In the late 1960s, Robert Langlands discovered a unifying principle in number theory providing a vast generalization of class field theory to include nonabelian extensions of number fields. This principle gives rise to a web of conjectures called the Langlands program which continues to guide research in number theory to the present day. For example, an important first instance of the Langlands program is the modularity theorem for elliptic curves over the rational numbers, an essential ingredient in the proof of Fermat's last theorem.
Despite its many successes, the Langlands program remains vague in many of its predictions, due in part to an absence of data to guide a precise formulation away from a few special cases. In this thematic program, we will experiment with and articulate refined conjectures relating arithmetic-geometric objects to automorphic forms, improve the computational infrastructure underpinning the Langlands program, and assemble additional supporting data. Such data has proven valuable for researchers in number theory, and it will continue to be made available at the L-Functions and Modular Forms Database.
During the semester we will focus on three specific aspects of the Langlands program. First, we will look at elliptic curves over number fields and genus 2 curves over the rationals and will consider their relationship to modular forms. Second, we will consider computational aspects of modular forms in higher rank. Specifically, we will examine K3 surfaces and their connections to modular forms on orthogonal groups. Our third topic concerns analytic aspects of L-functions, building upon and complementing the algebraic, arithmetic, and geometric data.
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Sharon Frechette, Holly Swisher, Fang-Ting Tu, A cubic transformation formula for Appell–Lauricella hypergeometric functions over finite fields, Research in Number Theory 4 (2017), 1-27.
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Angelos Koutsianas, Computing All Elliptic Curves Over an Arbitrary Number Field with Prescribed Primes of Bad Reduction, Experimental Mathematics 28 (2019), 1 - 15.
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Edgar Costa, Andreas-Stephan Elsenhans, J{\"o}rg Jahnel, On the distribution of the Picard ranks of the reductions of a K3 surface, arXiv 1610.07823 (2016).
John Cremona, Lassina Demb'el'e, Ariel Pacetti, Ciaran Schembri, John Voight, On rational Bianchi newforms and abelian surfaces with quaternionic multiplication, arXiv 1907.12103 (2019).
Chris Jennings-Shaffer, Holly Swisher, Mock modularity of the \$M\_d\$-rank of overpartitions, arXiv 1706.00521 (2017), 1144-1189.
Taylor Dupuy, Kiran S. Kedlaya, David Roe, Christelle Vincent, Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB, arXiv 2003.05380 (2020).
Stephen Samuel Gelbart, Ralph Greenberg, Stephen D. Miller, Freydoon Shahidi, Estimates on Eisenstein Distributions for Reciprocals of p-Adic L-Functions: The Case of Irregular Primes, arXiv 1611.09757 (2017), 193-208.
Frits Beukers, Henri Cohen, Anton Mellit, Finite hypergeometric functions, arXiv 1505.02900 (2015).
Joseph H. Silverman, Divisor Divisibility Sequences on Tori, arXiv 1511.09038 (2015).
Taylor Dupuy, Kiran S. Kedlaya, David Roe, Christelle Vincent, Counterexamples to a Conjecture of Ahmadi and Shparlinski, arXiv 2003.05368 (2020).
Alex J Best, Jonathan W. Bober, Andrew R. Booker, Edgar Costa, John Cremona, Maarten Derickx, David Lowry-Duda, Min Lee, David Roe, Andrew V. Sutherland, John Voight, Computing classical modular forms, arXiv 2002.04717 (2020).
Joseph H. Silverman, Arithmetic and Dynamical Degrees on Abelian Varieties, arXiv 1501.04205 (2015).
Ralf Schmidt, Alok Shukla, On Klingen Eisenstein series with level in degree two, J. Ramanujan Math. Soc. 34 (2019) no. 4, 373-388.
Steven Nugent, John Voight, On the arithmetic dimension of triangle groups, Math. Comput. 86 (2017), 1979-2004.
Cris Poor, Ralf Schmidt, David S. Yuen, Paramodular forms of level 8 and weights 10 and 12, Int. J. Number Theory 14 (2018) no. 2, 417-467.
Winnie Li, Tong Liu, Ling Long, Potentially $\textGL_2$-type Galois representations associated to noncongruence modular forms, Transactions of the American Mathematical Society 371 (2016).
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Andrew Sutherland, Sato-Tate distributions, Contemporary Mathematics (2019), 197–248.
Matija Kazalicki, Daniela Kohen, Supersingular zeros of divisor polynomials of elliptic curves of prime conductor, Research in the Mathematical Sciences 4 (2016), 1-15.
Michel B{\"o}rner, Irene I. Bouw, Stefan Wewers, The Functional Equation for L-Functions of Hyperelliptic Curves, Experimental Mathematics 26 (2017), 396 - 411.
John Cremona, The L-Functions and Modular Forms Database Project, Foundations of Computational Mathematics 16 (2016) no. 6, 1541–1553.
