Modern Algebraic Geometry and Analytic Number TheoryModern algebraic geometry vs. classical algebraic...
Modern Algebraic Geometry and Analytic Number Theory
Modern algebraic geometry vs. classical algebraic geometryAnalytic tools in algebraic geometry Stacks in modern number theory/arithmetic geometryAsymptotic formula in Analytic Number TheorySpinoffs of analytic number theoryIntroductions to modern algebraic geometryHow much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?Algebraic Geometry in Number TheoryComplex analytic vs algebraic geometryMotivation behind Analytic Number Theory
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I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (with the exception of probabilities). That is to say, to understand it, you really need to know everything. It also has extraordinary opportunities in the understanding of arithmetic (Pierre Deligne in the proofs of André Weil etc.).
However, I don't see any connection with the analytic number theory like the one undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on.
Does anyone have ideas of theorems, conjectures, or "approaches" that
combine these two points of view?
ag.algebraic-geometry at.algebraic-topology analytic-number-theory dirichlet-series
$endgroup$
add a comment |
$begingroup$
I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (with the exception of probabilities). That is to say, to understand it, you really need to know everything. It also has extraordinary opportunities in the understanding of arithmetic (Pierre Deligne in the proofs of André Weil etc.).
However, I don't see any connection with the analytic number theory like the one undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on.
Does anyone have ideas of theorems, conjectures, or "approaches" that
combine these two points of view?
ag.algebraic-geometry at.algebraic-topology analytic-number-theory dirichlet-series
$endgroup$
21
$begingroup$
I like the question at the end of your post, but I think that your claim "encompass all modern mathematics" is quite exaggerated.
$endgroup$
– EFinat-S
yesterday
1
$begingroup$
Drinfeld's "Finitely additive measures on S2 and S3, invariant with respect to rotations" solves a measure-theoretic problem using automorphic forms and a consequence of Weil conjectures (although the latter is not crucial to the argument, one can do with a weaker version which can be proved without algebraic geometry)
$endgroup$
– Aknazar Kazhymurat
23 hours ago
$begingroup$
If I remember rightly, one of Weil's original uses for the Riemann hypothesis for curves over finite fields was applications to bounding exponential sums. Grothendieck invented $ell$-adic cohomology to prove the Weil conjectures (including the Riemann hypothesis for varieties that was eventually proved by Deligne), so it's not totally surprising that there is a connection to exponential sums.
$endgroup$
– Robert Furber
16 hours ago
2
$begingroup$
The question would benefit from changing the speculative "I have the impression... everything." to something saying that it is connected with many subfields of maths, possibly with examples.
$endgroup$
– YCor
15 hours ago
add a comment |
$begingroup$
I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (with the exception of probabilities). That is to say, to understand it, you really need to know everything. It also has extraordinary opportunities in the understanding of arithmetic (Pierre Deligne in the proofs of André Weil etc.).
However, I don't see any connection with the analytic number theory like the one undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on.
Does anyone have ideas of theorems, conjectures, or "approaches" that
combine these two points of view?
ag.algebraic-geometry at.algebraic-topology analytic-number-theory dirichlet-series
$endgroup$
I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (with the exception of probabilities). That is to say, to understand it, you really need to know everything. It also has extraordinary opportunities in the understanding of arithmetic (Pierre Deligne in the proofs of André Weil etc.).
However, I don't see any connection with the analytic number theory like the one undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on.
Does anyone have ideas of theorems, conjectures, or "approaches" that
combine these two points of view?
ag.algebraic-geometry at.algebraic-topology analytic-number-theory dirichlet-series
ag.algebraic-geometry at.algebraic-topology analytic-number-theory dirichlet-series
edited yesterday
community wiki
lulu2612
21
$begingroup$
I like the question at the end of your post, but I think that your claim "encompass all modern mathematics" is quite exaggerated.
