Timeline for Modular forms from counting points on algebraic varieties over a finite field
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
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| Jan 22, 2017 at 11:08 | comment | added | Daniel Loughran | There a typo in my above comment regarding K3 surfaces; $H^3$ should be $H^2$. | |
| Jan 22, 2017 at 10:52 | vote | accept | Bruce Bartlett | ||
| Jan 22, 2017 at 10:18 | answer | added | David Loeffler | timeline score: 22 | |
| Jan 22, 2017 at 2:59 | history | edited | Michael Hardy | CC BY-SA 3.0 |
improved MathJax usage, including \tag{} and display
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| Jan 22, 2017 at 0:27 | history | edited | GH from MO |
edited tags
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| Jan 21, 2017 at 21:57 | history | edited | Bruce Bartlett | CC BY-SA 3.0 |
added 98 characters in body
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| Jan 21, 2017 at 21:01 | answer | added | Bruce Bartlett | timeline score: 2 | |
| Jan 21, 2017 at 17:42 | comment | added | Bruce Bartlett | Yes, I see my error, thanks for setting me straight. | |
| Jan 21, 2017 at 17:02 | comment | added | David Loeffler | ... [cont'd] This has dimension 4, but motivic weight 1, while the Galois representation associated to a weight k modular form with $k > 2$ has dimension 2 and motivic weight k-1, so there is no way to make these match up. | |
| Jan 21, 2017 at 17:02 | comment | added | David Loeffler | @BruceBartlett You seem very confident about your viewpoint, but I am afraid it is incorrect, as Daniel has pointed out; you are muddling together the dimension of the Galois representation and the dimension of the algebraic variety, which are totally unrelated. Take e.g. a nonsingular curve of genus 2. Then the $b_p$'s are (up to a small discrepancy coming from how you handle points at infinity) the trace of Frobenius at p on a 4-dimensional Galois representation, occuring as the etale H^1 of the curve, which is dual to the Tate module of its Jacobian. ... | |
| Jan 21, 2017 at 16:40 | comment | added | Bruce Bartlett | Yes - this is consistent with what I said, that the dimension of the variety will correspond roughly to the weight of the form. I agree that "higher dimensional Galois representations (conjecturally) correspond to automorphic forms on higher dimensional algebraic groups", but they also correspond (i.e. come from, on occasion) to higher-dimensional varieties. They correspond to more than one thing. | |
| Jan 21, 2017 at 16:02 | comment | added | Daniel Loughran | The weight corresponds to which degree of the cohomology it lives it. For example, singular K3 surfaces have a two dimensional piece of $H^3$ which corresponds to a modular form of weight 3. See arxiv.org/pdf/0809.0830.pdf | |
| Jan 21, 2017 at 15:59 | comment | added | Daniel Loughran | No. Higher dimensional Galois representations (conjecturally) correspond to automorphic forms on higher dimensional algebraic groups. | |
| Jan 21, 2017 at 15:35 | comment | added | Bruce Bartlett | Hi Daniel - thanks for your comments. You say "Modular forms roughly correspond to 2-dimensional Galois representations." Wouldn't it be more correct to say that "Modular forms of weight 2 roughly correspond to 2-dimensional Galois repressentations". As the dimension of the variety goes up, one expects the weight of the associated modular form to go up. | |
| Jan 21, 2017 at 14:51 | comment | added | Daniel Loughran | In fact, for a smooth projective curve $X$ over $\mathbb{Q}$, it seems that the trace of Frobenius $a_p$ should be the coefficients modular form if and only if $X$ has genus $1$. | |
| Jan 21, 2017 at 14:49 | comment | added | Daniel Loughran | It would help with your understanding if you read up on the Weil conjectures (this theory works best for smooth projective varieties). These relate the number of $\mathbb{F}_p$-points to the Galois representation on the etale cohomology groups. Modular forms roughly correspond to $2$-dimensional Galois representations, so one would only expect modular forms to occur if one of the pieces of the etale cohomology is $2$-dimensional, or has a $2$-dimensional subspace. This is very uncommon; as pointed out by Chris, most curves of higher genus will not have this property. | |
| Jan 21, 2017 at 13:31 | comment | added | Bruce Bartlett | To your first comment: For genus 1 (elliptic curves), the naive way I formulated it is precisely the way that Diamond and Shurman formulated on the second page of the Preface of their book, A First Course in Modular Forms. Is it not correct as it stands? For safety I have added a bad primes proviso. | |
| Jan 21, 2017 at 13:29 | history | edited | Bruce Bartlett | CC BY-SA 3.0 |
Took into account bad primes
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| Jan 21, 2017 at 11:33 | comment | added | Chris Wuthrich | First in your rather naive version, modularity holds if the curve is of genus 1 and has exactly one point on the line at infinity. You should better formulate it projectively. For curves of higher genus, the form that you are after must be more complicated as even the zeta-functions of the curve over $\mathbb{F}_p$ needs more information than just the $a_p$. | |
| Jan 21, 2017 at 11:31 | history | edited | Bruce Bartlett | CC BY-SA 3.0 |
added 38 characters in body
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| Jan 21, 2017 at 11:22 | history | asked | Bruce Bartlett | CC BY-SA 3.0 |