Frank Thorne

University of South Carolina
Department of Mathematics
447 LeConte College
1523 Greene Street
Columbia, SC 29201
thorne [at] math [dot] sc [dot] edu

Current Teaching (Spring 2023)

Math 141 , Calculus I.

Math 580 , Elementary Number Theory.

Math 552 (Complex Anal.), Math 788 (Ell. curves), Math 544H (Honors Lin Alg), Math 750 (Fourier Analysis), Math 532 (Modern Geometry), Math 550 (Vector), Math 788 (Ell. curves), Math 142 (Calc II), Math 374 (Discrete for CS), Math 701-2 (Alg I/II), Math 141 (Calc I), Math 735 (Lie Groups), Math 788 (Ell. curves), Math 141 (Calc I), Math 544 (Lin Alg), Math 547/702I (Alg II), Math 374 (Discrete for CS), Math 546 (Alg I), Math 142H, (Calc II), Math 788G (GON), Math 580 (Intro NT), Math 142 (Calc II), Math 788P (Alg NT), Math 531 (Geometry), Math 141 (Calc I), Math 574 (Discrete), Math 141 (Calc I), Math 782 (Anal NT).

I work in number theory. Some of my particular interests:

Classical analytic number theory. My first task in graduate school was to read all of Davenport. I am interested in sieve methods, L-functions, and related topics, and their classical applications -- as well as applications beyond their traditional settings.
The distribution of number fields, class group torsion, and related topics. I am especially interested in these from an analytic point of view involving Shintani zeta functions associated to prehomogeneous vector spaces. I am also interested in other perspectives on these questions: class field theory and Kummer theory; the Scholz reflection principle; algebro-geometric perspectives on these topics; and other related perspectives.

Current Graduate Students: Mohamed Wafik ElShiekh, Aditya Iyer, and Demmas Salim.

Algebraic Geometry, Arithmetic Geometry, and Commutative Algebra Seminar, on Friday afternoons.

Publications and Preprints:

1. Bounded gaps between products of primes with applications to elliptic curves and ideal class groups.
International Mathematics Research Notices (2008), 41 pp.

2. Irregularities in the distributions of primes in function fields.
Journal of Number Theory 128 (2008), 1784-1794.

3. Bubbles of congruent primes.
Mathematical Proceedings of the Cambridge Philosophical Society 157 (2014), no. 3, 443–456.

4. An uncertainty principle for function fields.
Journal of Number Theory 131 (2011), 1363-1389.

5. Maier matrices beyond Z.
Proceedings of the Integers Conference 2007.

6. Analytic properties of Shintani zeta functions.
Proceedings of the RIMS Symposium on automorphic forms, automorphic representations, and related topics (Kyoto, 2010).

7. The secondary term in the counting function for cubic fields, with Takashi Taniguchi.
Duke Mathematical Journal 162 (2013), no. 13, 2451-2508.

8. Shintani's zeta function is not a finite sum of Euler products.
Proceedings of the American Mathematical Society, 142 (2014), no. 6, 1943-195 2.

9. Orbital L-functions for the space of binary cubic forms, with Takashi Taniguchi.
Canadian Journal of Mathematics 65 (2013), no. 6, 1320-1383.

10. Four perspectives on secondary terms in the Davenport-Heilbronn theorems.
Integers Volume 12B, Proceedings of the Integers Conference 2011.

11. An error estimate for counting S_3-sextic number fields, with Takashi Taniguchi.
International Journal of Number Theory 10 (2014), no. 4, 935-948.

Supplementary files: CountS3.java, sextic-compare.gp, and an old note with some supplementary numerical data.

More supplementary files: css.gp, css-5.gp, and ParseBelabas.java.

12. Book review of Opera de Cribro and An introduction to sieve methods and their applications.
Bulletin of the American Mathematical Society 50 (2013), no. 2, 359-366.

13. On the existence of large degree Galois representations for fields of small discriminant, with Jeremy Rouse.
Pacific Journal of Mathematics 271 (2014), no. 1, 243-256.

14. Dirichlet series associated to cubic fields with given quadratic resolvent,with Henri Cohen.
Michigan Math Journal 63 (2014), no. 2, 253-273.

See below [11] for source code; also: a short list of cubic fields generated by Karim Belabas's program.

