of linear algebraic groups Serre's modularity conjecture, concerning Galois representations Serre's multiplicity conjectures in commutative algebra Ribet's...

In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre (1975, 1987), states that an odd, irreducible, two-dimensional Galois representation...

the conjecture became known as the modularity theorem. Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem...

(Javier Cilleruelo, Imre Z. Ruzsa, and Carlos Vinuesa, 2010) Serre's modularity conjecture (Chandrashekhar Khare and Jean-Pierre Wintenberger, 2008) Green–Tao...

height Serre group Serre's modularity conjecture Serre's multiplicity conjectures Serre's open image theorem Serre's property FA Serre relations Serre subcategory...

(which Serre had noticed early on: 1 ) became known as the epsilon conjecture (sometimes written ε-conjecture; now known as Ribet's theorem). Serre's main...

conjecture Kelvin's conjecture Kouchnirenko's conjecture Mertens conjecture Pólya conjecture, 1919 (1958) Ragsdale conjecture Schoenflies conjecture (disproved...

provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne. From 1959 onward Serre's interests turned towards group theory, number theory...

geometry) Perverse sheaf Riemann–Hilbert correspondence Serre's modularity conjecture Standard conjectures on algebraic cycles Abramovich, Dan; Graber, Tom;...

research. The Artin conjecture for odd, irreducible, two-dimensional representations follows from the proof of Serre's modularity conjecture, regardless of...

the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was...

2016. It also proved much of the Taniyama–Shimura conjecture, subsequently known as the modularity theorem, and opened up entire new approaches to numerous...

generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by Petersson (1930), is a generalization to other modular forms or automorphic...

\varepsilon (a,b,c,d)(cz+d)^{k}} which are used to generalise the modularity relation defining modular forms, so that f ( a z + b c z + d ) = ε ( a , b , c , d...

number theory, along with Chandrashekhar Khare, for his proof of Serre's modularity conjecture. Wintenberger earned his Ph.D. at Joseph Fourier University...

of Galois representations has dimension two, by the proof of Serre's modularity conjecture. The Consani–Scholton quintic provides a non-rigid example,...

stack of elliptic curves Modularity theorem Shimura variety, a generalization of modular curves to higher dimensions Serre, Jean-Pierre (1977), Cours...

allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine...

limited form of the modularity theorem (unproven at the time and then known as the "Taniyama–Shimura–Weil conjecture"). The modularity theorem involved elliptic...

Shimura posed the Taniyama–Shimura conjecture (now known as the modularity theorem) relating elliptic curves to modular forms. This connection would ultimately...

Fellow of the Royal Society. Khare, Chandrashekhar (2006), "Serre's modularity conjecture: The level one case", Duke Mathematical Journal, 134 (3): 557–589...

the Hasse–Weil conjecture follows from the modularity theorem: each elliptic curve E over Q {\displaystyle \mathbb {Q} } is modular. The Birch and Swinnerton-Dyer...

In number theory, the Stark conjectures, introduced by Stark (1971, 1975, 1976, 1980) and later expanded by Tate (1984), give conjectural information...

\mathrm {SL} _{3}(\mathbb {R} )} should always have the property. Serre's conjecture states that a lattice in a Lie group of rank one should not have the...

Taniyama–Shimura conjecture (now known as the modularity theorem), which states that every elliptic curve over the rational numbers is modular. This conjecture became...

extended this to modular elliptic curves over the rationals of analytic rank at most 1. (The modularity theorem later showed that the modularity assumption...

Khare Jean-Pierre Wintenberger for their remarkable proof of Serre's modularity conjecture 2014 Yitang Zhang for his work on bounded gaps between primes...

Khare "for his proof (with Jean-Pierre Wintenberger) of the Serre modularity conjecture in number theory" 2009 Elon Lindenstrauss "for his contributions...

all modular forms. Δ(z) is the first example of a modular form to be studied in this way. Deligne (in his Fields Medal-winning work) proved Serre's conjecture...

2000. Orient Blackswan. p. 469. Khare, Chandrashekhar (2006), "Serre's modularity conjecture: The level one case", Duke Mathematical Journal 134 Osmundsen...