Scientists / Open questions

Jean-Pierre Serre

Algebraic geometry; number theory; topology

1954 Fields Medal

29 questions

Glueing principally polarized abelian varieties in wild characteristic and explicit equations for extremal genus-3 curves over finite fields
Vortex-ring dynamics in hollow-cone swirl sprays: survival mechanism and tip-penetration growth-law transition
Explicit integral realizations of the quaternion group $Q_8$ : construct free $O_K$ -lattices in the two-dimensional representation over quadratic fields, with the companion Schur-index-2 groups $2A_4$ and $2^{-}S_3$
Hecke field degrees of cusp forms after Serre's equidistribution theorem: the level aspect without coprimality, the fixed-level weight aspect, and power-saving rates
Serre's uniformity question: is the mod- $\ell$ Galois representation of every non-CM elliptic curve surjective for all primes $\ell$ beyond a constant depending only on the base field?
Torsion structure of Hodge-Witt cohomology of abelian varieties in positive characteristic: the isogeny class of the unipotent part of the p-primary Brauer group
Karoubi's conjectures on Adams operations on Clifford modules
Removing the not-a- $p$ -th-power hypothesis on the trace formula and the structure-theorem dependency in Serre's characteristic- $p$ proof of the modulus theorem (Algebraic Groups and Class Fields, Chapter III)
Arithmetic determination of the coefficients $a_{ij}(p)$ of Hecke operators in the level-1 mod-2 Hecke algebra $A\simeq\mathbf{F}_2[[x,y]]$ , equivalently the image of the universal mod-2 Galois representation unramified outside 2
Cramer-type explicit determinantal solution formulas for the Fredholm equation $x - a\,u(x) = y$ with completely continuous endomorphisms of $p$ -adic Banach spaces (Serre, 1962)
Serre's uniformity problem for mod-p Galois representations of elliptic curves over Q and uniform finiteness of rational points on Y_0(N) over number fields: the open core of the four questions of Bourbaki expose 511
Open questions from Serre's 1984 paper on the Witt invariant of the trace form: a totally real Monster extension of $\mathbb{Q}$
Liftability of smooth projective varieties from characteristic p to characteristic zero: criteria and obstructions (the general question raised by Serre's 1961 non-liftable examples)
The congruence subgroup problem for anisotropic and low S-rank arithmetic groups: the remaining cases of the Bass–Milnor–Serre program
Decomposition matrices, positive-characteristic character formulas, and the extension-of-scalars map for non-split reductive group schemes (open questions from Serre 1968)
Exact 2-primary essential dimensions after Chernousov–Serre: the value of $\operatorname{ed}(E_8;2)=\operatorname{ed}(\mathrm{HSpin}_{16};2)$ , the equality $\operatorname{ed}(f)=\operatorname{ed}(f;2)$ for quadratic forms, and $\operatorname{ed}(\mathrm{Spin}_n)$ for $11\le n\le 14$ in characteristic 2
Self-dual normal bases for $G$ -Galois algebras over fields of $2$ -cohomological dimension at most $2$ : does vanishing of the degree-one and degree-two classes from $G$ suffice?
Pfister decomposition of the trace form of a degree-4 central simple algebra over fields without a primitive fourth root of unity
An a priori proof that projective objects are torsion-free in Serre's category of proalgebraic groups
The lacunarity of the powers of the Dedekind eta-function: the exceptional exponent 26, odd exponents, and lacunarity modulo m
The $d''$ homomorphism condition: characterizing closed range of the Cauchy-Riemann operator and Hausdorff Dolbeault cohomology in Serre duality
Exceptional modules for Serre's converse semisimplicity theorems for symmetric and exterior powers in positive characteristic
Open problems after Serre's embeddings of $\mathbf{PSL}_2(\mathbf{F}_p)$ into simple Lie groups: uniform prime-power lifting, conjugacy classes, $K$ -forms over $\mathbf{Q}(\sqrt{p^*})$ , and explicit integral constructions
Serre's integrality question for the quotient zeta function $ζ_k(s)/ζ_Q(s)$ at negative odd integers

Back to open problems

Glueing principally polarized abelian varieties in wild characteristic and explicit equations for extremal genus-3 curves over finite fields

Vortex-ring dynamics in hollow-cone swirl sprays: survival mechanism and tip-penetration growth-law transition

Explicit integral realizations of the quaternion group Q8Q_8Q8​: construct free OKO_KOK​-lattices in the two-dimensional representation over quadratic fields, with the companion Schur-index-2 groups 2A42A_42A4​ and 2−S32^{-}S_32−S3​

Hecke field degrees of cusp forms after Serre's equidistribution theorem: the level aspect without coprimality, the fixed-level weight aspect, and power-saving rates

Serre's uniformity question: is the mod-ℓ\ellℓ Galois representation of every non-CM elliptic curve surjective for all primes ℓ\ellℓ beyond a constant depending only on the base field?

