Jerrold Tunnell

Jerrold Bates Tunnell , also Jerry Tunnell (* 1950 ) is an American mathematician . He is an Associate Professor at Rutgers University .

Tunnell received his PhD in 1977 under John T. Tate at Harvard University ( On the local Langlands conjecture for GL (2) ). In 1983 he succeeded in connecting the problem of determining congruent numbers with the number theory of elliptic curves (and with modular forms of half-integer weight). He gave necessary conditions for a number to be congruent, which, given the conjecture of Birch and Swinnerton-Dyer, are also sufficient.

The problem of whether an integer D is congruent can be reduced to the problem of whether the elliptic curve

{\ displaystyle E_ {D}}

:

{\ displaystyle y ^ {2} = x ^ {3} -D ^ {2} x}

has an infinite number of rational solutions (x, y). According to the Birch-Swinnerton-Dyer conjecture, this is the case if and only if the value of the Hasse-Weil zeta function is at 1

{\ displaystyle L (E_ {D}, 1) = 0}

is. Based on the work of Gorō Shimura and Waldspurger , Tunnell constructed two modular forms of weight , whose coefficients interpolate the root of . With his theorem, Tunnell was able to reduce the problem of congruent numbers to the Birch-Swinnerton-Dyer conjecture. ${\ displaystyle {\ frac {3} {2}}}$ ${\ displaystyle L (E_ {D}, 1)}$

Results of Tunnell and Robert Langlands (a special case of Langlands' Reziprozitätsvermutung for Artin L-functions) also played a role in the proof of Fermat's conjecture by Andrew Wiles .

He is a fellow of the American Mathematical Society . In 1984 he became a research fellow of the Alfred P. Sloan Foundation ( Sloan Research Fellow ).

Fonts

A classical Diophantine problem and modular forms of weight 3/2 , Inventiones Mathematicae, Volume 72, 1983, pp. 323-334, online
Artin's conjecture for representations of octahedral type , Bulletin AMS, Volume 5, 1983, pp. 173-175, online

Web links

Individual evidence

^ Mathematics Genealogy Project
↑ Tunnell's theorem is z. B. in the textbook by Neal Koblitz Introduction to Elliptic Curves and Modular Forms , Springer Verlag, Graduate Texts in Mathematics, 1984, 2nd edition 1993, shown
^ The connection to elliptical curves was first suggested by Kurt Heegner in 1952.
↑ Stephen Gelbart Three lectures on the modularity of and automorphic representations of weight 1 ${\ displaystyle {\ bar {\ rho}} _ {E, 3}}$ , in Gary Cornell, Joseph Silverman , Glenn Stevens (Eds.) Modular forms and Fermat's last theorem , Springer Verlag 1997, chapter 6

personal data
SURNAME	Tunnell, Jerrold
ALTERNATIVE NAMES	Tunnell, Jerrold Bates (full name)
BRIEF DESCRIPTION	American mathematician
DATE OF BIRTH	1950

[1] Mathematics Genealogy Project

[2] Tunnell's theorem is z. B. in the textbook by Neal Koblitz Introduction to Elliptic Curves and Modular Forms , Springer Verlag, Graduate Texts in Mathematics, 1984, 2nd edition 1993, shown

[3] The connection to elliptical curves was first suggested by Kurt Heegner in 1952.

[4] Stephen Gelbart Three lectures on the modularity of and automorphic representations of weight 1 ${\ displaystyle {\ bar {\ rho}} _ {E, 3}}$ , in Gary Cornell, Joseph Silverman , Glenn Stevens (Eds.) Modular forms and Fermat's last theorem , Springer Verlag 1997, chapter 6