Sandro Bettin, S. Gonek, The $\theta=\infty$ conjecture implies the Riemann hypothesis, Mathematika 63 (2016).
Alex Bartel, Aurel Page, Torsion homology and regulators of isospectral manifolds, Journal of Topology 9 (2016) no. 4, 1237–1256.
Kamal Khuri-Makdisi, Upper bounds for some Brill–Noether loci over a finite field, International Journal of Number Theory 14 (2018) no. 03, 739–749.
Duncan A. Buell, Gregory S. Call, Class pairings and isogenies on elliptic curves, J. Number Theory 167 (2016), 31-73.
Irene I. Bouw, Angelos Koutsianas, Jeroen Sijsling, Stefan Wewers, Conductor and discriminant of Picard curves, Journal of the London Mathematical Society 102 (2020) no. 1, 368–404.
Sandro Bettin, Chantal David, Christophe Delaunay, Non-isotrivial elliptic surfaces with non-zero average root number, arXiv 1612.03095 (2016).
Andrew R. Booker, Jeroen Sijsling, Andrew V. Sutherland, John Voight, Dan Yasaki, A database of genus-2 curves over the rational numbers, LMS Journal of Computation and Mathematics 19 (2016) no. A, 235–254.
Matija Kazalicki, Daniela Kohen, On a special case of Watkins' conjecture, arXiv 1611.05671 (2016).
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Noriyuki Otsubo, Hypergeometric functions over finite fields, 2021.
Sharon Frechette, Holly Swisher, Fang-Ting Tu, A cubic transformation formula for Appell–Lauricella hypergeometric functions over finite fields, Research in Number Theory 4 (2017), 1-27.
O. Balkanova, D. Frolenkov, A uniform asymptotic formula for the second moment of primitive L-functions on the critical line, Proceedings of the Steklov Institute of Mathematics 294 (2016), 13-46.
Ralf Schmidt, Archimedean aspects of Siegel modular forms of degree 2, Rocky Mountain J. Math. 47 (2017) no. 7, 2381-2422.
John W. Jones, David P. Roberts, Artin L-functions of small conductor, Research in Number Theory 3 (2016), 1-33.
Ghaith A. Hiary, Asymptotics and Formulas for Cubic Exponential Sums, Proceedings of the Steklov Institute of Mathematics 299 (2017), 78-95.
Alejandro Argáez-García, John Cremona, Black Box Galois representations, Journal of Algebra 512 (2018), 526–565.
Alok Shukla, Codimensions of the spaces of cusp forms for Siegel congruence subgroups in degree two, Pacific Journal of Mathematics 293 (2018), 207-244.
Irene I. Bouw, Stefan Wewers, COMPUTING L-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES, Glasgow Mathematical Journal 59 (2016), 77 - 108.
David Harvey, Maike Massierer, Andrew V. Sutherland, Computing -series of geometrically hyperelliptic curves of genus three, LMS Journal of Computation and Mathematics 19 (2016) no. A, 220–234.
Jennifer S. Balakrishnan, Sorina Ionica, Kristin Lauter, Christelle Vincent, Constructing genus-3 hyperelliptic Jacobians with CM, LMS Journal of Computation and Mathematics 19 (2016) no. A, 283–300.
Fang-Ting Tu, Yifan Yang, Evaluation of Certain Hypergeometric Functions over Finite Fields, Symmetry, Integrability and Geometry: Methods and Applications (2018).
Andrew Sutherland, Fast Jacobian arithmetic for hyperelliptic curves of genus 3, The Open Book Series 2 (2019) no. 1, 425–442.
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John W. Jones, David P. Roberts, Mixed degree number field computations, The Ramanujan Journal 47 (2016), 47-66.
Jonathan W. Bober, Ghaith A. Hiary, New Computations of the Riemann Zeta Function on the Critical Line, Experimental Mathematics 27 (2018), 125 - 137.
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O. Balkanova, D. Frolenkov, Non-vanishing of automorphic L-functions of prime power level, Monatshefte f{\"u}r Mathematik 185 (2016), 17-41.
D. Kohen, Ariel Pacetti, On Heegner points for primes of additive reduction ramifying in the base field, Transactions of the American Mathematical Society 370 (2015), 911-926.
Christelle Vincent, A characterization of the U(Omega,m) sets of a hyperelliptic curve as Omega and m vary, arXiv 1903.08730 (2019).
Steve Donnelly, John Voight, A database of Hilbert modular forms, arXiv 1605.02637 (2016).
Ghaith A. Hiary, A fast amortized algorithm for computing quadratic Dirichlet \$L\$-functions, arXiv 1605.02637 (2016).
Sandro Bettin, Hung M. Bui, Xiannan Li, Maksym Radziwill, A quadratic divisor problem and moments of the Riemann zeta-function, arXiv 1609.02539 (2016).
Ghaith A. Hiary, An explicit hybrid estimate for \$L(1/2+it,\chi)\$, arXiv 1510.00950 (2015).
