# Dorian M. Goldfeld

Dorian Morris Goldfeld (born January 21, 1947) is an American mathematician working in analytic number theory and automorphic forms at Columbia University.

Dorian M. Goldfeld
Dorian Goldfeld at The Analytic Theory of Automorphic Forms workshop, Oberwolfach, Germany (2011)
BornJanuary 21, 1947
Marburg, Germany
NationalityAmerican
Alma materColumbia University
Known for
• Analytic number theory
• Automorphic forms
• Cryptography
AwardsFrank Nelson Cole Prize in Number Theory (1987)
Sloan Fellowship (1977–1979)
Vaughan prize (1985)
Fellow of the American Academy of Arts and Sciences (April 2009)
Scientific career
FieldsMathematics
InstitutionsColumbia University
Massachusetts Institute of Technology
Doctoral studentsM. Ram Murty
InfluencedShou-Wu Zhang

## Professional career

Goldfeld received his B.S. degree in 1967 from Columbia University. His doctoral dissertation, entitled "Some Methods of Averaging in the Analytical Theory of Numbers", was completed under the supervision of Patrick X. Gallagher in 1969, also at Columbia. He has held positions at the University of California at Berkeley (Miller Fellow, 1969–1971), Hebrew University (1971–1972), Tel Aviv University (1972–1973), Institute for Advanced Study (1973–1974), in Italy (1974–1976), at MIT (1976–1982), University of Texas at Austin (1983–1985) and Harvard (1982–1985). Since 1985, he has been a professor at Columbia University.[1]

He is a member of the editorial board of Acta Arithmetica and of The Ramanujan Journal.[2][3] On January 1, 2018 he became the Editor-in-Chief of the Journal of Number Theory.[4]

He is a co-founder and board member of SecureRF, a corporation that has developed the world’s first linear-based security solutions.[5]

Goldfeld advised several doctoral students including M. Ram Murty.[6] in 1986, he brought Shou-Wu Zhang to the United States to study at Columbia.[7][8][9]

## Research interests

Goldfeld's research interests include various topics in number theory. In his thesis,[10] he proved a version of Artin's conjecture on primitive roots on the average without the use of the Riemann Hypothesis.

In 1976, Goldfeld provided an ingredient for the effective solution of Gauss' class number problem for imaginary quadratic fields.[11] Specifically, he proved an effective lower bound for the class number of an imaginary quadratic field assuming the existence of an elliptic curve whose L-function had a zero of order at least 3 at ${\displaystyle s=1/2}$. (Such a curve was found soon after by Gross and Zagier). This effective lower bound then allows the determination of all imaginary fields with a given class number after a finite number of computations.

His work on the Birch and Swinnerton-Dyer conjecture includes the proof of an estimate for a partial Euler product associated to an elliptic curve,[12] bounds for the order of the Tate–Shafarevich group[13]

Together with his collaborators, Dorian Goldfeld has introduced the theory of multiple Dirichlet series, objects that extend the fundamental Dirichlet series in one variable.[14]

He has also made contributions to the understanding of Siegel zeroes,[15] to the ABC conjecture,[16] to modular forms on ${\displaystyle \operatorname {GL} (n)}$,[17] and to cryptography (Arithmetica cipher, Anshel–Anshel–Goldfeld key exchange).[18]

Together with his wife, Dr. Iris Anshel,[19] and father-in-law, Dr. Michael Anshel,[20] both mathematicians, Dorian Goldfeld founded the field of braid group cryptography.[21][22]

## Awards and honors

In 1987 he received the Frank Nelson Cole Prize in Number Theory, one of the prizes in Number Theory, for his solution of Gauss' class number problem for imaginary quadratic fields. He has also held the Sloan Fellowship (1977–1979) and in 1985 he received the Vaughan prize. In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley. In April 2009 he was elected a Fellow of the American Academy of Arts and Sciences. In 2012 he became a fellow of the American Mathematical Society.[23]

## Selected works

• Goldfeld, Dorian; Hundley, Joseph (2011). Automorphic Representations and L-Functions for the General Linear Group, Volume 1. ISBN 9780521474238.
• Goldfeld, Dorian; Hundley, Joseph (2011). Automorphic Representations and L-Functions for the General Linear Group, Volume 2. ISBN 9781107007994.
• Gerritzen; Goldfeld; Kreuzer; Rosenberger; Shpilrain, eds. (2006). Algebraic Methods in Cryptography. ISBN 0-8218-4037-1.
• Goldfeld, Dorian (2006). Automorphic Forms and L-Functions for the Group GL(n,R). ISBN 0-521-83771-5.
• Anshel, Iris; Goldfeld, Dorian (1995). Calculus: a Computer Algebra Approach. ISBN 1-57146-038-1.

