Timeline for Number theoretic sequences and Hecke eigenvalues
Current License: CC BY-SA 2.5
14 events
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| May 30, 2019 at 14:47 | comment | added | shehryar sikander | Namely, the target space is an abelian surface(T^4) such that its endomorphism ring is isomorphic to the ring of integers of a totally real quadratic field K. The moduli space of such surfaces is precisely the hilbert modular surface of K and indeed intersection numbers in this moduli space are coeffecients of the characters Are there explicit examples of character of Rational VOAs which are Eisenstein series of weight two and level a prime which is congruent to 1 mod 4(This is the condition in HZ)? Then one can try to work out the geometric interpretation in this example. | |
| May 30, 2019 at 14:47 | comment | added | shehryar sikander | Theorem 1 in Hirzebruch Zagier constructs weight two Eisenstein series and weight two cusp forms with coeffecients which are intersection numbers of T_N with T_0 in first case T_N with a curve orthognal to T_0 in the second case. The level of these functions is the dicriminant of the totally real field defining the hilbert modular surface in which T_N reside. If one of these Eisenstein series is the character of a Rational VOA, then indeed one can say that they are arising from the geometry of the moduli space of the target space. | |
| May 30, 2019 at 12:46 | comment | added | S. Carnahan♦ | You can form weight 2 modular forms using characters of vectors of weight 2, but I don't know any cuspidality criteria. | |
| May 30, 2019 at 4:08 | comment | added | shehryar sikander | Are the characters of Rational VOAs ever cusp forms of weight two? In your answer they are always of weight zero(modular functions). In Hirzebruch-Zagier the intersection numbers correspond to coeffecients of cusp forms of weight two. | |
| May 29, 2019 at 2:02 | comment | added | S. Carnahan♦ | @shehryarsikander For your first question, this is because the q-expansion coefficients are dimensions of vector spaces. People have tried in genus 2 (see papers of Mason and Tuite) and there seem to be some strange discrepancies. | |
| May 28, 2019 at 15:53 | comment | added | shehryar sikander | 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? | |
| May 28, 2019 at 5:59 | comment | added | S. Carnahan♦ | @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. | |
| May 26, 2019 at 18:06 | comment | added | shehryar sikander | 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. | |
| May 26, 2019 at 4:25 | comment | added | S. Carnahan♦ | @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. | |
| May 24, 2019 at 15:07 | comment | added | shehryar sikander | 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? | |
| Oct 31, 2009 at 17:30 | vote | accept | Jonah Sinick | ||
| Oct 31, 2009 at 17:30 | vote | accept | Jonah Sinick | ||
| Oct 31, 2009 at 17:30 | |||||
| Oct 28, 2009 at 7:33 | history | edited | S. Carnahan♦ | CC BY-SA 2.5 |
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| Oct 28, 2009 at 7:24 | history | answered | S. Carnahan♦ | CC BY-SA 2.5 |