Goldfeld's conjecture and the 2∞-Selmer converse

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Goldfeld's conjecture and the 2∞-Selmer converse

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Clase 11 de 100

Clase 11 / 100

Recursos

Take one elliptic curve over Q and its quadratic twists. This class shows you how to explain why half the twists have analytic rank zero and half have rank one, and where the proof gets its leverage: a cube of twists whose corners you control.

From Selmer counts to analytic ranks

Three numbers per curve: the analytic rank a(E), the Mordell–Weil rank r(E) and the 2^∞-Selmer corank c₂(E). The Kummer sequence gives c₂ = r + corank Sha[2^∞], so c₂ is an upper bound that Sha can inflate [0:57].
Gross–Zagier–Kolyvagin go from analytic to algebraic. Goldfeld needs the reverse arrow [1:13].
Theorem 1.1 (the 2-converse): if c₂(E) ∈ {0,1}, then a = r = c₂ and Sha is finite. It puts no condition on reduction, CM, torsion or isogenies. A bound on the rank alone is not enough [1:29].
Think-first: corank 0 gives L(E,1) ≠ 0, corank 1 gives a simple zero, and for corank 2 the theorem says nothing [2:00].
Smith's census gives corank 0 and 1 for half of the twists each. Because the converse translates every single twist, the counts carry over: densities 1/2, 1/2 and 0 for rank ≥ 2, with no BSD assumption [2:32].

The cube and interpolation

Twist by h_x = ∏(q*)^{λ_q(x)} for x ∈ F₂^b, with q* ≡ 1 mod 4. The vertex h₀ = 1 is your curve itself [3:29].
Arrange a unit coefficient at every nonzero vertex: a nonzero central specialization with uniformly bounded 2-adic valuation [4:00].
A coefficient on the cube expands into monomials, with 2^|S| dividing c_S. Chevalley–Warning then finds x ≠ 0 with A(x) ≡ A(0) mod 2^M. Bounded valuation plus a deep congruence forces the base to be nonzero [4:15].
What nonzero means at the base: at corank 0 it comes from Kato's zeta class along the real cyclotomic Z₂-extension (L(E,1) ≠ 0). At corank 1 it comes from a Heegner class over Q(√k) through Gross–Zagier, with a companion twist that has a nonzero central value [5:19].

When you read a rank statement about a twist family, look for a Selmer distribution theorem plus a pointwise converse, then find the cube that gives the converse its leverage (§11–13).

Research preprints

Milne's rationality conjecture for abelian varieties

BSD leading-term formula from low Selmer corank

Selmer converse and sums of two cubes

The two-primary BSD formula in rank one

Zero-free regions and the quasi-Riemann hypothesis

The quasi-Riemann hypothesis and zero-free regions

Landau–Siegel zeros: how a determinant could forbid a real zero near 1

Converse theorem for elliptic curves with 2-torsion

Hilbert's tenth problem over the rationals

Why Catalan's constant G is irrational

Goldfeld's conjecture and the 2∞-Selmer converse

Average rank of elliptic curve twists

Chowla–Elliott correlations via divisibility graphs

The Grothendieck–Teichmüller Lie algebra and the Deligne–Drinfeld conjecture

Milnor K-theory and field reconstruction mod p

Milnor K-theory and birational anabelian reconstruction

Bogomolov–Pop reconstruction: the uniform bound along a pencil

Fontaine–Mazur modularity at the prime 2

The dimension of the two-adic Hecke algebra

Pro-modularity at 2 without de Rham conditions

Beating the parity obstruction for primes p with μ(p − 1) = 1

Poisson-Dirichlet law for prime factors of p−1

Totient function fibers and smooth shifted primes

Independence of largest prime factors in consecutive integers

Ostmann's inverse Goldbach problem and prime indecomposability

Hecke eigensheaves with tame ramification

Frobenius structures on tame Hecke eigensheaves

Global Arthur parameters via cuspidal excursion data

Nilpotent orbit decomposition of cuspidal automorphic functions

Rationality of the canonical Arthur filtration

Hecke eigensheaves with ramification at marked points

Temperedness at ramified places for exceptional groups

Restricted geometric Langlands in positive characteristic

Equidistribution of prime-degree torus packets

Sextic torus packets: why degree six needs a new machine

Equidistribution of quartic torus packets as orders grow

Zilber–Pink for curves in A2: elliptic squares

Finiteness of quaternionic points on Siegel curves

Why E × CM points run out: the exponent race behind Zilber–Pink for curves in A2

The abelian Zilber–Pink conjecture: atypical intersections and height scales

Irrationality exponent of π and Flint–Hills convergence

Normal subgroups and the Margulis–Platonov conjecture

The Margulis–Platonov conjecture over number fields

Section conjecture for p-adic curves

Étale covers with a prescribed exterior sheet

Power-free values of irreducible quartic to octic polynomials

Jacobsthal's function and the quadratic bound

The weak inhomogeneous Duffin–Schaeffer problem

Kummer's bias in cubic Gauss sums

Counting totient values: Ford's asymptotic refined

Short Egyptian fraction expansions and Erdős's bound

Positive density of large prime gaps

Zariski density of integral points on character varieties

Gaussian moats: why bounded-step prime walks must stop

Artin's primitive root conjecture explained

Simultaneous primitive roots: conditional bound

Modularity of elliptic curves over imaginary quadratic fields

Uchida's conjecture: from local correspondences to Hoshi's cyclotomic test

A conditional reduction for algebraic Kuga–Satake correspondences

Hodge conjecture for self-powers via Gale duality

Hodge conjecture for K3 self-powers via Kuga–Satake

Hodge classes on CM abelian varieties

Hodge conjecture on abelian variety powers

Kähler canonical bundle formula and b-semiampleness

Logarithmic Kodaira dimension and whole-fiber variation

Iitaka subadditivity and Campana's orbifold conjecture

Iitaka's subadditivity via Hodge-theoretic techniques

Logarithmic Kodaira additivity and the reverse inequality

Abundance after nonvanishing for compact Kähler fourfolds

Bounded Stein degree for log Calabi–Yau boundaries

Good minimal models for Kähler fourfolds

Canonical nonvanishing for fourfolds via moving jets

Lifting sections for log abundance in fourfolds

Log abundance for compact Kähler spaces

Log abundance for log canonical pairs in characteristic zero

Minimal singular metrics on nef adjoint bundles

Uniform Cartier bounds in canonical bundle formulas

Pseudo-effectivity and Kodaira dimension of fiber spaces

Effective Iitaka fibrations and uniform pluricanonical degree