -converse to the theorem of Gross–Zagier and Kolyvagin.

Computing the Cassels-Tate Pairing on the Selmer group of a Richelot Isogeny

Computing the Cassels-Tate Pairing on the Selmer group of a Richelot Isogeny

Statistics for iwasawa invariants of elliptic curves, II

Statistics for iwasawa invariants of elliptic curves, II

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

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in residual characteristic p=2 obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as prescribed by the Galois cohomological method for GL(2): At each ramified place, the local Selmer condition is the tangent space of a smooth quotient of the local deformation ring. By methods of global class field theory, the Selmer group at the minimal level is computed explicitly. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192. The proof combines Wiles' formula and the global reciprocity law. The result has implications for the algebraic structure of even deformation rings and the distribution of their presentations in families.

On the $\lambda$-invariant of Selmer Groups Arising from Certain Quadratic Twists of Gross Curves

Let $q$ be a prime with $q \equiv 7 \mod 8$, and let $K=\mathbb{Q}(\sqrt{-q})$. Then $2$ splits in $K$, and we write $\mathfrak{p}$ for either of the primes $K$ above $2$. Let $K_\infty$ be the unique $\mathbb{Z}_2$-extension of $K$ unramified outside $\mathfrak{p}$. For certain quadratic and biquadratic extensions $\mathfrak{F}/K$, we prove a simple exact formula for the $\lambda$-invariant of the Galois group of the maximal abelian 2-extension unramified outside $\mathfrak{p}$ of the field $\mathfrak{F}_\infty = \mathfrak{F} K_\infty$. Equivalently, our result determines the exact $\mathbb{Z}_2$-corank of certain Selmer groups over $\mathfrak{F}_\infty$ of a large family of quadratic twists of the higher dimensional abelian variety with complex multiplication, which is the restriction of scalars to $K$ of the Gross curve with complex multiplication defined over the Hilbert class field of $K$. We also exhibit computations of the associated Selmer groups over $K_n$ in the case when the $\lambda$-invariant is equal to $1$; here $K_n$ denotes the $n$-th layer of $K_\infty/K$.

The Selmer groups of elliptic Curves $E_{n}: y^2=x^3+nx$

The Selmer groups of elliptic Curves $E_{n}: y^2=x^3+nx$

Iwasawa theory of plus/minus Selmer groups with non-co-free plus/minus local conditions

We often use the plus/minus Selmer groups to study Iwasawa theory for elliptic curves with good supersingular reduction at [Formula: see text]. But, if the plus/minus local conditions are not co-free, it can be difficult to use them effectively. In this paper, we introduce some technical improvements so that we can use the plus/minus Selmer groups even when the plus/minus local conditions are not co-free. In particular, we show that the plus/minus Selmer groups have no proper submodule of finite index and also they satisfy algebraic functional equations even when the said local conditions are not co-free, thus improve our previous results [B. Kim, The algebraic functional equation of an elliptic curve at supersingular primes, Math. Res. Lett. 15(1) (2008) 83–94; The plus/minus Selmer groups for supersingular primes, J. Aust. Math. Soc. 95(2) (2013) 189–200].

Size of the 2-Selmer groups for Heronian elliptic curves

Size of the 2-Selmer groups for Heronian elliptic curves

Cotorsion of anti-cyclotomic Selmer groups on average

For an elliptic curve, we study how many Selmer groups are cotorsion over the anti-cyclotomic Z p \mathbb {Z}_p -extension as one varies the prime p p or the quadratic imaginary field in question.

On the Cohomology of GL(N) and Adjoint Selmer Groups

Abstract We prove under certain conditions (local-global compatibility and vanishing of modulo $p$ cohomology), a generalization of a theorem of Galatius and Venkatesh. We consider the case of $\operatorname{\textsf{GL}}(N)$ over a CM field; we construct a Hecke-equivariant injection from the divisible group associated to the first fundamental group of a derived deformation ring to the Selmer group of the twisted dual adjoint motive with divisible coefficients and we identify its cokernel as the first Tate-Shafarevich group of this motive. Actually, we also construct similar maps for higher homotopy groups with values in exterior powers of Selmer groups, although with less precise control on their kernel and cokernel. By a result of Y. Cai generalizing previous results by Galatius-Venkatesh on the graded cohomology group of a locally symmetric space, our maps relate the (non-Eisenstein) localization of the graded cohomology group for a locally symmetric space to the exterior algebra of the Selmer group of the Tate dual of the adjoint representation. We generalize this to Hida families as well.

On the Iwasawa invariants of BDP Selmer groups and BDPp-adicL-functions

AbstractLetpbe an odd prime. Letf1{f_{1}}andf2{f_{2}}be weight 2 cuspidal Hecke eigenforms with isomorphic residual Galois representations atp. Greenberg–Vatsal and Emerton–Pollack–Weston showed that ifpis a good ordinary prime for the two forms, the Iwasawa invariants of theirp-primary Selmer groups andp-adicL-functions over the cyclotomicℤp{\mathbb{Z}_{p}}-extension ofℚ{\mathbb{Q}}are closely related. The goal of this article is to generalize these results to the anticyclotomic setting. More precisely, letKbe an imaginary quadratic field wherepsplits. Suppose that the generalized Heegner hypothesis holds with respect to both(f1,K){(f_{1},K)}and(f2,K){(f_{2},K)}. We study relations between the Iwasawa invariants of the BDP Selmer groups and the BDPp-adicL-functions off1{f_{1}}andf2{f_{2}}.

Structure of fine Selmer groups over -extensions

AbstractThis paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$ -extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$ , where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$ -extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$ -extension, then it holds for almost all $\mathbb{Z}_{p}$ -extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

Arithmetic statistics for the fine Selmer group in Iwasawa theory

Arithmetic statistics for the fine Selmer group in Iwasawa theory