$endgroup$
– EFinat-S
yesterday
1
$begingroup$
Drinfeld's "Finitely additive measures on S2 and S3, invariant with respect to rotations" solves a measure-theoretic problem using automorphic forms and a consequence of Weil conjectures (although the latter is not crucial to the argument, one can do with a weaker version which can be proved without algebraic geometry)
$endgroup$
– Aknazar Kazhymurat
23 hours ago
$begingroup$
If I remember rightly, one of Weil's original uses for the Riemann hypothesis for curves over finite fields was applications to bounding exponential sums. Grothendieck invented $ell$-adic cohomology to prove the Weil conjectures (including the Riemann hypothesis for varieties that was eventually proved by Deligne), so it's not totally surprising that there is a connection to exponential sums.
$endgroup$
– Robert Furber
16 hours ago
2
$begingroup$
The question would benefit from changing the speculative "I have the impression... everything." to something saying that it is connected with many subfields of maths, possibly with examples.
$endgroup$
– YCor
15 hours ago
add a comment |
21
$begingroup$
I like the question at the end of your post, but I think that your claim "encompass all modern mathematics" is quite exaggerated.
$endgroup$
– EFinat-S
yesterday
1
$begingroup$
Drinfeld's "Finitely additive measures on S2 and S3, invariant with respect to rotations" solves a measure-theoretic problem using automorphic forms and a consequence of Weil conjectures (although the latter is not crucial to the argument, one can do with a weaker version which can be proved without algebraic geometry)
$endgroup$
– Aknazar Kazhymurat
23 hours ago
$begingroup$
If I remember rightly, one of Weil's original uses for the Riemann hypothesis for curves over finite fields was applications to bounding exponential sums. Grothendieck invented $ell$-adic cohomology to prove the Weil conjectures (including the Riemann hypothesis for varieties that was eventually proved by Deligne), so it's not totally surprising that there is a connection to exponential sums.
$endgroup$
– Robert Furber
16 hours ago
2
$begingroup$
The question would benefit from changing the speculative "I have the impression... everything." to something saying that it is connected with many subfields of maths, possibly with examples.
$endgroup$
– YCor
15 hours ago
21
21
$begingroup$
I like the question at the end of your post, but I think that your claim "encompass all modern mathematics" is quite exaggerated.
$endgroup$
– EFinat-S
yesterday
$begingroup$
I like the question at the end of your post, but I think that your claim "encompass all modern mathematics" is quite exaggerated.
$endgroup$
– EFinat-S
yesterday
1
1
$begingroup$
Drinfeld's "Finitely additive measures on S2 and S3, invariant with respect to rotations" solves a measure-theoretic problem using automorphic forms and a consequence of Weil conjectures (although the latter is not crucial to the argument, one can do with a weaker version which can be proved without algebraic geometry)
$endgroup$
– Aknazar Kazhymurat
23 hours ago
$begingroup$
Drinfeld's "Finitely additive measures on S2 and S3, invariant with respect to rotations" solves a measure-theoretic problem using automorphic forms and a consequence of Weil conjectures (although the latter is not crucial to the argument, one can do with a weaker version which can be proved without algebraic geometry)
$endgroup$
– Aknazar Kazhymurat
23 hours ago
$begingroup$
If I remember rightly, one of Weil's original uses for the Riemann hypothesis for curves over finite fields was applications to bounding exponential sums. Grothendieck invented $ell$-adic cohomology to prove the Weil conjectures (including the Riemann hypothesis for varieties that was eventually proved by Deligne), so it's not totally surprising that there is a connection to exponential sums.
$endgroup$
– Robert Furber
16 hours ago
$begingroup$
If I remember rightly, one of Weil's original uses for the Riemann hypothesis for curves over finite fields was applications to bounding exponential sums. Grothendieck invented $ell$-adic cohomology to prove the Weil conjectures (including the Riemann hypothesis for varieties that was eventually proved by Deligne), so it's not totally surprising that there is a connection to exponential sums.
