Taniyama's problems are a set of 36mathematical problems posed byJapanesemathematicianYutaka Taniyama in 1955. The problems primarily focused onalgebraic geometry,number theory, and the connections betweenmodular forms andelliptic curves. Taniyama's twelfth and thirteenth problems were the precursor to theTaniyama–Shimura conjecture, also known as themodularity theorem, which would be used inAndrew Wiles'proof ofFermat's Last Theorem in 1995.
In the 1950spost-World War II period of mathematics, there was renewed interest in the theory ofmodular curves due to the work of Taniyama andGoro Shimura.[1] During the 1955 international symposium onalgebraic number theory atTokyo andNikkō,[a] Taniyama compiled his 36 problems in a document titled"Problems of Number Theory" and distributedmimeographs of his collection to the symposium's participants.[3][4] Serre later brought attention to these problems in the early 1970s.[1]
The most famous of Taniyama's problems are his twelfth and thirteenth problems.[1][3] These problems led to the formulation of theTaniyama–Shimura conjecture (now known as themodularity theorem), which states that every elliptic curve over the rational numbers ismodular. This conjecture played a major role inAndrew Wiles'proof ofFermat's Last Theorem in 1995.[3][4]
Taniyama's problems influenced the development of theLanglands program, the theory ofmodular forms, and the study ofelliptic curves.[3]
Taniyama's tenth problem addressedDedekind zeta functions andHecke L-series, and while distributed in English at the 1955Tokyo-Nikkō conference attended by bothSerre andAndré Weil, it was only formally published in Japanese in Taniyama's collected works.[1]
Let be a totally realnumber field, and be aHilbert modular form to the field. Then, choosing in a suitable manner, we can obtain a system ofErich Hecke'sL-series withGrößencharakter, whichcorresponds one-to-one to this by the process ofMellin transformation. This can be proved by a generalization of the theory of operator ofHecke toHilbert modular functions (cf.Hermann Weyl).[1]
According toSerge Lang, Taniyama's eleventh problem deals withelliptic curves with complex multiplication, but is unrelated to Taniyama's twelfth and thirteenth problems.[1]
Let be anelliptic curve defined over analgebraic number field, and theL-function of over in the sense that is the zeta function of over. If theHasse–Weil conjecture is true for, then theFourier series obtained from by the inverseMellin transformation must be anautomorphic form of dimension −2 of a special type (seeHecke[b]). If so, it is very plausible that this form is an ellipticdifferential of the field of associated automorphic functions. Now, going through these observations backward, is it possible to prove the Hasse–Weil conjecture by finding a suitable automorphic form from which can be obtained?[5][1]
Taniyama's twelfth problem's significance lies in its suggestion of a deep connection betweenelliptic curves andmodular forms. While Taniyama's original formulation was somewhat imprecise, it captured a profound insight that would later be refined into themodularity theorem.[6][3] The problem specifically proposed that theL-functions of elliptic curves could be identified with those of certain modular forms, a connection that seemed surprising at the time.
Fellow Japanese mathematicianGoro Shimura noted that Taniyama's formulation in his twelfth problem was unclear: the proposedMellin transform method would only work for elliptic curves overrational numbers.[6] For curves overnumber fields, the situation is substantially more complex and remains unclear even at a conjectural level today.[3]
To characterize the field of elliptic modular functions oflevel, and especially to decompose theJacobian variety of thisfunction field into simple factors up toisogeny. Also it is well known that if, aprime, and, then contains elliptic curves with complex multiplication. What can one say for general?[6]