Frank Thorne
University of South Carolina |

Math 788 (Elliptic Curves and Arithmetic Geometry), Spring 2016.

Reading course on analytic number theory, Spring 2016.

Previously, in reverse chronological order: (with complete notes in some cases!)

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.

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, to appear.*

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.*

18.
The number of ramified primes in number fields of small degree,
with Robert Lemke Oliver.

*
Submitted.*

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.

*
Submitted.*

Several more in preparation.

South Carolina State High School Math Contest, USC, January 30, 2016.

Automorphic Forms Workshop, Wake Forest University, March 7-10, 2016.

Arizona Winter School, University of Arizona, March 11-16, 2016.

University of Michigan, April 6-11, 2016.

Oberlin College, April 12-13, 2016.

Stanford University, May 9, 2016.

Chico State University, May 2016.

Arithmetic Statistics and the Cohen-Lenstra Heuristics, University of Warwick, June 27-July 1, 2016. (Tentative.)

Algebraic Number Theory and Related Topics, Congreso Latinoamericano de Matematicos, Barranquilla, Colombia, July 11-15, 2016.

Semester in analytic number theory, MSRI, Berkeley, CA, January-May 2017. (Tentative.)