$endgroup$
– Robert Furber
16 hours ago
2
2
$begingroup$
The question would benefit from changing the speculative "I have the impression... everything." to something saying that it is connected with many subfields of maths, possibly with examples.
$endgroup$
– YCor
15 hours ago
$begingroup$
The question would benefit from changing the speculative "I have the impression... everything." to something saying that it is connected with many subfields of maths, possibly with examples.
$endgroup$
– YCor
15 hours ago
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
There are lots of examples, so let me just tell one.
P. Deligne (1971) used Eichler–Shimura isomorphism to reduce the Ramanujan conjecture on the $tau$ function to the Weil conjectures, that he later proved by using the full strength of Grothendieck's machinery.
$endgroup$
8
$begingroup$
I don't know if this is tacit in your answer, but Deligne's proof of the Weil conjectures used, in addition to the Grothendieck machinery, the method of de la Vallee-Poussin from analytic number theory.
$endgroup$
– aginensky
21 hours ago
add a comment |
$begingroup$
Do you consider $L$-functions of elliptic curves over $mathbf Q$ (or other number fields) to be in the spirit of "analytic number theory undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on"? Those 19th and early 20th century folks did not have the definition, which only came much later in the 20th century, but the idea of defining such functions as an Euler product and then Dirichlet series, and seeking an analytic continuation and functional equation, is a task they would have understood. Deuring proved the analytic continuation and functional equation in a special case (CM elliptic curves) in the 1950s, but the case of all elliptic curves over $mathbf Q$ was settled using ideas coming from the proof of Fermat's Last Theorem, hence using modern algebraic geometry.
The Sato-Tate conjecture is an analytic conjecture somewhat in the spirit of the prime number theorem. It was formulated in the 2nd half of the 20th century but could have been appreciated earlier. Like the prime number theorem, which is equivalent to nonvanishing of the zeta-function on the line ${rm Re}(s) = 1$, the Sato-Tate conjecture was known to be a consequence of analyticity and nonvanishing of certain $L$-functions on vertical lines (boundary of right half-planes) and those $L$-function properties were proved about 10 years ago with algebro-geometric methods.
$endgroup$
add a comment |
$begingroup$
You can look at Lectures on applied $ell$-adic cohomology by Fouvry, Kowalski, Michel and Sawin : https://arxiv.org/abs/1712.03173
$endgroup$
add a comment |
Your Answer
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
There are lots of examples, so let me just tell one.
P. Deligne (1971) used Eichler–Shimura isomorphism to reduce the Ramanujan conjecture on the $tau$ function to the Weil conjectures, that he later proved by using the full strength of Grothendieck's machinery.
$endgroup$
8
$begingroup$
I don't know if this is tacit in your answer, but Deligne's proof of the Weil conjectures used, in addition to the Grothendieck machinery, the method of de la Vallee-Poussin from analytic number theory.
$endgroup$
– aginensky
21 hours ago
add a comment |
$begingroup$
There are lots of examples, so let me just tell one.
P. Deligne (1971) used Eichler–Shimura isomorphism to reduce the Ramanujan conjecture on the $tau$ function to the Weil conjectures, that he later proved by using the full strength of Grothendieck's machinery.
$endgroup$
8
$begingroup$
I don't know if this is tacit in your answer, but Deligne's proof of the Weil conjectures used, in addition to the Grothendieck machinery, the method of de la Vallee-Poussin from analytic number theory.
$endgroup$
– aginensky
21 hours ago
add a comment |
$begingroup$
There are lots of examples, so let me just tell one.
P. Deligne (1971) used Eichler–Shimura isomorphism to reduce the Ramanujan conjecture on the $tau$ function to the Weil conjectures, that he later proved by using the full strength of Grothendieck's machinery.
$endgroup$
There are lots of examples, so let me just tell one.
