View source for Shimura-Taniyama conjecture

''Shimura–Taniyama–Weil conjecture, Taniyama–Shimura conjecture, Taniyama–Weil conjecture, modularity conjecture''

A conjecture that postulates a deep connection between elliptic curves (cf. [[Elliptic curve|Elliptic curve]]) over the rational numbers and modular forms (cf. [[Modular form|Modular form]]). It has been completely proved thanks to the fundamental work of A. Wiles and R. Taylor [[#References|[a5]]], [[#References|[a4]]], and its further refinements [[#References|[a3]]], [[#References|[a2]]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s1201401.png" /> be the group of matrices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s1201402.png" /> which are upper-triangular modulo a given positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s1201403.png" />. It acts as a discrete group of Mobius transformations (cf. also [[Discrete group of transformations|Discrete group of transformations]]; [[Fractional-linear mapping|Fractional-linear mapping]]) on the Poincaré upper half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s1201404.png" /> (cf. also [[Poincaré model|Poincaré model]]). A cusp form of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s1201406.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s1201407.png" /> is an [[Analytic function|analytic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s1201408.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s1201409.png" /> satisfying the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014010.png" /></td> </tr></table>

together with suitable growth conditions on the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014011.png" /> (cf. also [[Modular form|Modular form]]). The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014012.png" /> is periodic of period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014013.png" />, and it can be written as a [[Fourier series|Fourier series]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014014.png" /> with no constant term: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014015.png" />. The [[Dirichlet series|Dirichlet series]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014016.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014018.png" />-function attached to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014019.png" /> (cf. also [[Fourier coefficients of automorphic forms|Fourier coefficients of automorphic forms]]; [[Dirichlet-L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014020.png" />-function]]). It is essentially the [[Mellin transform|Mellin transform]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014021.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014022.png" /></td> </tr></table>

The space of cusp forms of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014023.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014024.png" /> is a finite-dimensional [[Vector space|vector space]] and is preserved by the involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014025.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014026.png" />. E. Hecke has shown that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014027.png" /> lies in one of the two eigenspaces for this involution (with eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014028.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014029.png" /> satisfies the functional equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014030.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014031.png" /> has an [[Analytic continuation|analytic continuation]] to all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014032.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014033.png" /> be an [[Elliptic curve|elliptic curve]] over the rational numbers, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014034.png" /> denote its Hasse–Weil <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014036.png" />-series. The curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014037.png" /> is said to be modular if there exists a cusp form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014038.png" /> of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014040.png" />, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014041.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014042.png" />. The Shimura–Taniyama conjecture asserts that every elliptic curve over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014043.png" /> is modular. Thus, it gives a framework for proving the analytic continuation and functional equation for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014044.png" />. It is prototypical of a general relationship between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014045.png" />-functions attached to arithmetic objects and those attached to automorphic forms (cf. also [[Automorphic form|Automorphic form]]), as described in the far-reaching Langlands program.

A. Weil's refinement of the conjecture predicts that the integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014046.png" /> is equal to the arithmetic conductor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014047.png" /> (cf. also [[Elliptic curve|Elliptic curve]]). One now knows that every elliptic curve is modular. Wiles' method proceeds by viewing the Shimura–Taniyama conjecture in a wider framework which predicts the modularity of the (two-dimensional) Galois representations arising from the cohomology of varieties over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014048.png" /> (cf. also [[Galois theory|Galois theory]]).

The modularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014049.png" /> can also be formulated as the statement that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014050.png" /> is a quotient of the [[Modular curve|modular curve]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014051.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014052.png" />; this curve represents the solution to the moduli problem of classifying pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014053.png" /> consisting of an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014054.png" /> with a distinguished cyclic subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014055.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014056.png" />. Alternately, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014057.png" /> is modular, then there is a (non-constant) complex-analytic [[Uniformization|uniformization]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014058.png" />.

The importance of the Shimura–Taniyama conjecture is manifold. Firstly, it gives the analytic continuation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014059.png" /> for a large class of elliptic curves. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014060.png" />-function itself plays a key role in the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014061.png" />, most notably through the celebrated Birch–Swinnerton Dyer conjecture. Secondly, the modular curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014062.png" /> is endowed with a natural collection of algebraic points arising from the theory of complex multiplication (cf. also [[Elliptic curve|Elliptic curve]]), and the existence of a modular parametrization allows the construction of points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014063.png" /> defined over Abelian extensions of certain imaginary quadratic fields. This fact was exploited by B.H.. Gross and D. Zagier and by V.A. Kolyvagin to give strong evidence for the Birch–Swinnerton Dyer conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014064.png" />, under the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014065.png" /> is modular.

The Shimura–Taniyama conjecture admits various generalizations. Replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014066.png" /> by an arbitrary number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014067.png" />, it predicts that an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014068.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014069.png" /> is associated to an automorphic form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014070.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014071.png" /> is totally real, such an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014072.png" /> is often uniformized by a Shimura curve attached to a suitable quaternion algebra (cf. also [[Quaternion|Quaternion]]) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014073.png" /> with exactly one split place at infinity (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014074.png" /> is of odd degree, or when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120140/s12014075.png" /> has at least one prime of multiplicative reduction). In the context of function fields over finite fields, the Shimura–Taniyama conjecture admits an analogue which was established earlier by V.G. Drinfel'd using methods different from those of Wiles.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Breuil,   B. Conrad,   F. Diamond,   R. Taylor,   "Modularity of elliptic curves"  ''to appear''</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Conrad,   F. Diamond,   R. Taylor,   "Modularity of certain potentially Barsotti–Tate Galois representations"  ''J. Amer. Math. Soc.''  (to appear)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Diamond,   "On deformation rings and Hecke rings"  ''Ann. of Math.'' , '''144''' :  1–2  (1996)  pp. 137–166</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Taylor,   A. Wiles,   "Ring-theoretic properties of certain Hecke algebras"  ''Ann. of Math.'' , '''141''' :  2–3  (1995)  pp. 553–572</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Wiles,   "Modular elliptic curves and Fermat's last theorem"  ''Ann. of Math.'' , '''141''' :  2–3  (1995)  pp. 443–551</TD></TR></table>