## References

1. "Biographical Data Dorian Goldfeld" (PDF). Retrieved February 8, 2019.
2. "Acta Arithmetica". Polish Academy of Sciences. Retrieved February 8, 2019.
3. "Editorial board". Springer. Retrieved February 8, 2019.
4. "Editorial board". Elsevier. Retrieved February 8, 2019.
5. SecureRF Corporation, co-founded by Dorian Goldfeld
6. "从放鸭娃到数学大师" [From ducklings to mathematics master] (in Chinese). Academy of Mathematics and Systems Science. November 11, 2011. Retrieved May 5, 2019.
7. "專訪張壽武：在數學殿堂里，依然懷抱小學四年級的夢想" [Interview with Zhang Shou-Wu: In the mathematics department, he still has his dream from fourth grade of elementary school] (in Chinese). Beijing Sina Net. May 3, 2019. Retrieved May 5, 2019.
8. "专访数学家张寿武：要让别人解中国人出的数学题" [Interview with mathematician Zhang Shouwu: Let others solve the math problems of Chinese people] (in Chinese). Sina Education. May 4, 2019. Retrieved May 5, 2019.
9. Goldfeld, Dorian, Artin's conjecture on the average, Mathematika, 15 1968
10. Goldfeld, Dorian, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4
11. Goldfeld, Dorian, Sur les produits partiels eulériens attachés aux courbes elliptiques, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 14,
12. Goldfeld, Dorian; Szpiro, Lucien Bounds for the order of the Tate–Shafarevich group, Compositio Mathematica 97 (1995), no. 1-2, Goldfeld, Dorian; Lieman, Daniel Effective bounds on the size of the Tate–Shafarevich group. Math. Res. Lett. 3 (1996), no. 3; Goldfeld, Dorian, Special values of derivatives of L-functions. Number theory (Halifax, NS, 1994), 159–173, CMS Conf. Proc., 15, Amer. Math. Soc., Providence, RI, 1995.
13. Goldfeld, Dorian; Hoffstein, Jeffrey Eisenstein series of 1/2-integral weight and the mean value of real Dirichlet L-series. Invent. Math. 80 (1985), no. 2; Diaconu, Adrian; Goldfeld, Dorian; Hoffstein, Jeffrey Multiple Dirichlet series and moments of zeta and L-functions. Compositio Mathematica 139 (2003), no. 3
14. Goldfeld, Dorian, A simple proof of Siegel's theorem Proc. Natl. Acad. Sci. U.S.A. 71 (1974); Goldfeld, D. M.; Schinzel, A. On Siegel's zero. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4
15. Goldfeld, Dorian Modular elliptic curves and Diophantine problems. Number theory (Banff, AB, 1988), 157–175, de Gruyter, Berlin, 1990.
16. Bump, Daniel; Friedberg, Solomon; Goldfeld, Dorian Poincaré series and Kloosterman sums. The Selberg trace formula and related topics (Brunswick, Maine, 1984), 39–49, Contemp. Math., 53, Amer. Math. Soc., Providence, RI, 1986
17. Anshel, Iris; Anshel, Michael; Goldfeld, Dorian An algebraic method for public-key cryptography. Math. Res. Lett. 6 (1999), no. 3–4, Anshel, Michael; Goldfeld, Dorian Zeta functions, one-way functions, and pseudorandom number generators. Duke Math. J. 88 (1997), no. 2
18. Dr. Iris Anshel page at SecureRF Corporation Archived August 20, 2008, at the Wayback Machine
19. Dr. Michael Anshel page at City College of New York Archived August 7, 2008, at the Wayback Machine
20. Anshel, Iris; Anshel, Michael; Goldfeld, Dorian An algebraic method for public-key cryptography. Math. Res. Lett. 6 (1999), no. 3-4, Anshel, Michael
21. Braid Group Cryptography page Archived February 9, 2007, at the Wayback Machine
22. List of Fellows of the American Mathematical Society, retrieved 2013-01-19.