P. Deligne (1971) used Eichler–Shimura isomorphism to reduce the Ramanujan conjecture on the $tau$ function to the Weil conjectures, that he later proved by using the full strength of Grothendieck's machinery.
edited yesterday
community wiki
Francesco Polizzi
8
$begingroup$
I don't know if this is tacit in your answer, but Deligne's proof of the Weil conjectures used, in addition to the Grothendieck machinery, the method of de la Vallee-Poussin from analytic number theory.
$endgroup$
– aginensky
21 hours ago
add a comment |
8
$begingroup$
I don't know if this is tacit in your answer, but Deligne's proof of the Weil conjectures used, in addition to the Grothendieck machinery, the method of de la Vallee-Poussin from analytic number theory.
$endgroup$
– aginensky
21 hours ago
8
8
$begingroup$
I don't know if this is tacit in your answer, but Deligne's proof of the Weil conjectures used, in addition to the Grothendieck machinery, the method of de la Vallee-Poussin from analytic number theory.
$endgroup$
– aginensky
21 hours ago
$begingroup$
I don't know if this is tacit in your answer, but Deligne's proof of the Weil conjectures used, in addition to the Grothendieck machinery, the method of de la Vallee-Poussin from analytic number theory.
$endgroup$
– aginensky
21 hours ago
add a comment |
$begingroup$
Do you consider $L$-functions of elliptic curves over $mathbf Q$ (or other number fields) to be in the spirit of "analytic number theory undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on"? Those 19th and early 20th century folks did not have the definition, which only came much later in the 20th century, but the idea of defining such functions as an Euler product and then Dirichlet series, and seeking an analytic continuation and functional equation, is a task they would have understood. Deuring proved the analytic continuation and functional equation in a special case (CM elliptic curves) in the 1950s, but the case of all elliptic curves over $mathbf Q$ was settled using ideas coming from the proof of Fermat's Last Theorem, hence using modern algebraic geometry.
The Sato-Tate conjecture is an analytic conjecture somewhat in the spirit of the prime number theorem. It was formulated in the 2nd half of the 20th century but could have been appreciated earlier. Like the prime number theorem, which is equivalent to nonvanishing of the zeta-function on the line ${rm Re}(s) = 1$, the Sato-Tate conjecture was known to be a consequence of analyticity and nonvanishing of certain $L$-functions on vertical lines (boundary of right half-planes) and those $L$-function properties were proved about 10 years ago with algebro-geometric methods.
$endgroup$
add a comment |
$begingroup$
Do you consider $L$-functions of elliptic curves over $mathbf Q$ (or other number fields) to be in the spirit of "analytic number theory undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on"? Those 19th and early 20th century folks did not have the definition, which only came much later in the 20th century, but the idea of defining such functions as an Euler product and then Dirichlet series, and seeking an analytic continuation and functional equation, is a task they would have understood. Deuring proved the analytic continuation and functional equation in a special case (CM elliptic curves) in the 1950s, but the case of all elliptic curves over $mathbf Q$ was settled using ideas coming from the proof of Fermat's Last Theorem, hence using modern algebraic geometry.
The Sato-Tate conjecture is an analytic conjecture somewhat in the spirit of the prime number theorem. It was formulated in the 2nd half of the 20th century but could have been appreciated earlier. Like the prime number theorem, which is equivalent to nonvanishing of the zeta-function on the line ${rm Re}(s) = 1$, the Sato-Tate conjecture was known to be a consequence of analyticity and nonvanishing of certain $L$-functions on vertical lines (boundary of right half-planes) and those $L$-function properties were proved about 10 years ago with algebro-geometric methods.
$endgroup$
add a comment |
$begingroup$
Do you consider $L$-functions of elliptic curves over $mathbf Q$ (or other number fields) to be in the spirit of "analytic number theory undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on"? Those 19th and early 20th century folks did not have the definition, which only came much later in the 20th century, but the idea of defining such functions as an Euler product and then Dirichlet series, and seeking an analytic continuation and functional equation, is a task they would have understood. Deuring proved the analytic continuation and functional equation in a special case (CM elliptic curves) in the 1950s, but the case of all elliptic curves over $mathbf Q$ was settled using ideas coming from the proof of Fermat's Last Theorem, hence using modern algebraic geometry.
