Hodge classes on CM abelian varieties

Contenido del curso

Research preprints

Cursos Empresas Blog Live Conf Eventos Precios

0:005:49

Hodge classes on CM abelian varieties

OpenAI Math Preprints

Clase 62 de 100

Clase 62 / 100

Recursos

You are given four CM sign patterns and one tensor line built from them. Can you tell whether that line is a Hodge class? Read its sign table column by column. That check is the first step of a claimed proof of the rational Hodge conjecture for CM abelian varieties.

Why divisors are not enough

By Lefschetz (1,1), every degree-2 Hodge class is a divisor class, and cup products of divisors stay algebraic. The cohomology of an abelian variety is Λ*H¹, but that still does not make every Hodge class a product of divisors. Weil classes fall outside the divisor algebra, and this already happens on abelian fourfolds. Through two theorems of Milne, algebraicity of CM Hodge classes leads to the Tate conjecture for abelian varieties over finite fields and to the Hodge standard conjecture. The paper narrows the problem step by step. Deligne and André reduce CM Hodge classes to split Weil classes. Balanced tensors are then built from four-label switches. What is left is one line, U(v₁)₁ ⊗ U(v₂)₁ ⊗ U(v₃)_c ⊗ U(v₄)_c.

The column test [1:26]

An odd sign function v satisfies v(cτ) = −v(τ). It makes each embedding line of type (1,0) (sign +) or (0,1) (sign −). Conjugating by g moves the labels to g, g, gc, gc. So in column g the first two signs stay the same and the last two flip. The line has type (2,2) exactly when the four column signs sum to zero. A rational Hodge class needs this at every Galois conjugate, so the full condition is v₁ + v₂ = v₃ + v₄.

Worked case [1:44]: E = ℚ(ζ₇), c = σ³. With v₁ = (+,+,+), v₂ = (+,−,−), v₃ = (+,+,−), v₄ = (+,−,+), every column gives (2,2) [2:11]. The line is a Hodge class.
Think-first [2:56]: change v₄ to (+,+,−). The identity column still gives (2,2), but column σ gives (1,3). One good embedding proves nothing.

How the paper makes the line algebraic

Surface criterion [3:23]: suppose there are Hodge maps into H¹ of a surface S with a nonzero period ∫ h₁(α₁)∪…∪h₄(α₄). Then the image of S is a cycle that pairs nonzero with the line. Four divisor kernels of type (1,1), algebraic by Lefschetz, turn that functional into a vector on the line.
Theta one-forms [4:09] on a compact quotient of the complex 2-ball (signature (2,1) at one embedding) supply the nonzero period.
Frobenius slopes [4:47]: a Hecke–Frobenius relation forces slope 0 or 1. Weak admissibility turns these into the pure types (0,1) and (1,0), which are exactly the prescribed CM signs.

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