15. Dirichlet series associated to quartic fields with given resolvent, with Henri Cohen.
Research in Number Theory 2 (2016), Art. 29, 40 pp.

Appendix: Complete proof of Theorem 5.1, whose proof was only sketched in the paper.

16. Zeros of L-functions outside the critical strip, with Andrew Booker.
Algebra and Number Theory 8 (2014), no. 9, 2027-2042.

17. Identities for field extensions generalizing the Ohno-Nakagawa relations, with Henri Cohen and Simon Rubinstein-Salzedo.
Compositio Mathematica 151 (2015), no. 11, 2059-2075.

Supplementary file: nakagawa.gp.

18. The number of ramified primes in number fields of small degree, with Robert Lemke Oliver.
Proceedings of the American Mathematical Society 145 (2017), no. 8, 3201–3210.

19. Analytic number theory and zeta function methods.
This was submitted for inclusion in the Proceedings of the 2014 Summer School on Counting Arithmetic Objects, Montreal, but these proceedings seem unlikely to actually be published.

20. Orbital exponential sums for prehomogeneous vector spaces, with Takashi Taniguchi.
American Journal of Mathematics 142 (2020), no. 1, 177–213.

Computer code: bcf.gp, bqf.gp, tqf.gp, pbqf.gp, ptqf.gp,

21. On D_l-extensions of odd prime degree l, with Henri Cohen.
Proceedings of the London Math Society (3) 121 (2020), no. 5, 1171-1206.

22. Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves.
With Manjul Bhargava, Arul Shankar, Takashi Taniguchi, Jacob Tsimerman, and Yongqiang Zhao.
Journal of the American Mathematical Society 33 (2020), no. 4, 1087–1099.

23. Levels of distribution for sieve problems in prehomogeneous vector spaces, with Takashi Taniguchi.
Mathematische Annalen 376 (2020), no. 3-4, 1537-1559.

24. The distribution of G-Weyl CM fields and the Colmez conjecture, with Adrian Barquero-Sanchez and Riad Masri.
Submitted.

Computer code: g-weyl-residue.gp.

25. Uniform bounds for lattice point counting and partial sums of zeta functions, with David Lowry-Duda and Takashi Taniguchi.
Mathematische Zeitschrift 300 (2022), no. 3, 2571-2590.

26. Rank growth of elliptic curves in nonabelian extensions, with Robert Lemke Oliver.
International Mathematics Research Notices (2021), no. 24, 18411-18441.

27. Upper bounds on polynomials with small Galois group, with Robert Lemke Oliver.
Mathematika 66 (2020), no. 4, 1054–1059.

28. Improved lower bounds for the number of fields with alternating Galois group, with Aaron Landesman and Robert Lemke Oliver.
Bulletin of the London Mathematical Society 53 (2021), no. 4, 1159–1173.

29. Malle's Conjecture for G×A, with G=S_3,S_4,S_5, with Riad Masri, Wei-Lun Tsai, and Jiuya Wang.
Submitted.

30. Upper bounds on number fields of given degree and bounded discriminant, with Robert J. Lemke Oliver.
Duke Mathematical Journal 171 (2022), no. 15, 3077-3087.

31. Counting quintic fields with genus number one, with Kevin J. McGown and Amanda Tucker.
Submitted.

32. Asymptotic identities for additive convolutions of sums of divisors, with Robert J. Lemke Oliver and Sunrose Shrestha.
Mathematical Proceedings of the Cambridge Philosophical Society 174 (2023), no. 1, 59-78.

33. Improved error estimates for the Davenport-Heilbronn theorems, with Manjul Bhargava and Takashi Taniguchi.
Submitted.

34. Improved bounds on number fields of small degree.
With Theresa C. Anderson, Ayla Gafni, Kevin Hughes, Robert J. Lemke Oliver, David Lowry-Duda, Jiuya Wang, and Ruixiang Zhang.
Submitted.

35. On the asymptotics of cubic fields ordered by general invariants , with Arul Shankar.
Submitted.

Computer code: cm-test.gp, cubic-count.gp, rcf-1500k.gp, icf-1000k.gp.

36. What is the height of two points in the plane?, with Jesse Kass.
Submitted.

Several more in preparation.