Torsion structure of Hodge-Witt cohomology of abelian varieties in positive characteristic: the isogeny class of the unipotent part of the p-primary Brauer group

Karoubi's conjectures on Adams operations on Clifford modules

Removing the not-a-ppp-th-power hypothesis on the trace formula and the structure-theorem dependency in Serre's characteristic-ppp proof of the modulus theorem (Algebraic Groups and Class Fields, Chapter III)

Arithmetic determination of the coefficients aij(p)a_{ij}(p)aij​(p) of Hecke operators in the level-1 mod-2 Hecke algebra A≃F2[[x,y]]A\simeq\mathbf{F}_2[[x,y]]A≃F2​[[x,y]], equivalently the image of the universal mod-2 Galois representation unramified outside 2

Cramer-type explicit determinantal solution formulas for the Fredholm equation x−a u(x)=yx - a\,u(x) = yx−au(x)=y with completely continuous endomorphisms of ppp-adic Banach spaces (Serre, 1962)

Serre's uniformity problem for mod-p Galois representations of elliptic curves over Q and uniform finiteness of rational points on Y_0(N) over number fields: the open core of the four questions of Bourbaki expose 511

Open questions from Serre's 1984 paper on the Witt invariant of the trace form: a totally real Monster extension of Q\mathbb{Q}Q

Liftability of smooth projective varieties from characteristic p to characteristic zero: criteria and obstructions (the general question raised by Serre's 1961 non-liftable examples)

The congruence subgroup problem for anisotropic and low S-rank arithmetic groups: the remaining cases of the Bass–Milnor–Serre program

Decomposition matrices, positive-characteristic character formulas, and the extension-of-scalars map for non-split reductive group schemes (open questions from Serre 1968)

Self-dual normal bases for GGG-Galois algebras over fields of 222-cohomological dimension at most 222: does vanishing of the degree-one and degree-two classes from GGG suffice?

Pfister decomposition of the trace form of a degree-4 central simple algebra over fields without a primitive fourth root of unity

An a priori proof that projective objects are torsion-free in Serre's category of proalgebraic groups

The lacunarity of the powers of the Dedekind eta-function: the exceptional exponent 26, odd exponents, and lacunarity modulo m

The d′′d''d′′ homomorphism condition: characterizing closed range of the Cauchy-Riemann operator and Hausdorff Dolbeault cohomology in Serre duality

Exceptional modules for Serre's converse semisimplicity theorems for symmetric and exterior powers in positive characteristic

Open problems after Serre's embeddings of PSL2(Fp)\mathbf{PSL}_2(\mathbf{F}_p)PSL2​(Fp​) into simple Lie groups: uniform prime-power lifting, conjugacy classes, KKK-forms over Q(p∗)\mathbf{Q}(\sqrt{p^*})Q(p∗​), and explicit integral constructions

Serre's integrality question for the quotient zeta function ζk(s)/ζQ(s)ζ_k(s)/ζ_Q(s)ζk​(s)/ζQ​(s) at negative odd integers

Explicit integral realizations of the quaternion group $Q_8$ : construct free $O_K$ -lattices in the two-dimensional representation over quadratic fields, with the companion Schur-index-2 groups $2A_4$ and $2^{-}S_3$

Serre's uniformity question: is the mod- $\ell$ Galois representation of every non-CM elliptic curve surjective for all primes $\ell$ beyond a constant depending only on the base field?

Removing the not-a- $p$ -th-power hypothesis on the trace formula and the structure-theorem dependency in Serre's characteristic- $p$ proof of the modulus theorem (Algebraic Groups and Class Fields, Chapter III)

Arithmetic determination of the coefficients $a_{ij}(p)$ of Hecke operators in the level-1 mod-2 Hecke algebra $A\simeq\mathbf{F}_2[[x,y]]$ , equivalently the image of the universal mod-2 Galois representation unramified outside 2

Cramer-type explicit determinantal solution formulas for the Fredholm equation $x - a\,u(x) = y$ with completely continuous endomorphisms of $p$ -adic Banach spaces (Serre, 1962)

Open questions from Serre's 1984 paper on the Witt invariant of the trace form: a totally real Monster extension of $\mathbb{Q}$

Self-dual normal bases for $G$ -Galois algebras over fields of $2$ -cohomological dimension at most $2$ : does vanishing of the degree-one and degree-two classes from $G$ suffice?

The $d''$ homomorphism condition: characterizing closed range of the Cauchy-Riemann operator and Hausdorff Dolbeault cohomology in Serre duality

Open problems after Serre's embeddings of $\mathbf{PSL}_2(\mathbf{F}_p)$ into simple Lie groups: uniform prime-power lifting, conjugacy classes, $K$ -forms over $\mathbf{Q}(\sqrt{p^*})$ , and explicit integral constructions

Serre's integrality question for the quotient zeta function $ζ_k(s)/ζ_Q(s)$ at negative odd integers