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What are some number theoretic sequences that you know of that occur as (or are closely related to) the sequence of Fourier coefficients of some sort of automorphic function/form or the sequence of Hecke eigenvalues attached to a Hecke eigenform?

I know many such sequences, but am always looking for more.

Some examples

(1) The sequence a(n) deriving from the traces a(p) of the Frobenius elements in a Galois representation (Langlands reciprocity conjecture)

(2) Number of representations of a natural number as a sum of k squares (theta functions)

(3) The sum of powers of divisor functions (Eisenstein series)

(4) The central critical values of L-functions attached to all quadratic twists of a Hecke eigenform (Kohnen, Waldspurger)

(5) Intersection numbers of certain subvarieties of Hilbert modular surfaces (Hirzebruch-Zagier)

I'll end with a question that is ill-posed but nevertheless very interesting (at least to me personally): why do so many familiar and yet diverse sequences appear in this fashion? Note that many of them have a history of study that precedes the recognition that they are essentially coefficients of automorphic functions/forms.

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  • $\begingroup$ Theta functions aren't restricted to sums of squares. You can enumerate representations of natural numbers as the output of an arbitrary positive definite quadratic form. $\endgroup$
    – S. Carnahan ♦
    Commented Oct 28, 2009 at 6:55

2 Answers 2

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Characters of rational vertex operator algebras tend to yield modular functions. This is due to the space of torus partition functions in a chiral conformal field theory being a complex moduli invariant. The standard example is the monster vertex algebra, whose character is j-744. Other examples come from lattice CFTs (presumably describing a bosonic string propagating in a torus), and have the form of a theta function divided by a power of eta. The characters are never Hecke eigenforms, because of the pole at infinity, but traces arising from higher-weight vectors may be. In some cases, the vertex algebra structure is supposed to arise from geometry of a target space, so this phenomenon may be related to Hirzebruch-Zagier (number 5).

Characters of highest weight representations of affine Kac-Moody algebras yield modular forms. One can reasonably argue that this is a special case of the previous paragraph, since (I think) they come from Wess-Zumino-Witten. The Weyl-Kac character formula for such representations is one way to get Macdonald identities, and the smallest case (trivial rep of affine sl2) yields the Jacobi triple product.

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  • $\begingroup$ Since this question was asked such a long time ago i dont really expect an answer but if you find time can you please elaborate on "In some cases, the vertex algebra structure is supposed to arise from geometry of a target space, so this phenomenon may be related to Hirzebruch-Zagier (number 5)" Is there an example of this phenomenon? $\endgroup$
    – shehryar sikander
    Commented May 24, 2019 at 15:07
  • $\begingroup$ @shehryarsikander This was mostly a reference to the Chiral de Rham complex and Chiral Differential Operators. I do not know of a specific situation where intersection numbers yield a Hecke eigenform in this context. $\endgroup$
    – S. Carnahan ♦
    Commented May 26, 2019 at 4:25
  • $\begingroup$ Much thanks! How does one see intersection numbers in the context of Chiral de Rham complex and Chiral Differential Operators, weather or not they are eigenvalues of Hecke operators being a secondary question. $\endgroup$
    – shehryar sikander
    Commented May 26, 2019 at 18:06
  • $\begingroup$ @shehryarsikander I do not know an immediate answer, but I seem to recall there is an additional differential on the Chiral de Rham complex that yields a quasi-isomorphism with the de Rham complex. $\endgroup$
    – S. Carnahan ♦
    Commented May 28, 2019 at 5:59
  • $\begingroup$ Thanks again! Is there a reason why the q-expansion of characters of Rational VOAs have integer coeffecients? Perhaps a physical reason? Also, is it completely hopeless to play this game in genus two and hope that the characters are Siegel modular forms? $\endgroup$
    – shehryar sikander
    Commented May 28, 2019 at 15:53
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Another example: various subsequences of the integer partition function p(n) occur as coefficients of (sometimes half-integer weight) modular forms. One reference is Ahlgren and Boylan "Arithmetic properties of the partition function".

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