Algebraic modular forms

These were developed by Gross in the late 1990s. They are similar to automorphic forms, but they are defined on reductive groups for which the relevant double coset spaces are finite sets, and hence have a less analytic and more combinatorial flavour. Nevertheless, they are conjectured to retain sufficient arithmetic information that there are Galois representations attached to simple Hecke submodules. I think there is a lot of interesting stuff to explore here, in terms of theory, computations, and applications/related topics.

References

  1. Foundational
    1. Gross, Algebraic modular forms, Israel Journal of Mathematics 113 (1999)
    2. Gross, Modular forms (mod p) and Galois representations, IMRN (1998)
    3. Gross, Modular Galois representations, unpublished (1996)
  2. General
    1. Lansky--Pollack, Hecke algebras and automorphic forms, Compositio Math 130 (2002)
    2. Gross, Algebraic modular forms, notes from talk at University of Melbourne (2009)
    3. Gross, Algebraic modular forms mod p, notes from talk in Montreal (1998)
    4. Gross, Letter to Serre (1995)
    5. Gross, Odd Galois representations
    6. Dummigan, A simple trace formula for algebraic modular forms, Exp. Math. 22 (2013)
  3. Relations to usual modular forms
    1. Serre, Two letters on quaternions and modular forms, Israel Journal of Mathematics 95 (1996)
    2. van Hoften, A geometric Jacquet--Langlands correspondence for paramodular Siegel threefolds, Math. Z 299 (2021)
  4. Computational issues
    1. Loeffler, Computing with algebraic automorphic forms (2011)
    2. Greenberg--Voight, Lattice methods for algebraic modular forms on classical groups (2011)
    3. Dembele, Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms, Math. Comp. 76 (2007)
    4. Loeffler, Explicit calculations of automorphic forms for definite unitary groups, LMS J. Comput. Math. 11 (2008)
    5. Cunningham--Dembele, Computing genus-2 Hilbert-Siegel modular forms over Q(sqrt(5)) via the Jacquet-Langlands correspondence, Experiment. Math. 18 (2009)
    6. Dembele, On the computation of algebraic modular forms on compact inner forms of GSp4, Math. Comp. 83 (2014)
  5. Constrained ramification (prompted by 1998 IMRN paper by Gross)
    1. Brueggeman, Septic number fields which are ramified only at one small prime, J. Symbolic Comput. 31 (2001)
    2. Lesseni, Nonsolvable nonic number fields ramified only at one small prime, JTNB 18 (2006)
    3. Dembele, A non-solvable Galois extension of Q ramified at 2 only, C. R. Math. Acad. Sci. Paris 347 (2009)
    4. Roberts, Nonsolvable polynomials with field discriminant 5 to the power A, Int. J. Number Theory 7 (2011)
    5. Dembele--Greenberg--Voight, Nonsolvable number fields ramified only at 3 and 5, Compos. Math. 147 (2011)
    6. Dieulefait, A non-solvable extension of Q unramified outside 7, Compos. Math. 148 (2012)
    7. Pollak, Ramification in the inverse Galois problem, JNT 220 (2021)