Frank Thorne
University of South Carolina |

Math 580, Elementary Number Theory.

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The Mathematics of Game Shows (with course notes -- feel free to use!)

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.

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

Number Theory, on Thursday mornings.

The colloquium.

Graduate students are also encouraged to attend the graduate colloquium.

See here for a list of other seminars run at USC.

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.

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.

15.
Dirichlet series associated to quartic fields with given resolvent, with Henri Cohen.

*
Research in Number Theory*
**2** (2016), Art. 29, 40 pp.

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.

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.

*
Submitted.*
Solicited for inclusion in the Proceedings of the 2014 Summer School
on Counting Arithmetic Objects, Montreal.

20.
Orbital exponential sums for prehomogeneous vector spaces,
with Takashi Taniguchi.

*
American Journal of Mathematics*.

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,* to appear.

23.
Levels of distribution for sieve problems in prehomogeneous vector spaces,
with Takashi Taniguchi.

*
Mathematische Annalen,* to appear.

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

*
Submitted.*

25.
Uniform bounds for lattice point counting and partial sums of zeta functions,
with David Lowry-Duda and Takashi Taniguchi.

*
Submitted.*

26.
Rank growth of elliptic curves in nonabelian extensions,
with Robert Lemke Oliver.

*
International Mathematics Research Notices,* to appear.

27.
Upper bounds on polynomials with small Galois group,
with Robert Lemke Oliver.

Mathematika,*
to appear.*

28.
Improved lower bounds for the number of fields with alternating Galois group,
with Aaron Landesman and Robert Lemke Oliver.

*
Submitted.*

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.

*
Submitted.*

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.

*
To be submitted shortly.*

Several more in preparation.

International Conference on Number Theory and Discrete Mathematics, Kochi, India, December 11-14, 2020.