The Sato-Tate conjecture is an analytic conjecture somewhat in the spirit of the prime number theorem. It was formulated in the 2nd half of the 20th century but could have been appreciated earlier. Like the prime number theorem, which is equivalent to nonvanishing of the zeta-function on the line ${rm Re}(s) = 1$, the Sato-Tate conjecture was known to be a consequence of analyticity and nonvanishing of certain $L$-functions on vertical lines (boundary of right half-planes) and those $L$-function properties were proved about 10 years ago with algebro-geometric methods.
$endgroup$
Do you consider $L$-functions of elliptic curves over $mathbf Q$ (or other number fields) to be in the spirit of "analytic number theory undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on"? Those 19th and early 20th century folks did not have the definition, which only came much later in the 20th century, but the idea of defining such functions as an Euler product and then Dirichlet series, and seeking an analytic continuation and functional equation, is a task they would have understood. Deuring proved the analytic continuation and functional equation in a special case (CM elliptic curves) in the 1950s, but the case of all elliptic curves over $mathbf Q$ was settled using ideas coming from the proof of Fermat's Last Theorem, hence using modern algebraic geometry.
The Sato-Tate conjecture is an analytic conjecture somewhat in the spirit of the prime number theorem. It was formulated in the 2nd half of the 20th century but could have been appreciated earlier. Like the prime number theorem, which is equivalent to nonvanishing of the zeta-function on the line ${rm Re}(s) = 1$, the Sato-Tate conjecture was known to be a consequence of analyticity and nonvanishing of certain $L$-functions on vertical lines (boundary of right half-planes) and those $L$-function properties were proved about 10 years ago with algebro-geometric methods.
answered 4 hours ago
community wiki
KConrad
add a comment |
add a comment |
$begingroup$
You can look at Lectures on applied $ell$-adic cohomology by Fouvry, Kowalski, Michel and Sawin : https://arxiv.org/abs/1712.03173
$endgroup$
add a comment |
$begingroup$
You can look at Lectures on applied $ell$-adic cohomology by Fouvry, Kowalski, Michel and Sawin : https://arxiv.org/abs/1712.03173
$endgroup$
add a comment |
$begingroup$
You can look at Lectures on applied $ell$-adic cohomology by Fouvry, Kowalski, Michel and Sawin : https://arxiv.org/abs/1712.03173
$endgroup$
You can look at Lectures on applied $ell$-adic cohomology by Fouvry, Kowalski, Michel and Sawin : https://arxiv.org/abs/1712.03173
answered 15 hours ago
community wiki
François Brunault
add a comment |
add a comment |
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21
$begingroup$
I like the question at the end of your post, but I think that your claim "encompass all modern mathematics" is quite exaggerated.
$endgroup$
– EFinat-S
yesterday
1
$begingroup$
Drinfeld's "Finitely additive measures on S2 and S3, invariant with respect to rotations" solves a measure-theoretic problem using automorphic forms and a consequence of Weil conjectures (although the latter is not crucial to the argument, one can do with a weaker version which can be proved without algebraic geometry)
$endgroup$
– Aknazar Kazhymurat
23 hours ago
$begingroup$
If I remember rightly, one of Weil's original uses for the Riemann hypothesis for curves over finite fields was applications to bounding exponential sums. Grothendieck invented $ell$-adic cohomology to prove the Weil conjectures (including the Riemann hypothesis for varieties that was eventually proved by Deligne), so it's not totally surprising that there is a connection to exponential sums.
$endgroup$
– Robert Furber
16 hours ago
2
$begingroup$
The question would benefit from changing the speculative "I have the impression... everything." to something saying that it is connected with many subfields of maths, possibly with examples.
$endgroup$
– YCor
15 hours ago